Radiation Heat Transfer
Prof. J. Srinivasan
Centre for Atmospheric and Oceanic Sciences
Indian Institute of Science, Bangalore
Lecture - 4
Spectral and Directional Variations
(Refer Slide Time: 00:20)
In the last lecture, we looked at variation of emissivity of metals. And we saw that if we
assume that the directional spectral absorped, which is equal to their expert emissivity is
assumed to vary as square root of the temperature and inversely proportional to wave
length. Then, we saw that the directional total emissivity varied linearly as the
temperature and the directional total absorptivity varied as root of T s and T i, the T i is
the source temperature. So, we clearly saw that the emissivity where it depends only on
its surface temperature, where the absorptivity depends both on the surface temperature
and the source temperature. So, this decision was brought clearly, we also saw that if by
chance T s was equally T i then alpha prime is equal to epsilon prime as per Kirchhoff’s
law.
But this is a special case, in general we saw that the ratio of alpha prime to epsilon prime
goes as root of T i by T s. Clearly showing that in many cases example, if T i is close to
6000 Kelvin, the suns temperature and T s is 300 Kelvin then alpha prime by epsilon
prime is root of 20 which is around 4.5. So, in the case of metals you see that the solar
absorptivity is about 4 to 5 times larger than h infrared emissivity. So, this is could be
called as selective surface, a surface which whose property is depends on the wave
length and which enables you to, so we draw it graphically.
(Refer Slide Time: 03:31)
You can write, alpha prime lambda equal to epsilon prime lambda and it varies as 1 over
square root of lambda. So, can really see that in the wave length below 1 micron
absorptivity is high while at higher wave length, let us say around 10 micron emissivity
is low. So, you such a material is put out the sun is solar radiation impinging on it, it will
reach temperature higher than a surface boost properties do not vary with wave length.
And in we saw just now that the absorptivity of metal is about 4 times higher in the solar
region compared to the infrared region. So, that advantage has, but you can improve this
advantage further, if we take a metal substrate and put a coating, you put a
semiconductor coating. For example, it can be copper oxide on copper then this copper
oxide will absorb strongly in the solar region and then it will rapidly, this region and then
it will come down rapidly in the infer region.
So, in the infer region you will see, the effect of the substrate. So, let me just redraw this,
this, how the metal behave this semiconductor coating gives you a higher absorptivity is
solar region and then it mergers with the, so this additional absorptivity is given by the
coating. So, coating typically a micron are less thick, improves the absorptivity in the
solar region, while keeping the emissivity in the infrared region and changed. So, by that
the ratio of alpha prime to epsilon prime can be as high as 10 can twice that of pure
metal. So, these selective coatings are used in solar collectives to achieve higher
temperatures and we will show some examples little later in this lecture. Now, let us look
at some other examples of materials.
(Refer Slide Time: 07:07)
And see how there emissivity is vary with temperature. In this figure, you see that the
emissivity of graphite varies roughly linearly with temperature all those not metal is a
nonmetal. And see tungsten, which is a metal, has a linear variation with emissivity,
polished gold has roughly linear variation of emissivity with temperature and sodium
magnesium except in kernel shows somewhat non-linear behavior. These various
temperature, while oxide layers as one you see here magnesium oxide the emissivity
decreases with temperature. So, generally you see that for metals measured in metal
show, and increase in emissivity with temperature, now in the next figure.
(Refer Slide Time: 08:12)
You see the dependence of the directional spectral emissivity of the material. And the
data is shown forms of these squares and circles, while the line here in, in blue or green
is really shows the theoretical relation that we had assume which is proportional to
square p and inversely proportional to square root of lambda. So, that behavior is the
theory, so called Hagen Rubens relation and when the metal is oxidized this is titanium is
oxidized the emissivity increases that you can see here. They are called the oxide layer
will examine this issue little further in today’s lecture. So, you can see that although the
data does not agree precisely with the simple theory, if the show as similar trend overall
or at decrease in emissivity as the wave length decreases. And, so this is the important
reason why the emissivity of metals increases with temperature.
Now, there are few more results here showing a variational’s spectral emissivity wave
length. For example, we have molybdenum, we have and we have platinum nickel they
all show at decrease in emissivity with wave length. Although, the slope is somewhat
different for different materials, but interestingly copper does not show that effect copper
seems to have an emissivity which does not vary that much with wave length. Now, one
proper reason, why some metals do not show the predictions of simple theory is because
there could be a thin oxide layer. Because you see these are the metals that high
temperatures.
So, unless you keep them in an inert atmosphere and oxide layer forms on a metal and
the emissivity can be then control by the oxide layer and not metal. So, so whenever we
try to look for data from various sources here to be show whether the metal, whose
emissivity is given is really a pure metal with no impurities. And whether it is really free
of oxide layers the oxide layers have a huge impact on the emissivity of materials, and
we will see that further in the subsequent slides.
(Refer Slide Time: 11:15)
Now, here gain we show the normal total emissivity that all emissivity in the normal
direction as a function of temperature in 1000s of degrease Kelvin. And you see clearly
that, in the case of tungsten the theory, and in the case of tungsten the theory expand or
not very good agreement tungsten is the red line here. And the blue line shows the value
of platinum, and you see clearly that platinum theory for platinum agrees quite well with
the data of platinum. And the fact that indicates the platinum the theory expand agree
while in the case of tungsten they do not agree could be on account of a small oxide layer
on the tungsten surface, which is not there in the case of platinum. So, these issues must
be always we looked into when trying to use data of emissivity of metals. Because oxide
layers are usually there they are not easy to avoid unless the metal is heated in an inert
environment like organ. Otherwise oxide layers will form and one has to account for the
oxide layer effects.
(Refer Slide Time: 12:46)
Now, here we are showing you the variation of the spectral emissivity of tungsten as one
of wave length. You see that beyond 1 micron this decreasing effect of emissivity is
wave length is seen clearly is follows the theory we which we discussed. But at low
wave length around the visible region, you see that emissivity is almost constant. This,
this shows that the simple assumption we made that the epsilon prime lambda. I am
sorry, the simple assumption we made that the emissivity is a function of wave length
and various as one over square wave length is using that only beyond 1 micron, below 1
micron other effects coming to play. And so, the so the simple theory that we have
proposed may not be valid.
(Refer Slide Time: 13:56)
Now, here we show you the directional variation emissivity at various wave length; this
is pure titanium with a certain surface finish. And you see that in the visible wave length
here the emissivity we shown function of the angle normal to the surface of the plate.
And you see emissivity is almost constant, from 0 degrees earlier data the almost 60 and
then stars decreasing. So, this is the typical behavior of many materials their emissivity
depends roughly constant up to around 60 degrees and then they start decreasing. And
you see this tendency is seen at also 1 micron and 1.62.
But beyond that at 4 8 and 20 micron, you see that emissivity is continuously increasing
up to about 75 degrees and then it start decreasing. So, the behavior of emissivity with
respect to angle is wave length dependent. At low wave length, it has 1 behavior and at
high wave length, it has a different behavior. So, when you are modeling the variation
emissivity of metals, this feature of metals that the angular dependents of emissivity is a
function of wave length must be accounted for. And you can see that variation of some
very different between low and high wave length.
(Refer Slide Time: 15:39)
Now, here is interest example of the effect of coating on metals. Here, what we shown in
reflectivity of white paint that is coated on aluminum. And what you see that white paint
is very good reflector in the visible, it has a value around 0.9 here. So, that is what we
explain white paint to be nice in the reflective in the visible. But when you go to the
infrared beyond about 3 micron you see that the emissivity the reflectivity is very, very
low. So, reflective is around 0.1; that means, it absorbs 90 percent of the radiation, so is
almost black in this region. So, this is the interesting factor, you look at surface and it
looks white to you in the visible. And hence you know it is a reflective surface from
there you cannot conclude anything about the behavioral surface in the infrared. In the
infrared is not a good reflective it is a co-reflective.
So, white paint is a selective surface, whose behavior below 1 micron and above 1
micron are very different. And so this factor must be accounted for, when you are trying
to model the emissivity of white paint aluminum. You have to account for the fact that
the reflectivity of the material is very different is around 0.9 in the visible around 0.1 in
the infrared. So, when white paint is coated on aluminum, it is still in the infrared. So,
that is why, it is a very good paint, use to paint outdoor containers, which will contain let
say is a petrol or other flammable material and you want to keep the temperature low.
So, you coated with white paint, white paint is all reflective in the region where solar
radiation is impinging on the tank. So, it will not absorb a solar radiation.
At the same time, whatever radiation it absorb is able to emit very efficiently in the
infrared. And hence, it will keep the tank temperature sufficiently low, so that it can be
safe, most story. So, this is the good example where a selective coating like white paint
keeps the temperature was surface low. With example, I coated little earlier was where
we coat copper oxide and copper, which makes the surface a very strong absorber in the
visible and a very poor emitter that. So, in that case the selective coating of copper oxide
and copper, actually enables it reach very high temperature.
So, one can a selective coating there is semiconductor on metals are copper oxide and
copper helps you to get high temperature when subject is solar radiation. But if we coat a
metal with white paint it enables you to keep the surface at a low temperature. So, we are
able to achieve both by suitable selective coating a flux selective coating, like copper
oxide and copper or a cool coating, like white paint on aluminum. So, this is a way
temperatures are control in satellite or in solar energy application or other situations in
engineering, where you want to control this surface temperature, so radiation.
(Refer Slide Time: 19:39)
Now, here is an example of effect of surface vanish on the directional variation of
emissivity. Here, we are seen example of 3 different conditions ground surface honed
and lapped. So, as you go ground to honed to lapped this the surface getting smooth and
smoother the roughness height is going down from around 0.4 micron to point naught 5
micron. So, smooth of the surface of course, higher the reflectivity and hence lower the
emissivity. So, you see lower emissivity overall lapped surface and higher emissivity for
ground surface. But in spite of the honing and grinding or lapping, the angular variation
of emissivity has not changed much, it remains almost similar.
So, the state of the surface has not altered its angular dependent, it has altered the value
of emissivity, but not its dependents on angle. So, for at least this pure titanium you can
have a single function which will account for the angular variation of emissivity with,
with data, without worrying about the type of surface finish the object has. And you must
notice here that this measurements have made for a wave length of 2 micron. So, the
reason why this grinding honing or lapping has not had and large impact on the
functional variation here, is because the wavelength of the radiation we have dealing
with is larger than the roughness side that is depicted here, now this figure.
(Refer Slide Time: 21:46)
Now, this figure now shows how they hemispherical total emissivity there is a after
integration over both angle and wave length varies with temperature for copper, where if
you have pure copper polish copper with no oxide layer then the emissivity is very low
around point naught 5. But situation will not prevail for long time as a real world, very
soon an oxide layer will form we very thin oxide layer. And you see the emissivity only
jumps for a naught point 5 to around 0.55.
So, emissivity goes up 11 times on a kernel oxide layer. And if the oxide layer, layer
increase thickness it can go to around 0.8 and you make really thick oxide layer when
even get a black oxide. This is the standard coating used on solar collectors to increase
the temperature of the collector. And so, you can see their oxide layer plays a very
important role can change the emissivity from point naught 5 to 0.9. So, oxide layer
should always recorded for in while estimating the emissivity of metals.
(Refer Slide Time: 23:09)
Now, this debate depicted in a somewhat different way in this figure to shows various of
the hemispherical total emissivity as a function of coating thickness for aluminum. And
you can see that initially, the emissivity increase rapidly from point naught 5 to around
0.5. When the oxide coating is about 2 micron, beyond 2 micron, it increase, but
somewhat slowly goes whom around 0.5 to around 0.8. So, this gives a rough idea as to
the impact of the thickness of the oxide layer on the emissivity of metals. So, it is
important to have some information about oxide layer thickness, before you assume
emissivity of metals, because in engineering practice pure metals with no oxide are very
rare. So, you should be looking out for a certain amount of oxide layer thickness, which
was substantially impact on the emissivity of the metal.
(Refer Slide Time: 24:27)
Now, this an illustrations of impact of surface finish and oxide layer on in kernel x. For
example, you see that newly polished in conel x has an emissivity close to 0.2. But if it is
not polish, it is just you see the clean is small like point 0.4, but once the oxide layer
forms on it it jumps to around 1 7 5. So, here again, you see the important role plays the
oxide layer here and important role plays by polishing material after is received. So, I
have received polished its emissivity goes down, wherever some time an oxide layer
forms emissivity jumps back to 0.8. So, this again is important issue in making estimate
of emissivity metals. Now, you go back to our discussion on selective surfaces, a
selective surfaces one whose absorptivity or emissivity is a strong question of wave
length. Now, here is picture of the observed normal spectral absorptivity of silicon oxide,
a symmetric factor on aluminum. And you can see that it falls around 0.9 in the reason
below 1 micron rapidly to around less than 0.1 beyond 2 micron.
So, this is the kind of material we will need for solar collectors to absorb the suns
radiation strongly. And the same time not to emit radiation that is coming in the infrared
on the surface. Now, real surface of course, are complex behavior for modeling
prospective, you can replace this by a surface with absorptivity of 0.9 or 0.95 in the
region below 1.5 micron. And assume it to be rapidly changing to an emissivity around
point naught 5 beyond 1.5 micron. This is simplification you can make or you want to be
little more realistic, you can you make an audio approximation, you can assume 1 in the
solar region and then rapidly considering to almost 0, in the reason beyond 1.1 micron.
So, these kind of approximation are made to calculate and we will take up 1 example in
this lecture.
(Refer Slide Time: 27:12)
Now, here is realistic depiction of the incoming solar radiation, which is around 1000
watts square meter. And you can see peaks around 0.5 to 0.6 micron. In the same figure
we have shown, you the emissions of black body at 100 degree centigrade, 300 degree
centigrade and 400 degree centigrade. And what you find is that up to 400 degree
centigrade most of the emission from the surface is confined to the region where the
reflectivity in the material is quite high or alternatively emissivity is very low. So, after
400 degree centigrade, the fact in emissivity is low in this region is advantages in not
losing heat. At the same time, when the same material in the solar wavelength region that
is around 0.5 to around 1.1 micron received that the reflective is very, very low.
So, it is absorbing strongly here and it is has a very low emission here, because high
reflectivity. So, this is molybdenum aluminum oxide, its already absorb a. And you can
see that it is very useful to get a flux surface to reach temperatures of the order of 400
degree centigrade. Without losing too much radiation is the region where the solar
radiation exist. So, this kind of coating have, are now big routine in use in solar
collectors all over the world. And the the basic principle here of course, is to have this
coating thin enough not to alter the emissivity of the material basis, already quite low.
And at same time ensure that we absorb all the radiation that is coming from the sun.
(Refer Slide Time: 29:52)
Now, one more example we will discuss in this lecture or the next lecture is the
temperature variation from at tungsten filament lamp. And you can see that a 65 wall
bulb, 100 wall bulb and 450 wall bulb. As you go to higher and higher voltage of the
bulb, the temperature or the filament is increasing slightly from 2800 kelvin to 2900
degrees Kelvin. And if you plot the black body emissivity power at those temperatures,
because see that the tungsten filament lamp peaks at a wave length of around 1 micron.
So, the tungsten filament lamp is not peaking in the visible.
(Refer Slide Time: 30:52)
So, tungsten filament lamp has maximum emission at around 1 1 micron. So, which is,
what we would not like, because tungsten filament lamp it is used for purpose of
lighting. Ideally should I had its peak emission somewhere in the wave length range from
0.42. So, peak emission is not between 0.4 and 0.7 micron. So, this means that tungsten
filament lamp is not a very efficient device for porosity light. For example, a typical 100
watt lamp will produce only around 8 watts of like radiation that is emission in the range
0.4 micron to 0.7 micron. So, it is efficiency is around 8 percent for 100 watt of electrical
energy consumption only 8 watts is coming out of the light.
So, that is why, in the last 10 years should be the same that most people, who are energy
conscious are switching over from tungsten lamps to light emitting diodes. This
occurring, because efficiency of the L E D’s is much higher than that of tungsten
filament lamps. And hence they are more efficient creational energy tungsten filament
lamp ideally is most suited for infrared heating, because it generates the peak radiation
around 1 micron in the infrared. So, if there are application where infrared heating is
desirable then you can go for tungsten filament lamp. But for purpose of lighting
tungsten filament lamp is not a good choice, because only about 8 percent of the energy
is consumed, where the filament lamp actually comes out a light energy. So, this is a
weakness, which you cannot do much about with reference tungsten filament lamp. But
other device like, L E D take a different approach to generate this light source. And it is
not L E D is not govern by loss of equilibrium thermodynamics.
So, tungsten filament lamp is of course, involved invention goes back to the days
addition. But it is now not planning a lot of favor with people who want to improve
energy efficiency, because a lamp by where it is design cannot really increase efficiency
substantiation. So, this must be kept in mind, now little later we will tell you why, the
you should realize by now, that if the fraction of radiation lying putting 0.4 micron to 0.6
micron is around 8 percent. We had cause was somebody thinking, but how to utilize
filament lamp, because 92 percent of the radiation emissivity filament lamp comes out as
heat and all of us are observing that heat the room also absorb that heat.
So, in a hot climate, having tungsten lamp actually will add to your air conditional load,
because your putting over more and more heat and that heat has to ultimately appear will
ultimately heated room unless there are ways to now the air from the there is here is. So,
the tungsten filament lamp can never achieve high efficiency, because the fact is that it
cannot go above it percent, because it temperature the filament lamp cannot exceed 3000
kelvin. So, that is the basic limitation of this lamp, it cannot exceed 2000 Kelvin, because
the melting point of tungsten is close to 3001 degree Kelvin. And as you approach the
melting point of the material the evaporation rates go up dramatically. Now, I have given
little example of the empirical data, we have.
(Refer Slide Time: 36:41)
For sublimation that is conversion of tungsten to vapor and a so this is a value which can
be shown to be around 20 power of minus 6 into T by 3000, the power of 30 grams per
centimeter square per second. This is this rate at which tungsten sublimation for the wire
and the key issue here is not the numerical value, but this power. So, notice that the
sublimation increases almost exponentially with temperature and it is going of the rate of
T to power 30. So, even a few degree rising temperature below a huge impact on the life
of the tungsten filament lamp, as most of you realize the tungsten filament is already
very thin. And as the sublimation occurs and parts of the material become vapour, there a
point where there will be 1 point in the filament where the amount metal. So, small that it
will break and all of you a use tungsten filament lamp good experience of fact that as a
lamp gets older and you switch on the light and the voltage some more high and normal
then you will see the filament breaking.
And so, because of high voltage the temperature the filament goes up by 10 to 20 degrees
and that is enough to big about a catastrophic failure. So, that is why people designed it
as filament lamp cannot afford to increase temperature beyond 3000. Now, if the
temperature tungsten filament lamp is kept below 3000 then you can see that it can have
only 8 percent of the radiation in the visible. So, the upper limit of the efficiency of the
tungsten filament lamp is controlled by the highest temperature that can be achieved
there keeping in mind that you need a lamp to have at least the life of 1000 hours why.
So, if you typically give a lights on at night for 2 to 3 hours a day you expect it loss
about year.
So, the design of the tungsten filament lamp has to account for the fact that you want
million life 1000 hours. And if that is a case you have to keep a temperature somewhat
below 3000 kelvin. And because of that your conversion of the electro energy to light
energy is only 8 percent. So, when you put tungsten filament lamp it is room 92 percent
of the electro energy that is consumed lamp coming on its heat. So, in country like India,
tungsten filament lamp is heating a room more than lighting. And that is why most
people are now switching over to light emitting diodes. Although, they are more
expansive, but their efficiency being higher in the long term they will achieve a better in
the basis your higher efficiency lower consumption of electric energy.
Now, let me take a few example to illustrate, how one computes total absorptivity and
emissivity for materials. Now, let us take a typical example here of any material. And let
us say the variation of characteristic spectral wave length is like this. Now, in today’s
world we will replace that by series of values steps. So, let us say this is i th and this is i
plus 1. So, the direction total absorptivity will be nothing, but some over all i, there is
different regions here from 1 to n. And this value let us call it as alpha lambda i has alpha
lambda i that is the absorptivity of the material between lambda i and lambda i plus 1.
Then you have to know the fraction radiation lying between lambda i to lambda i plus 1.
So, we have to know how much of the incoming radiation is laying between these 2
wavelengths, then once you know that you can compute this. Now, the information about
the fraction of radiation lying between lambda i and lambda i plus 1 can either be
provided directly to you wave transfer measurement. Or if you know that the income
radiation is proportional to a blackbody radiation as a certain temperature, then you can
compute this from standard tables available in most heat transfer text books. The fraction
radiation lying between 2 wave lengths for a given temperature for blackbody, for the
same material we know that alpha prime lambda is equal to epsilon prime a kirchhoff
law. The same material, you can calculate the emissivity, total emissivity again as sum
over i equals to 1 to n, now alpha prime lambda and epsilon lambda same. So, you can
use alpha prime lambda i, now we will use fraction of blackbody radiation that is waiting
function for emissivity.
Then lambda i T S to lambda i plus 1 T S this is available from tables in all text books.
So, the main difference between they calculate absorptivity in emissivity is the value F
we use. The value F for absorptivity is based on the how the incoming radiation varies
wave length. Where in the case of emissivity we go by the temperature of the surface and
we look at the blackbody function and compute the fraction of radiation lying between
lambda i to lambda i plus 1 from the blackbody function formula. Now, to make all this
very clear, I will take a very simple example to illustrate this.
(Refer Slide Time: 44:37)
Imagine material, whose emissivity varies its wavelength in a very simple way. Below 1
micron, let us assume it has 1 value and bring 1 and 10. It has is another value and
beyond 10, it is third value. The numbers are assumed for this example is 0.9, 0.5 and
0.2. Its variation more complicated you can use more number of intervals. So, let us
assume that this material is subject to solar radiation. And solar here we can assumed to
be blackbody of the temperature of 5800 Kelvin. So, we know from the blackbody
function that from F equal to 0 to 2 8 9 8 micron Kelvin. The radiation amount fraction is
0.25 where you go from F 0 to 5800 micron Kelvin; it is around 0.72, this available from
tables in the back of most books on heat transfer. Even this, 2 information you can
estimate alpha prime as fraction of radiation lying between 0 to 1 micron there will be 0
to 5800 micron Kelvin 0.72.
So, 0.9 into 0.72 plus remaining radiation for sun lies between 1 and 10 micron that is
remaining 0.28. So, 0.5 into 0.28 radiation beyond 10 micron was sun. So, these two will
lead to a value of 0.788 for absorptivity. For same material, we look at directional total
emissivity. We have to use an assuming surface temperature material is 289.8 degrees
kelvin assumed for convenience closed to room temperature. And for this value at 1 up
to 1 micron, there will be no radiation, will 1 and 10 will be our 0.25. So, we take 0.5
value from here multiplied by 2 5 plus, the remaining 0.75 is here. So, 0.2 into 0.75 and
this will give you a value of 0.275.
And so, you can clearly see that for this material alpha prime, epsilon prime is close to 3
not quit, but close to 3. So, this is some more selective like the, we saw in the case of
metal. And so, this was just illustrate how you compute the total properties from
properties, if we know piecewise values of these functions this can be of course,
extended to any number of intervals. And this can be automated you can add computer
program calculate emissivity in absorptivity given a temperature of the surface and also a
temperature of the incoming radiation if it is a blackbody. Now, let me give another
illustration, now as you saw.
(Refer Slide Time: 48:41)
The directional emissivity of many materials that is values like cos theta, let us assume is
1 into cos theta that is a theta equals 0, epsilon prime is 1 and at theta equals 90 epsilon
prime is 0 very simple variation. So, what will be the hemispherical total emissivity. This
we solved from earlier lectures is nothing, but e prime cos theta d omega. Now, this can
be written as 0 to 2 pi for the pi variation. And 0 to pi by 2 e prime is cos theta, there is
another cos theta you get cos square theta. And the, from the definition of d omega you
get sin theta d theta d phi. So, this integrates to 2 pi, so this becomes 2 times 0 to pi by 2
cos square theta sin theta d theta. And we assume sin theta is x. Sorry, you can assume, i
must below 1, you assume cos theta is x cos theta is x and d x d x nothing, but minus sin
theta d theta. So, this will come out and when you go from x equals theta goes 0 x goes
1. But because of minus sin, this will become 0 to 1 x square d x which will integrate to
2 into x cube by 3. So, you will get answer of two third. So, what you find is that the
directional emissivity varies as cos theta. The hemispherical emissivity is two third of the
value of the emissivity normal. So, directional emissivity at data equals 0 is 1. The
hemispherical emissivity now comes out a two third.
So, you can see typically many materials the hemispherical emissivity is about two thirds
of the directional normal emissivity. So, this is a useful to have, because you can
measure the directional normal emissivity fairly easily. Because for an opaque material
the some of directional term normal emissivity and the directional normal reflectivity is
equal to 1. So, if you mention the reflectivity you get the emissivity. And once you know
that if the material has wave has similar to what is given here as cos theta then your
hemispherical value will be two thirds of the normal value. So, this can information
useful for many applications, where in; where in we have to estimate the hemisphere
value from the value. Now, one more example give just omega point, we saw that for
many materials.
(Refer Slide Time: 52:32)
Emissivity is constant equal to less than 1 for theta between 0 and 60 degree. We saw
that example in the in the figures, I go earlier and after that goes rapidly 0. So, let us
assume that goes to 0 for theta greater than 60. So, if you have this kind of behavior then
your epsilon will come out as 1 by pi, again integral over omega epsilon prime cos theta
d omega. And once you do the integer get to in to 0 to 60 degree, because that is up to
which emissivity is there you will get sin theta d theta. Because the value is there is no
emissivity is constant comes out cos theta sin theta d theta. And so, you merely integrate
this now, and may integrate this now as before you get a value of 3 4.
So, in this case where the material had emissivity variation which was a constant from 0
to 60 and went to 0 after that, for such material you can see that the hemispherical value
is three fourth of the directional normal value. And this is some more higher than one we
saw in the previous example, where it was two third. So, that gives a roughly the typical
expectation that the hemispherical value emissivity is somewhere between two thirds to
three fourths of the normal emissivity. So, this information is useful for application
where in you may not get the emissivity value. And you have to make do with the value
of normal emissivity measured in the laboratory. And from there you want to infer the
value of the hemispherical emissivity.
So, what you have done so far is we have highlighted the fact that for most real materials
both emissivity and absorptivity varies strongly with wavelength and somewhat with
angle. And we have showing you how Kirchhoff’s law can be applied for these
materials, carefully not casually as is done in some sub books. And we also showed how
the spectral variation emissivity of many materials like, metals. And several surfaces can
be very useful in the design of tanks to keep the tank cool or in designing solar collectors
to increase the absorptivity solar radiation, at the same time keeping the emissivity low.
So, a good knowledge of the directional and spectral variation emissivity is very useful
in the design a many engineering systems. And the challenge for the designer very often
it should get the right data about this materials. We saw that for metals the radiative
property is very sensitive to the oxide layer thickness. The oxide layer is there, the
emissivity is much larger well oxide layer is not there emissivity is quite low. So, before
we can assume of value emissivity for metals we must be quite sure about the amount of
oxide that is there in that metal and an oxide layer is inevitable. Because many metals are
it easily oxidized and you should expect you to have a certain amount of oxide layer
unless its freshly cut material. So, with this we complete our discussion on the very
important surfaces. And so, we will be going on now to next understand, how we
estimate the fraction of radiation that is there in one region how much reach another
region. So, that will be the topic of the next lecture.
Thank you.