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Welcome to lecture 2 on module 9. In module 9, we are discussing on radiation heat transfer.
In the last lecture, we have discussed on various basic concepts, and various laws of
radiation and we introduced Also the all these laws and the concepts. Now, in this lecture,
we are going to discuss on the radiation intensity and radiation view factor. These are the two
important things that particularly we should know particularly, for any radiation calculations,
radiation heat exchange calculation and these are very important.
To start with the radiation intensity, we will see that how to get and expression for
the radiation intensity in this case. Just before that to define the radiation intensity,
it is the the rate of radiation by a surface, I should say that it is not the rate; rather the rate
of radiation by a surface is expressed in terms of radiation intensity. Now, this radiation
intensity can be divide two parts. There are two types; one is called spectral intensity
and I will denote it as I d lambda; I for intensity, lambda indicates the monochromatic
that is the spectral and b is for the black body. And Another one is the total intensity
like total emissivity power and this; basically I will write it as I b.
We will start with that spectral intensity. If we say, this is the rate of emission of
radiant energy or radiation energy, in a particular direction per unit area of the emitting surface,
which is normal to the direction. The surface is normal to the direction and per unit wave
length lambda, per unit solid Angle d w; per unit and the per unit solid Angle d w. So,
it is the rate of emission of radiation energy in a particular direction per unit area of
the emitting surface, but normal to the direction. That means the emitting surface is taking
normal to the direction. So, there will be a projection of the emitting
surface normal to the direction of the emission and that is emitted And per unit wave length
and per unit solid Angle d w. Now, d omega not d w, I should say d omega. So, unit of
I blambda will be equal to watt per meter square per micrometre per s r. So, d omega
is the solid Angle, lambda is the wave length and so here micrometre is for per unit wave
length and s r specifically steradian. This is the unit of solid Angle.
Let us take a small say first, I will draw something like this. This is the x-direction,
this is y-direction and this is the z-direction. Let us take a small differential area d A.
This is a hemisphere surrounding this. This is the circle of the hemisphere and this is
the hemisphere surrounding this. So, this is the hemisphere and we will consider any
point somewhere here say these. So, if we see this, it will make like this. Now, this
is the centre point and here if I take this point and this is at an angle theta and this
is d theta. So, this angle is theta and this says d theta and this is phi, this one is
d phi. So, this is area, this will be and if i say this is the r, this is also the r,
which is nothing but this is also r. So, this is the hemisphere, so this portion
will be this part, will be r d theta r sin theta d theta and this portion will be this
portion basically this is r d phi. So, area of this section is this section, say d A s,
will be equal to r square sorry r sin theta d theta into r d phi. So, this is becoming
r square sin theta d theta d phi. So, this angle is phi and ranges zero to two pi, and
theta ranges zero to pi by two. So, these are the two angles, this is its spherical
coordinate. Now, this is say d A, so a differential area d A is surrounded by a hemisphere of
radius r. So, this is the radius of hemisphere is equal to r and incidentally this is the
centre point O. O is a centre of the differential area d A, as well as the centre of the hemisphere.
So, area of hemisphere; surface area basically is two pi r square, total area of this sphere
is four pi r square. This is the half of this and is equal to two pi r square. Now, projection
of d A that is the base area position of d A normal to r direction is d A cos theta.
Now, the differential rate of radiant energy emission from the surface d A around A wave
length lambda should be like this, d q dot for black body, for lambda wave length is
proportional to, this is the differential energy emitted and should be d A cos theta
that is in the direction of the projection in the direction of the surface.
The d omega is the solid angle into d lambda that is emitted by this from small area d
A, and d q dot b lambda is given as I b lambda d A cos theta into d omega d lambda. This
is I b lambda and this also can be written as I b lambda d A cos theta and d omega can
be written as d A s by r square, that is the solid angle; d A s by r square, r square is
the distance, d A s is the surface area on the hemisphere, into d lambda. So, this is
called solid angle d omega is equal to d A s by r square. So, d q dot b lambda is given
by this expression and and we can say that I b lambda, here it is called, it is the proportionality
constant and it is called as this is called as spectral radiation intensity.
Now, for the total intensity, we have to do the integration that is I b; I b will be equal
to zero to infinity I b lambda d lambda, this is called total radiational intensity. Now,
if we just see the spectral emissive power, now this is what the relationship we have
got for is. So, I b can be obtained from this, so I b is called the proportional and I b
is called proportionality constant and it is also called the spectral radiation intensity,
which is written like this. And I b total radiation intensity is that is this, zero
to infinity I b lambda d lambda. Now, if we have to develop a relationship between emissive
power and the intensity, then spectral emissive power is given as d q dot differential amount
b lambda that is equal to say E b lambda dashed, that means E b lambda, we know the spectral
emissive power and not only that and E b lambda dashed means, it is in a particular direction.
So, E b lambda dashed is a directional spectral emissive power, this is called directional
spectral emissive power and we know that d q dot also again can be written as d A d omega
d lambda. So, this is also we know that it is d I b lambda d A cos theta d A s d lambda
by r square. Here, we know that E b dashed is equal to this and all other terminologies
are known, d A s is we have just seen r square sin theta d theta d phi we have seen at the
beginning.
We can write is that d w, sorry, d omega is that this is called solid angle is d A s by
r square and that is equal to sin theta d theta d phi. So, E b lambda dashed d omega
is equal to I b lambda cos theta sin theta d theta d phi and now hemispherical emissive
power
can be obtained by integrating the directional emissive power e dashed b lambda over the
hemispherical surface. So, E b lambda is equal to for over the solid angle E b dashed d omega,
for the whole angle, so this is equal to zero to two pi, phi equals to zero to two pi, theta
is equal to zero to pi by two, I b lambda cos theta sin theta d theta d phi and then
after integration and putting the values.
We will get E b lambda is equal to pi I b lambda. So, this is called hemispherical emissive
power of the black body. So, this is equal to pi times or so hemispherical emissive power
of a black body is pi times the spectral intensity, hemispherical also spectral emissive power
we can say, spectral intensity of radiation. Now, again further integrating over the whole
range, so E b there will be, hemispherical emissive power for the whole wave length,
if we say, E b will be equal to say zero to infinity for all the angles sorry all the
wave lengths. This will be equal to pi zero to infinity I b lambda d lambda, and this
is equal to E b is equal to pi into I b. Again the emissive power, the total emissive power
of a black body is pi times the total intensity of the black body. So, this is the relationship
that can be derived. Now, coming to next part; this is the radiation intensity and radiation
intensity and emissive power relationships.
Now, coming to the relation view factor, which is very important thing, which is radiation
view factor. What is meant by that is that suppose a surface is radiating some energy
and that it can as, because it is a diffuse surface. Normally, the surfaces are diffusive
in nature, so that the radiational energy goes to in all the directions, so the point
is that a surrounding surface, which is recieving the energy, may not be the whole energy that
is being emitted by a surface is being received by the other surface, nearby surface.
Therefore, there must be a fraction of the energy that is being received by that particular
surface. So, to find out that how much fraction or what fraction it can receive that is being
expressed in terms of view factor, that means to what extent or what fraction of wall surface
is viewing the other surface and that is the basic idea. So, that is also called a view
factor. So, a definition of a view factor, if we say that it is also called say that
F i j is the view factor, it is also called shape factor and it depends upon the shape
of the surface. It also called as configuration factor, so there are several names.
So, between two surfaces, surface i and surface j; so what is this? The fraction of energy,
the total energy, again the radiation energy or radiant energy, that is emitted by the
surface i and is intercepted by the surface j; so, this is called as view factor. Also,
alternatively, it is the
fraction of energy, radiational energy, radiant energy leaving surface i, which reaches surface
j and that means, from ith surface that fraction of energy that reaches to the surface j. That
is called view factor or radiational shape factor or radiational configuration factor.
Now, to develop a mathematical expression
expression for view factor, this is the target. So, let us see that this is one object and
this is another surface; object of a surface. Now, small surfaces are these, so I will say
that this is surface say A i and this is surface A j. Here, this is d A i differential area
and this is d A j differential area of the surface and they are at a distance r from
each other. But, r is the distance connecting them, but
r makes an angle say theta i, with the normal of the surface A. Say, angle theta, so if
i say like this, say this is say theta j. This one theta j with a normal, this is the
normal for the surface. This is the normal for this surface, so r is the distance between
the two surface, r distance between two surfaces of d A i and d A j; theta i is the angle between
d A i and its between r, r and the normal of d A i, similarly, theta j is the angle
between d A j and the normal to d A j.
Now, if I b i is the total intensity of radiation that means it includes all the wave lengths
as we have seen that, emitted by the surface say d A i, and intercepted by the surface
d A j. Then we say that d q dot i j equals to I b i; I b i is the intensity of the radiation
emitted by the surface. So, it is better to write like this; intensity of the radiation
emitted by the surface A i, then the amount of radiation emitted by d A i and intercepted
by d A j. Now, it is in case of black bodies. So, both bodies are black bodies. We can write
that d q dot i j, so amount of energy that is coming from d A i and going to and it is
being intercepted, received by d A j and that is equal to I b i into d A i into cos theta
i d omega i j. This is the solid angle subtended by them by d A j on this d A i. So, d omega
i j is basically the solid angle subtended by area d A j into cos theta j by r square;
r square is the distance between these two points. Then d q dot i j can be written as
I b i d A i d A j cos theta i cos theta j divided by r square.
Now after integrating we get that q dot i j, the total for the total surface area A
i, it will be integral of A i integral of A j, for both the surface between the total
surfaces A i and A j, it will be I b i d A i d A j cos theta i cos theta j divided by
r square. This can be written as, now integral I b i is taken as the proportionality. So,
this will be constant A i integral A j and then cos theta i cos of theta j divided by
r square into d A i into d A j. Now, the total amount of radiation q dot emitted
by surface; the total amount of radiation q dot i that is emitted by the surface A i.
This is the emitted, but it may not be intercepted by surface j; this is emitted by surface j,
but may not be intercepted by surface j, whatever is intercepting by surface is this one. This
one is intercepted by the Q i j, but total amount emitted is Q dot j sorry Q dot i and
that is given as Q dot i and that is equal to that is based on the emissive power E b
i. That already we have seen that this is A i into pi into intensity that this we have
already proved that E b i is equal to pi into I b i.
Now, from the definition of view factor F i j will be equal to Q dot i j by Q dot i,
which the view factor or the definition of the view factor and that will be one by pi
A i integral A i integral A j cos theta i cos theta j by r square d A i d A j. Similarly,
view factor F j i will be one by pi A j into integral A j integral A i into cos theta j
cos theta i by r square d A j d A i; similarities. Now, if we see here so A i into F i j will
be equal to A j into F j i. So, comparing these two equations we get A i F i j is equal
to A j F j i. So, this is a very very well known relationship, this is called reciprocity
relationship. So, this derivation is being made for black body.
Though the derivation is made for black body, it is well applicable for any other bodies.
Also, it is well applicable for any other bodies provided the radiation is diffused
radiation. So, it should not be specular radiation, it should be diffuse radiation. Now, we have
seen the so called relationship for F i j or F j i. So, we got that relationship also,
we have also got a relationship for reciprocity relationship and that also we have got. Now,
another point is that the number of view factors for say suppose for a black body enclosure
of N surfaces. The total view factors or number of view factors
should be would be N square and they are represented by a matrix called view factor matrix. Typically,
say if i say that for Nth surface like F 1 1, F 1 2, F 1 3 and then F 1 N, then again
F 2 1, F 2 2, F 2 3, F 2 N, then like this F 3 1, sorry F N 1, F N 2, F N 3 and like
this, F N N; this is called the view factor matrix, which is by N by N matrix, square
matrix so this is called view factor matrix.
Now, there are different ways, we can find out this view factors. If you see that the
various charts are available to find out the view factors for defined geometries. This
is a particular geometry. You can see here the view factor for parallel rectangles; these
are the rectangles, which are one and two. These are the two rectangles, they are in
parallel and we see that for different parameters like one is b, view factor one to two, the
fraction of one that is being received by surface two, F 1 2, there is the fraction
of one that is being received by two is represented by this figure, where B is given by w by d.
For different values of A and A is given by l by w, l by d, and d is the distance of separation
between these two. So, from this chart we can find out the view factor, which is very
important to find out the view factor.
Similarly, there is another figure is there. So, standard charts are available in the literatures,
in the books, so one can use this standard chart to calculate the heat exchanger problem,
radiational heat exchanger problems. Here, it is that we can see that these are perpendicular,
this is the 90 degree angle, and this is the common edge between two A 1 and A 2. These
are the two surfaces and these are the defined dimensions of the surfaces and A is b by c
and B is a by c, so there is B and for different values of A, this F 1 2, is again calculated.
There is another case like this and it is known to us. This is the radius, which is
nothing but R 2, and this is the radius for surface one which is capital R 1 and there
are two parameters. One is called l by r one and this is called r dashed, which is R two
by l, based on that F 1 2 again is calculated and these are the different platform, different
values of these things. So, from the charts, this is somebody else
done, and these charts can be obtained by doing that already we have developed a relationship
that F 1-2, can be found out between two surfacesm by this. This kind of relationship can be
used to find out the values of this view factors and then this can be plotted to generate this
kind of charts. So, theoretically we can have an expression for view factors and then those
expressions can be solved to generate the charts for view factors, for different kind
of geometries. That is also possible. However, when we have the geometries available with
us we can use those geometries and try to find out this view factors.
A typical application of the equation that what we have developed just for view factors,
if we try to see for the case of say two concentric circles or concentric disks, if we see that
this is a bigger disk and this is a very small disk or little bigger, and a little smaller
disk and they are concentric in nature. So, this is the central line, for then the central
line should match, so these are the concentric one, say for a differential region of this
and so this is the distance of separation between them, centres of this.
At a distance say r, it is from the centre, this is r and this is say at a point this
r naught and this is d A 1. Now, if this whole one is say A 2, so if F d A 1 to A 2, if i
try to find out F d A 1 to A 2, that will be 1 by pi d A 1. From the equations, we have
seen A 2 into cos theta 1 cos theta 2 by r square d A 1 d A 2. We are telling that these
are two concentric disks. So, theta 1 will be equal to theta 2 and r square is equal
to l square plus r square. So, at any point, this is r and this is l, so l square plus
r square is equal to, r square like this, and we can find out that cos of theta is equal
to theta, say will be equal to l by root over of l square plus r square and d A 2 at a distance,
d A 2 differential area at a distance r, will be equal to twice pi r into d r 2 pi r into
this distance is d r.
Now, because this is a differential area, so d A 1 and this d A 1 cancels with each
others. then what we can get is that F d A 1 r 2 A 2, is equal to one by pi, d A 1, d
A 1 cancels and this integration will be zero to r naught; r naught is the radius of the
A 2 surface and this is l square by l square plus r square into one by l square plus r
square into two pi r d r. So, if we do the integration, we will get it to be r naught
square plus l square plus r naught square. So, this is becoming the view factor from
smaller d A 1 to A 2. So, if i want to find out for FA 2 to d A
1, how to get it? It is practically simple by putting, by applying reciprocity relationship.
That means we know that A 2 d A 2, F A 2 d A 1, will be equal to d A 1 F d A 1 A 2. So,
from here we can find out, if F A 2 to d A 1 will be equal to d A 1 by A 2 into F d A
1 to A 2, so F d A 1 to A 2 already we have calculated. So, we can now calculate F A 2
to d A 1 pretty easily. So, this is a demonstration of the mathematical expressions that we develop
for view factor. Now, one more thing I just want to say here is that the view factor matrix
we have seen so a reciprocity relationship already we have seen.
So, if i say that there is a four surface enclosure then number of view factor matrixes,
view factors, so number of view factors will be 16. So, the matrix should be is 4 by 4
matrix. Now, number of reciprocity relationship will be for all the surfaces, there will be
one like F 1-1 plus f two-two f one -one plus F 1-2, plus F so, that is what I wanted to
see that say number of reciprocity relationship would be, It is four into N into or rather
should say four into four minus one by two that means, if i have N numbers of surfaces
say for one surface, I will have N minus one, such thing, so A 1 F 1-2, A 1 F 1-3, like
this so, and then this will be repeated. Therefore, we will be having N into N minus
1 by 2 reciprocity relationship and this becomes now three into two equal to six and then another
thing is there is that another relation say summation of F i j is equal to one. What I
say is that there is a surface one and surface one to say we can say that all the energy
supplied from one to different surfaces, summation of all these should be equal to one.
So, this kind of relations will be having for N surfaces, we will have N relations.
In this s present case, it is four such relations, typically, say F 1-1 plus F 1-2 plus F 1-3
plus F 1-4 equal to one, F 2-1 plus F 2-2, plus for surface two, is F 2-3 plus F 2-4
is equal to 1 and so on. Like this there will be four, if write that F 3-1 plus F 3-2 plus
F 3-3 plus F 3-4 is equal to one and F 4-1, plus F 4-2 plus F 4-3 plus F 4 -4 is equal
to 1.0. There will be such relationships and these are being available for this.
The relationship between shape factors and some more problems we will discuss. The relationship
between shape factors, for example, say this is a surface say A 3 and let us say there
is another surface say say A 1 and this is A 2. Now, shape factor for radiation from
A 3 to combined area A 1-2, shape factor from A 3 to combined surface A 1-2 is desired.
Then what we will write is that A 3 to 1-2, that is what is needed and that is shape factor.
This should be equal to and can be written as F 3-1 plus F 3-2, so between the two plates
A 3 to 1 and A 3 to 2, that can be so and from there we can say that A 3 F 3 to 1-2,
will be equal to A 3 F 3 1 plus A 3 F 3 to 2. From reciprocity relation A 3 F 3 to 1-2
would be equal to A 1-2 F 1 to 2-3 and A 3 F 3 to 1 is equal to A 1 F 1 to 3 and A 3
F 3 to 2 equals to A 2 F 2 to 3.
If we do the analysis then we will get that A 1-2 to F 1 2 to 3 that is equal to A 1 F
1-3 plus A 2 F 2 to 3 and F 1 to 2-3 equals to F 1 to 2 plus F 1 to 3. So, total radiation
arrived at surface 3; total radiation arriving at surface 3, is the sum of radiation from
surface one and two. Now, total radiations arriving at surface three is the sum of the
radiation from surface one and surface two.
We like to see that another case like say if we would be interested to know the radiation
for this shape factor say like this. so, this is A 1 and this is say A 2 and this is A 3.
The interest is to find out F 1 to 3, because they are not adjacent. So, for that we can
find out F 1 to 2-3, we can find out. From figure discussed and this figure are perpendicular
rectangles. Similarly, F 1 to 2, also can be found out from figure. Now, our interest
should be to find out F 1 to 3, so we can say that 9 F 1 to 2-3 is equal to F 1 to 2
plus F 1 to 3. So, we can find out from here that F 1 to 3 and that is my target F 1 to
2-3 minus F 1 to 2. So, we know the value of this and we know the value of this, then
we can find out the value of F 1 to 3 from the figure.
Some problems, say we have say a hemisphere and we want to determine the view factor matrix
for the hemispherical enclosure. The hemispherical enclosure is like this, this is the hemispherical
enclosure, and thus this is r. So the radius is r. So, this part is surface 2 and this
part is surface 1. So, view factor matrix is is equal to F 1-1, F 1-2 F 2-1 F 2-2 and
we know from the relationship F 1-1 plus F 1-2 is equal to one, and F 1-1 is equal to
zero. The reason is that for a flat surface or any point of a flat surface cannot see
the other point of the same flat surface. Therefore, the F 1-1 will be equal to zero
and this is true for a convex surface also; flat surface and the convex surface, any point
on a flat or convex surface cannot see the other point the same convex and flat surfaces.
Therefore, it is zero and then F 1-2 is equal to one and then we can find out A 1 F 1-2
will be equal to A 2 F 2-1 and then F 2-1 will be equal to A 1F 1-2 by A 2. This is
pi r square by two, pi r square; two pi r square is the hemispherical area. So, this
is equal to half and that is equal to 0.5. So, F 2-2 is also equal to 0.5. So, view factor
matrix is equal to matrix (0, 1, 0.5, 0.5).
There is say another problem is that find the view factor matrix for the two concentric
long cylinders exchanging radiation heat. So, it is like this, so this is a big cylinder,
this is the small, and if I say that this is surface one and inner surface is surface
two and say r one, this is say r two. So, r one is equal to outer radius of inner pipe
cylinder of long hollow cylinder, and r two is equal to inner radius of outer cylinder.
What we will get is for inner cylinder F 1-one plus F 1-2 is equal to one and F 1-1 is equal
to zero, so F 1-2 again is equal to one. Now, F 2-1 will be equal to A 1F 1-2 by A 2 and
that is equal to one. I will say that twice pi r one l into one by twice r two into l.
This is equal to r one by r two. So, F 2-2 is equal to one minus F 2-1 is equal to one
minus r one by r two. So, V F M is equal to matrix of zero, r one by r two, one, one minus
r one by r two. So, this is the view factor matrix and so this way we can find out the
view factors for different case. We can use that analytical expressions and we can use
that graphical representations whatever is the available and then we can go for the calculations
of radiational heat exchanger which we are going to discuss in the next class thank you.