Systems and apparatus for concentrating cell light
are often used in the field of photovoltaics.
So for example, shown here is this parabolic curve disk.
And it collects the sunlight coming from the sun.
And then it concentrates into this narrow beam.
Which is made incident on the solar cells
and many times these concentrated photo voltaic systems,
they use these multi-junction solar cells which have these
multiple junctions which have higher efficiency for solar conversion.
Shown here is the another such system.
So, over here you have this Fresnel lens which is placed over
here, and this is collecting again all the sunlight coming from the Sun.
It concentrates it into this narrow beam, which is then made
incident on this solar cell. So these systems, currently they have
concentrations of 500x or 1,000x or 1,500x.
What I'm interested in finding out in this video is what is the fundamental limit.
Or, you know, how much x of concentration,
you can achieve for these for these systems.
So in this video I want to investigate,
is there a fundamental limit, or is there a thermodynamic limit for this light
concentration? So, let's get started, so The
first thing I'll do is essentially I'll consider that I have this
receptor over here and this has a surface area of S.
So how much power of sunlight can be made incident
on this receptor?
And I'll relate it to the brightness of my source, which is sun.
So I'll relate it to the brightness of the source, which I'll denote by B
with a subscript S, standing for source or Sun, whichever way you want to take it.
And I want to relate it to this angle which the sun subtends on Earth.
So, many a times we, we consider the light which is
incident from the Sun just as a source of For these parallel rays.
But, in fact the sun has a finite radius. And it has
there's a finite distance from which the sun and the earth are separated.
So, this angle is a very small angle. But nonetheless it's a finite angle.
And it's equal to approximately 0.27 degrees.
So this is the angle which sun subtend
subtends at a point source located on the Earth.
So this brightness from the Sun, is is incident
on this point source, which has a surface area dS.
Within this solid angle.
So I can relate this this power, to essentially the brightness of the source.
And it's related by this angle, by this factor of a sine squared theta.
And there's a factor of pi, which comes from this geometric averaging, for
[INAUDIBLE]
Integrate this over this small solid angle, and you get this factor of pi.
If you're to understand more of where this this formula comes from, you can
read this Wikipedia article, and many a times this principle
of, you know, expressing light in this way Is called the Principle of
Etendue conversate, Conservation, or Conservation of Etendue.
And so we can express the power incident as essentially
this pi times the area of this Area of this area
of this disk which is S, and then sine square of
this angle, sine square alpha, into the brightness of my source.
And you might you might recall that this this sine
sine alpha or sine square alpha I can also express
it as a numerical aperture.
So I can also express this power
as essentially square of my numerical aperture.
Times my area times the brightness of the source.
So this is another way to remember this formula.
So now let's see how, how, you know, how can I concentrate it, and then
What is the power incident on my disk on which I'll place my solar set?
So what I've done over here is the shown that initially
I had this surface which was collecting the sunlight, facing the sun.
And this had an area of S.
And now I have a place that's concentrated optics over here.
So this will essentially take this light incident on the, on the
surface S and concentrate it to this disc which has an area
of, let's say, sigma over here.
So just like I did for the sub,
bigger disc, I can express the Total power incident
on the smaller disc, which has a area sigma,
as essentially pie times the area of the disc.
Sigma multiplied by this angle, sine square of this angle.
Which my receptor is now substending on this modern disc.
So it's it's substending
this angle of two beta, then.
The factor of sine squared beta,
multiplied by the brightness of the receptor.
So this is the power which is now concentrated on this smaller disk.
[SOUND]
And now what I'm interested in finding is this
concentration, or how many times I've concentrated my light.
And I can express that as essentially the area of
this bigger disk, divided by this area of the smaller disk.
And that gives me the concentration of how many times I
have reduced the area, that many times I've increased my concentration.
Now important thing to note is that the power, which was incident
on this bigger disk Is now concentrated into this power, into this smaller disk.
But I'll have a conservation of power, assuming that
there's no power which is lost during this concentrating process.
The power which was incident on this bigger disc has
to be equal to the power incident on this smaller disc.
So I can equate this power incident on the smaller disc.
I can equate it to the power
which was incident on the bigger disc which was pi times The area of the bigger
disk, multiplied by sine squared times this alpha,
into the brightness of my source which was sun.
So now I, I can equate these two things, and I see that if
I equate I can essentially take take get a ratio of this S by sigma.
So I can express this S,
times Sin square alpha multiplied by
the brightness of the source. This must be equal to sigma
times this equation which is sigma times sin square beta.
Multiplied by the brightness of the receptor.
So I get this relationship for my concentration.
Now what I'm interested in finding out is the maximum value
of this concentration.
So let's see when this concentration would be maximum.
So I have this Br by Bs term over here.
And this concentration will be maximum when this ratio is maximum.
So Br is the brightness of my receptor and Bs
is the brightness of my source which is the sun.
So what is the maximum value of this ratio?
So the maximum value of this ratio could only be 1, because my receptor,
it cannot be more brighter as compared to the Sun.
Because that would violate my second law of
thermodynamics, that would say essentially that my receptor has
become more hotter, it's emitting out more brightness
as compared to the Sun, which is not possible.
So the maximum value of this ratio has to be essentially equal to 1.
Similarly, what is other thing I can maximize over here?
So this sine squared alpha
is pretty much fixed by the geometry of the
sun and how far the sun is from the Earth.
But this sine squared beta, so this this is something I can optimize, but the
maximum value of this sine squared beta, can essentially be only equal to the 1.
And that would be the case when this On
this slide this beta is equal, equal to 90 degree.
And in that case, the maximum value of
the concentration we'll now get is essentially this
term will be equal to 1, this term will be equal to 1, and the
maximum value of the concentration would be
essentially 1 divided by sine square of alpha.
Now I've done this analysis so far assuming that
my concentrated system It had a refractive index of one.
But I could as well as make this conservative system,
in a material which has a refractive index of of n.
And, in that
case what will happen is that my
numerical aperture, it will essentially increase from From
sin theta to essentially n sin theta,
or my numerical aperture will increase n time.
As a result of that, this power intensity
which is incident with this power which is incident
on this on this on this smaller disk
will essentially increase by a factor of n squared.
You know, I can Now
essentially get, er, I can concentrate light from a larger acceptance angle.
Which is, increased by a factor of n.
So what I'll see in this power, I'll essentially see a factor of, n squared.
Sine square beta instead of, just sine squared beta.
As a result of that, in this concentration as well,
I see a factor of n squared, and this, maximum concentration
will instead become scaled by a factor of n squared
[INAUDIBLE].
I'll get the value of a maximum concentration to be this.
If I make my this receptor and this concentrated system,
in a medium which has a refreactive index of n.
So now I see that the maximum value of my concentration,
it essentially given by N square divided by this sine square alpha.
And I know that alpha is essentially fixed to be equal
to approximately 0.27 degrees. So I, if I substitute that
and into this formula, I get a value of this maximum concentration to be
approximately equal to n Square times 45,032.
So if I make this this concentrated system in a medium
of refractive index of one, then I can get a maximum concentration
of approximately 45,000. And as I mentioned earlier, most.
Of the concentrated systems today they operate at a concentration
of maximum concentration of 1000x or 1500x, which is limited
by, if you go higher than this concentration, you tend
to heat up your cell, and your efficiency starts to degrade.
But you see that from a fundamental perspective there's a lot of headroom
in terms of how much we can increase this concentration.
So there's a lot of headroom till we hit the fundamental limit.