Hello and welcome to lecture number twenty-seven
of this lecture series on turbo
machinery aero dynamics. Last few lectures,
we have been discussing extensively on
turbines, and we have actually started with
an introductory lecture on turbines, different
types of turbines, and then, we emphasized
on the axial turbines, and we had quite lot
of
discussion, and lot of lectures devoted exclusively
to the various aspects of axial flow
turbines, starting from very basic thermodynamics
principles and working of axial
turbines, two-dimensional flow and analysis
in axial turbines, and then, the losses and
their estimation, the efficiencies and then
moving towards the 3 D flows and 3 D design
of turbine blades and so on.
So, and also of course, we had a tutorial
session on turbines. So, all these lectures
I
believe have been quite interesting as well
as educational for you in understanding the
basic working of axial turbines. What we going
to do today is basically to a initiate some
discussion on a very important aspect which
is associated with turbine blades and that
is
to do with the turbine blade cooling, and
I think you are very well aware that the turbine
inlet temperature plays a significant role
in the overall performance of the engine,
and it
is in this context that we talking about Turbine
blade cooling and the various methods
that are employed in turbine blade cooling.
Just to keep you informed turbine blade cooling
continuous to be a very active area of
research all over the world, where all the
engine manufactures and universities where
research is going on and research labs. Turbine
blade cooling and associated problems
which, which, of course, I am going to highlight
today are bring extensively studied and
modified and improve upon every year. With
the sole objective that the Turbine inlet
temperature can be increased to as higher
level as possible without increasing the
associated penalties. Some of these things
of course, I will be discussing in today’s
class.
So, there are two distinct aspects that I
am going to talk about in today’s class.
We will
start with some introduction to the turbine
blade and balding, blade cooling requirements
where I will kind of explore the requirements
of turbine blade cooling and why we need
blade cooling and so on. Subsequently I will
spend some time in elaborating the
fundamentals of heat transfer. I guess you
must have undergone a course in heat transfer,
but let me, I will portably just touch up
on a few topics which according to me would
be
very much essential for deeper and proper
understanding of the Turbine blade flows.
So, we will also spend a few, some minutes
on discussion on very fundamentals of heat
transfer and those aspects of heat transfer
which are of significance in reference to
Turbine blade cooling. So, I, what I will
do is I will start with a slide which I had
shown
in a couple of lectures earlier where we had
discussed about the, the, two-dimensional
blades and blade geometries and so on. So,
I think I mentioned that the Turbine inlet
temperature is a very significant parameter
when you consider the overall engine
performance. I think I had mentioned some
number with reference to this, that is, if
you
have one percent increase in the Turbine inlet
temperature, it is likely to result in two
to
three percent increase in the overall engine
output.
So, that is the level of importance that the
turbine inlet temperature has for the whole
cycle, and therefore, it is necessary that
we arrange or elaborate methods by which,
we
can increase the turbine inlet temperature,
but what is the limit? Why are we not
increasing the temperature to as higher level
as possible? The basic reason is that the
current day materials have a certain temperature
up to which they can function properly;
beyond that temperature, the material would
fail, and therefore, and add to this the fact
the Turbine blades also undergo extreme levels
of stresses because of the higher
rotational speeds, and so, there are centrifugal
stresses, there are bending stresses
because of the blade loading, and of course,
because of high temperature, there are there
are thermal stresses. So, and are all theses
stresses the Turbines rotor also there said
lot
of limitations in terms of the temperature
levels which, which, we can use for a typical
Turbine blades given the current materials
that we have.
So, with this in mind, we need to still ensure
that we can or at least make an attempt to
increase the Turbine inlet temperature to
as high level as possible. One of the ways
of
doing that is to use blade cooling techniques.
The other ways of course, you could give a
ceramic coating to the blades, some research
is also going on in this direction that you
can coat the blade with ceramics. Ceramics
as you know can which stands extremely
high temperatures, but the main disadvantage
is that ceramics are highly brittle. So, they,
on they are down, they probably would it be
able to which stands the stresses, and
therefore, there are there are lot of research
work going on, on, trying to coat a standard
Turbine blade with ceramics and possibly use
much higher temperatures in the Turbine.
So, but we are not going to look at ceramic
coating as our discussion here. We will look
at pure the method which is currently used,
that is to used blade cooling method, and
so,
they are lot of blade cooling methods which
I think we will be discussing in next class.
Today’s a class is basically aimed looking
at the requirement why do we need blade
cooling and the thermo dynamics benefits of
blade cooling and also some heat transfer
related issues which are associated inherently
with blade cooling techniques.
So, what I mentioned was that, for a given
pressure ratio and efficiency, the turbine
work
per unit mass obviously is proportional to
the inlet stagnation temperature, and I
mentioned that typically of course, is adjusted
ball for figures one percent increase in the
turbine inlet temperature can cause about
two to three percent increase in the engine
output, and therefore, we would like to use
extensive techniques or methods to increase
the turbine inlet temperature, because that
is the amount of significance that Turbine
inlet
temperature has with reference to the engine
performance.
Now, if you have undergone course in basic
course in thermo dynamics which I
presumed you would have, then I am I am sure
you would have carried out a bray ton
cycle analysis. Bray ton cycle as you know
is the basic Fundamental cycle of a gas
Turbine engine, and in bray ton cycle, you
know that there is a significant effect of
the
maximum cycle temperature on the work output
and efficiency, and in the case of gas
turbine engines, the maximum cycle temperature
is the turbine inlet temperature, and
therefore, that is the kind of significance
that bray ton cycle has, well, bray ton cycle
depends on the turbine inlet temperature.
Current day materials cannot really which
stands temperature greater than thirteen
hundred Kelvin and that kind of put a limitation
on the maximum efficiency that one can
get, because as you know the efficiency is
in some sense, a function of the ratio of
the
maximum temperature to the minimum temperatures.
So, if a maximum temperature is
fixed, minimum temperature you cannot change
because that is the ambient temperature
and that is fixed, and once you fixed the
max temperature, then that also puts a limit
on
the max efficiency of the engine. You would
not want to have a certain limitation; you
would like to extend that limit further take
the efficiency to higher levels. Now, there
are
obviously inherent benefits to blade cooling
techniques which can permit as to use much
higher turbine inlet temperatures.
At the same time, there also lot of disadvantages
an associated here. Disadvantage in the
sense that it, it, increase the complexity
of the whole process by orders of magnitude.
It
leads to mechanical complexities; because
you need to incorporate methods which can
leads to blade cooling; you would also leads
to aerodynamics complexities because you
have a cooling flow which is interacting with
the hot gases. So, that leads to
aerodynamics issues and obviously the thermodynamics
issues because of stresses and so
on. So, it, it, changes the whole boil game
of design an analysis of a Turbine which has
these techniques being used. The turbine blade
cooling obviously will increase the
complexities by orders of magnitudes. So,
that is one important message that you need
to
keep in mind that of course, you get a lot
of benefit in the same time that is with a
cost
and that cost is the extreme complexity that
is associated with the whole aspect of turbine
blade cooling.
So, so with the gain in performance we have
mechanical aerodynamics thermodynamic
complexities which are involved in design
and analysis of these cooling techniques and
ah
So, if you now look at a Turbine stage, you
know it consists of nozzles and rotors. They
operate in very extreme environment and the
extreme environment is because of hot
gases which are at very high temperatures.
At the same, time they are also at very high
velocities, because a nozzle accelerates the
flow to very high speeds. Therefore, the flow
is coming in the very high velocity; it also
has high temperature, and possibly, the
compensation of the gases itself is not just
air. So, all kinds of combustion products
are
involved there and that makes it highly, very
highly extreme operating conditions for the
nozzle as well as the rotor, and therefore,
it is this one of the challenges of cooling
designers is to take care of these complexities
in terms of high temperatures, high
velocities and gas mixture. Now, the other
aspect of the, well, the others higher level
of
the complexity is the fact that the high temperatures
are really fixed.
There are significant variation or fluctuation
in these temperatures because of the fact
that the flows highly unsteady and highly
turbulent, and therefore, random fluctuations
like the way coming from the rotor interacting
with the stator or the nozzle downstream.
So, the flows extremely unsteady, it is highly
turbulent, and therefore, the random
fluctuations in temperature which you cannot
really take care of while designing turbine
blade of course, with the modern design tools
like CFD and so on. It is possible to
partially take care of this complexity, but
it still is a very challenging area of designing
and continuous to be very significant area
of importance in terms of research.
Now, if you compare a nozzle with rotor, you
may be surprised to know that a nozzle is
subject to a slightly higher level of extremity
in terms of temperature. The basic reason
here is that because of the relative motion
between the nozzle and the rotor, the rotor
actually sees the stagnation temperature which
is in the related frame, which is slightly
lower than that the nozzle. It is probably
about 200 to 300 Kelvin lower than the
temperature which is the nozzle faces.
Therefore, nozzle is actually facing temperature
environment which is more severe than
that of an a rotor, but this is only in the
in terms of temperatures being look at others
complexities like stresses, the bending stress,
centrifugal stress and the thermal stress,
then of course the rotor obviously has much
more to endeavor than nozzle, and so, it is
in
this context that I just mentioned that nozzle
is subject to more severe operating
conditions. The rotor is also subject to severe
operating conditions, but if you look at
simply the temperature, the nozzle actually
faces slightly higher temperature.
Now the main reasons why the nozzle and rotor
faced different levels of complexities in
terms of temperature is because of the fact
that rotor experiences slightly lower
stagnation temperature probably 200 to 300
Kelvin lower than the nozzle, but of course,
it experiences far most stresses due to high
rotational speeds, and the highest
temperatures are basically felt in the first
stage, and as you move towards this and later
stages, the cooling problems become lesser
and lesser complicated in the later stages,
which is obvious because the later stages
do not really have that higher temperature
as
the case with the initial stages of the turbine
now.
What we will do is let us take a look at what
are the modes of failure. Let us say you do
not employ any cooling technique. Then in
the first place, why should we worry about
cooling at all? I think I mentioned that the
material image that will fix the turbine
temperature. Given a certain material, you
cannot go beyond a certain temperature unless
one uses artificial methods of keeping the
temperature, metal temperature lower than
the
gas temperature itself.
So, there are different modes by which turbine
blade can failed where we can categorize
them into three distinct classes - one of
them is to do with the oxidation or corrosion
or
erosion of the blades which is because of
the chemical and particulate attacks from
the
gases, that is, the combustion products which
enter the turbine may have some amount of
particular matter unburned fuel and so on,
which might damage the blades because they
are coming in an extremely high speeds. We
may also have certain chemical reaction
taking place due do oxidation on the blade
surface. So, that is one of the modes of
failure, that is, eventually if this is allowed
to grow obviously, it will lead to failure.
The other mode is because of creep that is
has Turbine blade is exposed to high
temperature for prolong periods of time, then
the blades will undergo what is known as
creep failure, and one may also have the third
mode of failure which called a thermal
fatigue because of the repeated cycling. As
the turbine operates through a cycle, that
is, it
is started and then taken to max temperature
and eventually it stops and it turbine blades
cool down, and then, it again, after sometimes
it is started and so on. So, as the Turbine
undergoes these cycles of operation, it undergoes
fatigue, and therefore, it leads to high
thermal stresses during these cyclic loading
in terms of temperature and that also leads
to
failure in terms of a thermal fatigue.
So, these are three different modes of failure
which are possible in the case of a turbine,
and one may have, if one does not employee
any cooling techniques, one is likely to
encounter either of these modes or a combination
of these modes which can lead to a
early failure of the turbine blades; obviously,
you cannot really operate a turbine at a
temperature which is higher than the material
limit itself. Let us say material limit says
that max temperature is 1300 Kelvin; obviously,
you cannot design a blade for operating
at 1400 or 1500 Kelvin. It has to be lower
than thus limit. Even then the turbine blades
eventually will undergo one or more of these
modes of failure, and that is something that,
as a designer one would like to avoid and
prolong the life of a turbine blade by using
some of these blade coding techniques.
So, I think I mentioned in couple of slides
earlier that a turbine blade will undergo
a
variation in temperature, and in the sense
that it if the combustion chamber is operating
at a certain temperature, the combustion products
eventually pass through the nozzle, and
then, subsequently put the rotor, and if there
are subsequent stages, obviously through
those stages as well. Now, in most of the
common turbines, it is seen that of course,
during a simplistic a beginning level design,
one would like to assume that the turbine
faces a temperature distribution which is
kind of uniform.
Well that is still an idealistic scenario
where, you know, one may have a uniform
temperature profile, but what is seen is that,
most of the cases because of the unsteady
transient nature of the flow, the flow is
hard; the temperature distribution is hardly
ever
uniform.
Just you give me some idea if you look at
this particular schematic where we have an
average temperature profile which has been
plotted. This is an average radial temperature
profile. So, this is a typical average temperature
profile, radial temperature profile, that
is, from the hub of the blades to the a shroud
and combination products enter the stator
typically with an a certain temperature profile.
Now, it is one can always assume that the
profile has a shape like this, but of course,
depending upon the operating conditions, one
may or may not really have a uniform
profile like what you seen here. It may have
substantial variation in the temperature
though the combustion designers would, combustion
chamber designers would like to
keep these profiles as uniform as possible,
but under extreme operating conditions, one
may have significant variations in the temperature
profile from what we shown here.
And thus the flow passes from the inlet of
the stator to the exit. It again undergoes
a
change in it is profile, temperature profile
entering the rotor can also be quiet different,
and that is one of the reasons what, why the
design of a cooling system becomes even
more complicated, because how do you take
care of these non uniformities in the
temperature profile which obviously depend
upon the flow, because there is as we will
see very shortly, there is a very strong coupling
between the velocity field and the
temperature field.
For normal low speed applications, one would
kind of like to assume that the velocity
field and temperature field are decoupled
and thus hardly any linking between them,
but
in, in, a high temperature, high speed flow
like that of these turbine flows. The coupling
is inevitable; one cannot simply neglect the
coupling between temperature and velocity.
So, design of a cooling system for a flow
which is likely to be highly unsteady and
turbulent is extremely complex because you
cannot really predict the temperature
variations, because the flow itself is unsteady,
and therefore, design and that is why I am
mentioned that turbine blade cooling continues
to be inactive, very active research and
designers all over the world are trying to
and researchers are trying to develop better
methods of designing cooling techniques for
a turbine blade.
So, with this background, I guess now you
must have understood the significance of the
this particular topic of turbine blade cooling,
and why I said that one needs to employee
a
blade cooling techniques, and now, that we
have understood or had some background of
the requirement of turbine blade cooling.
I think it is about time that we also look
at
methods by which, one can estimate the cooling
requirements.
I mentioned that turbine blade cooling is
inherently heat transfer problem which, which,
also involves certain coupling with the fluid
mechanics. So, it is an aerodynamic and
heat transfer problem where both of these
vast areas have to come together to arrive
at a
certain configuration which can serve this
particular purpose.
So, what we will do is to have an overview
of the heat transfer, fundamentals of heat
transfer and with this specific relevant to
this particular topic of turbine blade cooling.
So, Turbine blade cooling inherently involves
a application of concepts of heat transfer.
Heat transfer as you know is a very well established
area like fluid mechanics or aero
dynamics and substantial knowledge base is
available in the form of books journals and
other form of literature.
What we will do is to take a brief overview
of the different concepts of heat transfer
which will be required for an efficient design
of a cooling system. So, let us go through
some of the very fundamental aspects of heat
transfer. I think you must have learn this
several times in this earlier on in heat transfer
courses thermodynamics and so on.
You probably aware that, I am sure you were
aware that there are three modes of heat
transfer – conduction, convection and radiation.
So, what are these different modes of
heat transfer? Conduction basically involves
heat transfer between two bodies or two
parts of the same body through molecular level
and which are more or less stationary,
that is the body stationary conduction is
heat transfer between the molecules of the
body
or between two bodies which are actually stationary.
So, conduction is something that
occurs in basically because of the molecular
motion; we are not talking about any mass
motion of the fluid itself. It is just a result
of energy interaction or energy transfer
between molecules.
And so, and in the case of gases and liquids,
conduction basically results from transport
of energy by molecular motion near the walls,
and in solids, it takes place by a
combination of lattice vibration and electron
transport.
Conduction as I mentioned is energy transfer
at a molecular level. There is no mass
movement or macroscopic movement of matter
relative to one another. Now, the other
mode of heat transfer is convection, and convection
is something that involves motion,
mass motion of fluid and it is not something
which occurs on a molecular levels. So,
there is much more than just molecular motion;
it involves basically mass motion of
fluids either liquids or gasses.
Now, you may have conduction again taking
place in different modes. You could have
conduction taking place just because of change
in density which is again as a result of
temperature difference and that is called
free convection, that is, when heat transfer
takes
place as a result of temperature difference,
and therefore, density difference and a result
of that, it is to what is known as free convection.
Now, if you use an artificial mode of a inducing
convection, that is known as forced
convection. Let us say use of pump or a blow
or compressor, then that mode of heat
transfer is known as forced convection. And
heat transfer in turbine blade which involves
blade cooling techniques is essentially a
forced convection problem because we are
actually introducing external air, which is
basically air taken from the later stages
of a
compressor which is used for cooling a turbine
blade.
Now, the third mode of heat transfer is radiation
and radiation is basically energy transfer
taking place through electromagnetic waves,
and obviously, it does not need any
medium. For example, sun radiates heat to
the earth and there is no medium between the
sun and the earth and does not require a medium,
but for the temperatures that we are
looking at in a turbine, it, the major modes
of heat transfer are through conduction and
convection and radiation is of course present.
We cannot say it is 0, but compare to the
heat transfer taking place to conduction and
convection, radiative wave transfer is
usually negligible and it, it, is not usually
considered that significant in the case of
turbine blade cooling in heat transfer in
a turbine blade.
So, we will be restricting our discussion
on heat transfer in turbine blades to conduction
and convection. Now, let us take a, first
look at the conduction little more detail,
and
subsequently, will locate convection and both
of these of course, in the context of heat
transfer in a turbine blade.
Now, one of the Fundamental loss of conduction
is the Fourier conduction law as I am
sure you must have leaned, which basically
relates the rate of heat transfer to the
temperature gradient and that is through the
thermal conductivity. So, rate of heat
transfer per unit area q by a or it is denoted
by q is proportional to the temperature
gradient, and temperature gradient obviously
in the y direction normal to the surface and
that is the function proportionality constant
is the thermal conductivity which is basically
the defined as a amount of heat conductor
per unit time, per unit area per unit negative
temperature gradient. So, thermal conductivity
obviously is a property of the surface
itself of these solids and it basically is
a constant which relates the rate of heat
transfer to
the temperature gradient.
Now, this equation we can generalize and we
can write a generalize governing equation
in a three-dimensional Poisson equation form
and which is basically stated as by rho cp
del square t is equal to del t by the rate
of change of temperature are with time. So,
there
is transient temperature term here. The temperature
denser here and the parameter that
you see here, that is, k by rho cp is known
as thermal diffusivity which is again a
property of the conductivity material.
So, this is basically known as the Fourier
equation of course, the generalized version
of
the Fourier equation. It is used in simplified
versions with lot of assumptions in normal
design level calculations where one would
like to carry out design of let us say cooling
system in a simplified fashion to begin with
and therefore, simplified versions of these
equations away extensively used by researches
working in the area of heat transfer of
turbine blade cooling methods.
Now, so, the first form of heat transfer that
we have just discussed is conduction and
described very well by the Fourier’s equation,
which relates the heat transfer to the
temperature gradient. Now, the other mode
of heat transfer which is the, which is
basically to do with interaction between the
fluid and the solid itself, and as a result
of
mass motion of the fluid, that is known as
the convective heat transfer.
So, in convective mode of heat transfer, we
have seen that in, in, solids for example,
the
mode of heat transfer in just a solid is purely
by conduction, that is, there is no mass
motion of fluid if you look at just a solid
as a hole and heat transfer takes place just
because of transfer of energy from one molecule
to another. So, conduction is the only
mode of heat transfer that is possible in
a solid in and of course, you may have radiation
depending upon the temperature.
But if you look at a fluid, the liquid or
gases, both these modes, that is, conduction
as
well as convection heat transfer are possible.
Conduction is possible because molecules
can interact with each other and transfer
energy from one to another and convection
is
possible because liquids and gases can, would
involve mass motion of molecules and
that leads to convective heat transfer as
well.
Now, the other important aspect of convective
heat transfer is the fact that there is a
very
strong coupling between temperature and the
velocity fields, which is especially true
for
high velocity, high temperature like in case
of turbines that way currently talking about.
In low speed in compressible flow, normally
it is a practice to decouple temperature and
velocity field and just calculate the velocity
field, because we are primarily interested
in
the velocity field. In this case of turbines,
that is not possible, that we, it is not correct
to
decouple temperature and velocity fields,
but because they are strongly coupled as we
going to see very soon in, in, the side of
equation which will reveal the fact that this
fields are very much strongly coupled and
it is not possible to decouple them.
Now, in ah modern day turbine scenario, we,
the coupling is even more significant
because of the fact that velocity as well
as temperature gradients are very high, and
in
that scenario, the coupling between the temperature
and velocity fields will have a very
strong influence on each other, and turbine
blade which is and do which is being design,
which has been design for cooling methods
will involved forced convection and that is
the dominant phenomenon of a heat transfer
in turbine flows.
Now, in a typical turbine blade, the boundary
layer developing on the blade surface is
also of significant interest, because boundary
layer sort of acts as a buffer between the
solid blade and the hot free stream and it
offers resistance to heat transfer between
the
blade and the free stream. Now, the heat transfer
that is taking place in this boundary
layer, the thin discuss layer is both by conduction
as well as convection. Conduction it
basically transfers heat from the fluid which
is at much higher temperature to be solid
that is the blade, and at the same time, it
also transfer heat to the solid through
convection.
So, there is heat transfer mechanism involving
both conduction as well as convection
between the free stream which is at a much
higher temperature as well as the solid,
which is the turbine blade and this heat transfer
is a largely dependent on the nature of
the boundary layer, that is, the boundary
layer laminar or turbulent, the nature of
the heat
transfer is quite different depending upon
the type of boundary layer that one is
encounter.
So, on a typical turbine blade which will
see very soon, there is change in the nature
of
boundary layer from the leading edge that,
say the stagnation point all the way up to
the
trailing edge. Boundary layer changes from
initially its stagnation point. There is growth
of boundary layer it is initially laminar,
then it transaction and become trouble.
So, as the flow becomes are as the flow becomes
laminar and transaction, and then, the
finally, become turbulent. The nature of the
heat transfer through each of these layers
is
quite different and there are separate method
of calculating heat transfer through each
of
these distinct element of the boundary layer.
Whether it is laminar or transactional or
turbulent, the heat transfer or calculation
of heat transfer is quite different and that
is
handle separately by separate tools or methods,
and what we will say in next slide is
typical distribution of the heat transfer
rates in a typical turbine blade.
So, let us take a look at a typical turbine
blade and how heat transfer way can vary
around the turbine blade. So, what is indicated
by these arrows are the rates of heat
transfer and why it is high will be clear
in a fluid slides from now. So, if you look
at a
stagnation point, this is the stagnation point
of the blade. The boundary layer begins
development from the stagnation point. It
is initially laminar and then it becomes
transitional and eventually it becomes turbulent
and this is on the section surface. On the
pressure surface, of course depending upon
the nature of the blades, some of the modern
blades may also have substantially high levels
of acceleration leading to re
laminaraisation of the flow, that is, the
turbine, the flow is initially laminar. Then
it
transition and possibly become turbulent and
then eventually it might become than
laminar again.
So, what is, what are indicated by this distinct
point? One is of course, the stagnation
point have will highlight the significant
of stagnation point little later, because
that is
where the maximum heat transfer is going to
take gradient. I will explain that little
later.
Now, on the section surface, one might have
presence of shocks defending upon the
Mach number at which the blades are operating.
I mentioned it one of my earlier lectures
that it turbines designs of usually would
want to delay the occurring of shocks towards
the latter half of the blade, and so, you
may have shocks in the, towards trailing edge
of
the blade especially on the section surface
and. So, there is a possibility of a shock
boundary layer interaction here which can
of course in complicate the heat transfer
substantially.
And one may have an unsteady weight flow at
the trailing edge there. That again is a
very challenging area of estimating heat transfer
in an unsteady flow, because that also
affects the temperature distribution substantially.
So, these are the different distinct
regions of a typical turbine blade, and where
in the method of estimating heat transfer
rates in all these distinct areas whether
it is laminar or transition turbulent or it
has
become laminar again through lamenarisation
or stagnation point of the turbulent way.
The heat transfer rates are quite different
in all these distinct areas. So, there is
a separate
method of calculating heat transfer through
that is possible through each of these
different layers are region of the boundary
layer.
Now, I mentioned that there is a very close
coupling between the fluid mechanics and
heat transfer especially in the context of
the turbine blades and Turbine with cooling
with
essentially.
So, analysis of the flow around a blade requires
special analysis which is valid for that
particular region. For example, if you are
looking at laminar flow that is leading edge
of
turbine blade, then one can analysis the heat
transfer through a laminar boundary layer
and has been transition and go to the later
part of the turbine blade. The boundary layer
is
Turbulent and heat transfer through that boundary
layer is quite different.
Now, in general, one can write the overall
heat transfer which is related to the
temperature difference between the fluid and
the solid through the Newton’s law of
cooling which relates the heat flux, which
is mention here as a q subscripts w, that
is,
heat transfer from the wall is proportional
or is equal to heat transfer co efficient
h
multiplied by the temperature difference and
this is again related to k times del t by
del y
which is the temperature gradient.
Now, this heat transfer coefficient that we
have seen can be non-dimensionalized by the
thermal conductivity. So, and that is through
what is known as the Nusselt number. So,
Nusselt number is the heat transfer coefficient
multiplied by a characteristics line usually
the god of the blade in these case divided
by the thermal conductivity of the blade.
This
is also equal to l by the temperature difference
t minus T w multiplied by del t by del y at
the wall.
So, Nusselt number is one of the non-dimensional
parameters which is the extensible
used in heat transfer. There are numerous
other non-dimensional groups like Reynolds
number obviously, you were aware of is the
Prandtl number which will seen very
shortly. There is a Eckerts number, Grashof
number, these two in and one may have
Richardson number and Stanton number. So,
these are some of these non dimensional
number of groups which play very significant
role in heat transfer analysis in a turbine
blades, and depending upon the nature of heat
transfer, one are more of these non
dimensional groups well play significant role
in the heat transfer characteristics.
So, what we will do next is to take two examples
- one is to do with laminar flow; other
is to do Turbulent flow. Both force convection
because Turbine blade with cooling is a
post convection problem. So, we will look
at a laminar boundary layer and subsequently
turbulent boundary layer, both undergoing
post convection, and then, look at how we
can
analyze heat transfer in both these different
boundary layer scenarios.
So, let us consider a very simple case of
an incompressible Laminar flow over a flat
plate. So, for this kind of an application,
we can write the transport equation as del
u phi
by del x plus del V phi by del y is equal
to alpha del square phi by del y square. So,
here,
phi could be either u or theta; alpha could
be either mu by rho or k by rho C p and theta
is the temperature differential t minus T
w by T u minus T w.
So, for this case, the boundary condition
could be at y is equal to 0 phi by could be
V phi
and v is equal to 0, that is at the wall,
the velocity y direction is 0, and at has
y tends to
infinity, phi is equal to u is equal to theta
is 1. So, what you can see here is that in
this
kind of a transport equations that you see
the both the velocity and temperature equation
are quite similar, and which means that the
coupling between the temperature and
velocity fields becomes very obvious that
the complete between velocity and temperature
field simply cannot be ignore especially for
high temperature and high velocity flows.
And so, in a laminar flow, people have come
up with empirical correlation of course,
depending upon the application. Right now
we are simply talking about flat plate which
means there is no pressure gradient, and for
a very simple application like this one can
relate some of the non dimensional numbers
that I mention that Nusselt number to
Reynolds number and Prandtl number through
some empirical co relations.
So, what is been demonstrated what has been
root is that the Nusselt number can be
related to Reynolds number and Prandtl number
for a typical 0 pressure gradient flat
plates application. Nusselt number is 0.332
times Reynolds number raise to 1 by 2 into
Prandtl number raise to 1 by 3 this is all
so related to the skin friction coefficient
C f by
2; Prandtl number raise to one by three into
Reynolds number. So, what you can see is
that heat transfer is indeed a function of
this square root of Reynolds number and Prandtl
number is to 1 by 3 as well as this skin friction
coefficient.
What is all so interesting is that a thin
boundary layer has a larger heat transfer.
Therefore, the thinner the boundary layer
heat transfer obviously is larger, because
you
do not have a buffer which well separate the
surface from the free stream, and therefore,
the maximum heat transfer would take place
at this stagnation point where the boundary
layer thickness is close to 0.
So, that is where the boundary layer begins
development, and since there is no buffer
between the boundary layer, that buffer between
the free stream which is the high
temperature and this surface which is at lower
temperature heat transfer rate is the
maximum. Which is why if you recall the heat
transfer distribution, I was showing
around blade surface the maximum was at this
stagnation point that is because, that is,
at
that point, that we have the boundary layer
thickness which is at it is minimum. Thinner
the boundary layer more is the heat transfer.
Let us now move on to turbulent boundary
layer for a very similar application that
is calculate, and if you look at turbulent
boundary
layer, how do you calculate the heat transfer?
So, heat transfer to which is owing to turbulent
fluctuation is can be written in the form
of this equation, that is, q subscript t is
equal to row C p into v prime t prime, and
average of that the end symbol average of
that this is minus C p into epsilon subscript
t
del t by del y which is at temperature gradient.
Here, epsilon subscript t is the eddy
diffusivity which is basically owing to the
turbulent fluctuation which of present in
a
turbulent boundary layer.
In a turbulent flow, there is all very close
coupling between the momentum transfer and
heat transfer which in turn translates to
the coupling between the heat flux and shear
stress, because in an turbulent boundary layer,
we know that there is momentum
exchange between the different layers of the
flow which is absent in the case of the
laminar boundary layer, and therefore, there
is a close coupling between the momentum
transfer and heat transfer.
So, as we will see very shortly in the turbulent
boundary layer, one would have much
higher levels of heat transfer. That is because
there is momentum exchange between the
different layer of the boundary layer, which
is unlike in the laminar flow where the
different layer do not really mix and interact,
and so, the momentum transfer between the
layers in laminar flows is much less than
that in a Turbulent flow. So, in turbulent
boundary layer, we would define what is known
as a turbulent Prandlt number and
Turbulent Prandlt number is basically defined
as the ratio of mu subscript t over the eddy
diffusivity. So, ratio of the viscosity to
the eddy diffusivity.
So, what is the significance of the Turbulent
Prandlt number? We can express the ratio of
heat flux and momentum flux as what is given
here, that is, the heat transfer q t to the
momentum flux is equal to the mines C p into
temperature gradient and the turbulent
Prandlt number multiply by the velocity gradient.
So, we can relate the heat transfer and or
heat flux and momentum flux through the
temperature gradient velocity gradient, and
therefore, the total rate of heat transfer
due to
both molecular and turbulent motions, that
is because of conduction as well as the
convection involved is the some of the molecular
heat transfer and the turbulent heat
transfer and that is expressed as minus C
p into mu by plus mu t by Turbulent Prandtl
number multiply by del t by del y, and there
is indeed a very clear difference between
Prandtl number and the Turbulent Prandtl number.
Prandtl number is a physical property
of the fluid, whereas the Turbulent Prandtl
number is a physical property of the flow
field and not just the fluids.
So, depending upon nature of the flow whether
it is Turbulent, Prandtl number is indeed
a function of the flow field. Turbulent Prandtl
number is the function of the flow field as
against Prandtl number which is just a function
of the, or it is just the property of the
flow itself. Now, same as we have defined
for a laminar boundary layer, we can now
relate the Nusselt number to the Reynolds
number and Prandtl number to again an
empirical correlation which is of course,
here for a flat plate.
Nusselt number is related to the Reynolds
and Prandtl number through this equation that
is 0.029 Reynolds number is to four by five
Prandtl number raise to 1 by 3. So, we can
see the quite similar they both equation are
very similar, the, that we have written for
the
laminar flow and that for a turbulent flow.
In general, we can write Nusselt number is
related to Reynolds number and Prandtl number
through three constants and this of
course, would depend upon the nature of the
flow itself.
So, this is basically know as Nusselt equation,
and here, this constant is indeed depend
upon the particular flow and nature of the
flow and the whether it is flat plate or if
it is a
flow with ah pressure gradient which is adverse
or favorable pressure gradient, one could
actually come up with empirical correlation
for the Nusselt number or the Nusselt
equation and relate that to the Reynolds number
at the Prandtl number.
So, we have very quickly had an overview of
transfer, fundamental heat transfer, of
course, heat transfer itself a very vast subject;
obviously, not possible to cover all the
aspect of heat transfer in a few slides that
I have done. This was just to give you an
overview of the heat transfer. The Fundamental
concept of heat transfer which are used
in analysis of Turbine blade cooling.
So, let me just quickly recap the different
points I have mentioned on discussion on
laminar and Turbulent flows. We have seen
that heat transfers is higher for a thin
boundary layer than a thick boundary layer
as the temperature gradient of this leaves
higher for a thin boundary layer, and the
buffer between the solid surface and the free
stream is lower in a thinner boundary layer,
and heat transfers in a turbulent bound layer
is higher than that of a laminar boundary
layer, and this is also coming from the fact
that
we can see that it, then we actually defined
what is known as a turbulent Prandtl number
for Turbulent flows which is quite difference
from the conventional Prandtl number.
And heat transfer in thin viscous region near
the stagnation point of leading edge is very
high, and in this region’s, the velocity
and temperature gradients are extremely high
and
that result in extremely high levels of heat
transfer especially in the regions close to
the
stagnation point, and that explains why we
have seen the high level of heat transfer
taking close to the stagnation point. In fact,
the maximum heat transfer actually takes
place near the stagnation point, and as the
flow progresses from the stagnation point
to
the training edge, the level of heat transfer
also changes depending upon the nature of
the
flow itself, and that of course possesses
a lot of challenge for the, the, blade cooling
designers would like to place cooling holes
at different location on the blade surface.
So,
how does one decide these cooling whole locations.
One is of course, if you look at a very steady
state flow, it probably is easier to estimate
the cooling whole distribution for such a
flow, but as we have seen, turbine blade flows
are extremely unsteady with lot of turbulent
fluctuations, and because of the coupling
between the velocity field and temperature
field, there is a substantial variation in
the
temperature flow, in temperature field around
a turbine blade and that is essentially not
steady and one cannot really take up a steady
state analysis of a turbine blade to
determine the cooling blade hole distributions.
And that is where the challenge lies designing
an optimum cooling blade distribution for
two reason, because one requires an substantially
high amount of cooling mass flow that
is required in cooling a turbine blade. Modern
turbine one might has use as highest
twenty percent of the compressor mass flow
for cooling Turbine blades, and this mass
flow; obviously, does not result in much thrust
because it is not really contributing to the
overall pressure rise of the engine, and,
and, therefore, that is a certain amount of
mass
flow which is not really contributing to work
output. The second reason is that this
cooling mass flow also interferes with aero
dynamics of a turbulent flow and that can
lead to substantial losses.
So, on one hand, we would like to employee
cooling methods to enable us to use higher
turbine inlet temperature. On the other hand,
the cooling methods also lead to loss in
performance of the turbine in terms of increase
in losses. As a result of a cooler mass
flow interacting with the hot combustion flow
which is there in the turbine, and so, it
can
lead to problem in terms of the aero dynamics
of the turbine itself, and that is where I
emphasize the fact that blade cooling continuous
to be a very active vary of research
because one would there is still enough scope
for improving the cooling methodologies
which are used. To ensure than, one can minimize
the amount of compressor mass flow
that is used for cooling and also ensure for
cooling that monitor of compressor mass
flow, that is used for cooling and else ensure
that cooling mass flow does not adversely
affect of the aero dynamics of the turbine
blades and does not really affect the turbine
blade efficiency.
So, just to summarize the various points regarding
with reference to blade cooling
methods, needless to say of very strong understanding
of the heat transfer mechanisms is
essential because cooling a turbine blade
is essential heat transfer problem, of course,
there is also strong coupling of heat transfer
with the fluid mechanics here.
Turbine blade cooling obviously requires the
significant amount of compressor layer
which might firstly lead to losses in the
turbine leading to poor a lower efficiency
of the
turbine, and also it leads to a loss in overall
loss in thrust, but that of course, is
compensated by the fact that you can actually
a get higher turbine inlet temperature with
cooling probably that kind of compensates,
but the effect of cooling on the aero dynamic
performance is something that will need a
greater attention and analysis to be able
to
achieve have cooling methodology which does
not significantly affect the turbine
performance.
Let me just conclude today’s lecture with
an overview of what we are discussed. We had
discussed about two distinct aspects of turbine
blade cooling. We began our lecture here
with an overview of wide turbine cooling is
required, and what is the significance of
turbine blade cooling and why I said that
one needs to employee elaborate methods of
cooling a turbine blade.
We also had very quick overview of the heat
transfer Fundamentals which are required
in analysis of turbine blade and some of the
concepts which are used in turbine blade
cooling analysis. So, we will be continuing
discussion on turbine blade cooling and also
the different types of turbine blade cooling
methods which are used in modern day gas
turbine engines and this, these topics of
course we will be taken up for discussion
in the
next lecture.
