In this screencast, I'll discuss what view factors are and why they're important for
radiation. So a view factor is written as F sub ij, where i are integers indicating
surfaces that radiation leaves from or is intercepted by. In particular, view factor
is the fraction of radiation that leaves area i - so that's where the radiation comes from
- and is intercepted by area j. Although many of these view factors for different shapes
are tabulated, it's still important to understand how to calculate them. So there are a few
methods by which to do so. The first one is just inspection, and what that means is that
you can look at one surface and tell right away how much radiation is being intercepted.
This is often all of it or none of it. The second is to use relationships developed for
view factors. So one of them is that A sub i times F sub ij is equal to A sub j
F sub ji. And this is known as the reciprocity relationship. Finally, we can use what's known
as the summation rule, and what it says is that the summation of all these view factors
from j equals one to n, where n equals the number of surfaces, that summation has to
equal one. This by the way includes F sub ii, which represents the fraction of radiation
that leaves the one area and is intercepted by that same area. So for example, if you
have a concave surface, F sub ii does not equal zero. However, if you have a plane,
or you have a convex surface, F sub ii is going to equal zero because it can't see itself.
One thing to remember is that the reciprocity rule only applies to two surfaces while the summation
rule applies to an enclosure, whether it's two, three or more surfaces. We're going to
look at a couple of examples where we'll only have two surfaces, so we'll have four different
view factors. Just as a note, the number of view factors equals the number of surfaces
squared. So first let's look at a hemispherical duct. Our surfaces are inside the duct and
we'll call the floor right here Area 1, and we'll call this wall (the curved surface)
being Area 2. It's easy I think, even though I mentioned it before, to see that F11 equals
zero because A1 cannot see itself and therefore cannot intercept any radiation. So by the
summation rule, if F11 equals zero, then F12 has to equal one. Now we use the reciprocity
rule to find F21. So A2 times F21 equals A1 times F12. So our F21 is just equal to F12
A1 divided by A2, or in this case, because F12 equals one, this is just A1 divided by
A2. So remembering this is a duct or a tube - and the length of the tube we'll just call
L - so the area of the floor (or Area 1) is going to equal 2 times the radius of the hemisphere,
times the length of the tube. Our A2 is going to equal half the area of a circular tube
which is one half times pi D L, or just pi r L. And so if we take the ratio of the two,
we get that our F21 is equal to 2 over pi. And if we calculate that out, that is equal
to 0.637. One thing to recognize here is that F22 does not equal zero and by the summation
rule, F22 equals 1 minus 0.637, or 0.363. And what that means is that's the amount of
radiation that leaves Area 2 that is then intercepted by Area 2. So now let's look at
a slightly harder example. Here we have a duct with a triangular groove in it, and we'll
call the area here, of the groove, right here all the way down the length of the pipe, that's
going to be Area 1, where this is also going to be the radius of the circle. We'll call
Area 2 the wall that hangs over the duct. So we'll start with F12. You might be able
to see from inspection that F12 equals one since it intercepts all the radiation from
A1. Another way to look at it is to look at F11. Those arrows indicate the radiation
that's leaving from the surface, and what you can see from that is none of the surface
intercepts any of the radiation. So F11 is going to be zero, and by the summation rule
F12 has to equal one. And now again we use the reciprocity rule. So our A1 F12 is equal
to our A2 F21, and since we're looking for F21, it's again going to equal A1 over A2.
However, as you can probably see, the areas of these surfaces are different. So let's
start with A1. The area of that is going to be 2 (because there are two planes) times
r (which is the length of each one) times L (which is the length of the pipe). Now if
you take a look at Area 2, it's basically a circular tube with a quarter of it cut out.
So we can look at this as three quarters times 2 pi R L. And so when we look at A1 divided
by A2, it is just going to equal 4 divided by 3 pi. And if we calculate this out, this
is 0.424. And like the previous shape, F22 does not equal zero. Our F22 is going to equal
1 minus 0.424, or 0.576.