 Okay, so in the last segment what we did is we came up with expressions for the amount of heat transfer from a fan for three different cases. One case was for a very long fan, the second case was for where we have convection from the tip, and then the third was for a tip that has insulation on it. And what we're now going to do, we're going to plug numbers into the equations that we came up with. So we're going to be considering an example of a rod fan. And what a rod fan is, it's a fan that has a round cross section. So if this was our fan, it would look something like that. This here is the base, and the base recall has temperature T base. So this is our fan, X is going in this direction, but essentially what it means is that the cross sectional area at any point in the fan is round, and so we will have a diameter that describes the fan, and we will also have some length that we're dealing with, and that would tell us what the length of the fan is. So that is what a rod fan looks like, not the greatest drawing, but you can figure out what's going on there, hopefully. And so for a particular example problem what we're looking at is a rod fan. The diameter we're told is 0.005 meters, thermal conductivity of the fin material 398. The base temperature we're told is 100 degrees C, and the free stream convective environment which would be the fluid in here, coming through T infinity H. We're told that the free stream convective environment is 25 degrees C, and we're told to consider a fin that the length is 20 diameters. So that is the length for the problem that we're looking at, and what we're going to do, we're going to consider this for two different convective environments. Convective environment one will be 100 watts per meter squared degrees C, and convective environment two is 1000 watts per meter squared degrees C. So what we're going to do, we're going to take these numbers, we're going to plug them into the three different cases that we looked at, and what we'll be coming up with essentially is the temperature distribution along the fin. So we're going to be solving for T of X, although it's going to be in forms of this theta of X that we had from our equation. Okay, so let's take a look at the solution for a rod fin, and we'll begin by considering the convective environment H1 equals 100 watts per meter squared. So this is the result of that, and what we have here. We have three different cases that the first one, you won't see it here, but this was for the long fin, so that is black, and then we had the convecting tip, and then finally we had the insulated tip. And what you can see is that the long fin is here, and the convecting and insulated tip are here. So those two solutions are quite similar, and what we're looking at here, this is actually over D on the horizontal, so this would be the base of the fin, and then we're plotting temperature of X minus T infinity. And so what we can see is that the solutions for the convecting and the insulating tip are really quite similar to one another, even though the insulated tip solution was quite a bit simpler analytically. It only had a hyperbolic tan, whereas the convecting tip had hyperbolic sines and cosines in it. Another thing that we have plotted on here is this line here, and what this is showing is, let me erase that, it is showing the slope at the base of the fin. So that is DT by DX at X equals zero, so what we're doing is we're quantifying the slope down here, and through Fourier's law, we know that everything that flows into the base is what is going to be dissipated to the air by the convective system of the fin. And so through Fourier's law, if you know that slope, you can then determine the amount of energy being withdrawn, specifically QF, and that is the heat transfer from the fin. So that is the case of H equals 100. Now let's take a look, let me see here, did I, I had the right units there, that's good. Let's go on to H equals 1000, and that's what we have here, and first of all what you notice is our slope is a lot different. We have a much larger convective environment, so a lot more convective heat transfer is taking place. The other thing to notice is that the long fin, the insulating fin, and the, or the insulated fin tip, and the convection from the tip, they're all pretty similar here, with the exception of a little tiny disagreement at the end. And so with the higher convective environment, we have a scenario where all three solutions are very similar to one another. This slope is dt by dx at x equals zero, and again we can determine the heat being withdrawn from the fin using the slope at the base at x equals zero. And so this here is the base of the fin, and this here is the tip. Okay, so looking at the two different solutions, comparing them, first of all we notice that our slope is changing significantly, and so with the higher convective environment at 1000 we would assume that there would be more heat drawn away, and certainly there is, because the slope is greater and consequently through Fourier's law we would have increased heat transfer. The other one is that for the lower convective environment, the convecting tip and the insulating tip are really quite similar, and we're going to take advantage of that in the next lecture, and we kind of come up with a shortcut correction technique that enables us to use the solution for the insulated tip, which was very simple by just correcting the length of the fin itself, and that is a bit of a correction that will apply, and it's a bit of a shortcut for doing fin calculations. But anyways, that gives you an idea as to what these solutions look like for the case of a rod fin.