 Okay, we're now going to solve an example problem using the finite difference technique and the application that we're going to look at uses Excel. And what we're going to do, we're going to begin by solving a fairly simple problem, that of a square flat plate with four different temperatures on the boundaries. And so for this plate, we have a temperature of 100 degrees C at the top. We have one of 75 degrees C on the left, 90 degrees C on the right and 25 degrees C on the bottom. And what we'll be doing is the plate itself is 0.6 meters wide by 0.5 meters high. And we're going to assign a grid spacing a delta x delta y of 0.05 meters. And with that, we will have 13 cells in the horizontal direction and 11 cells in the vertical direction. So here we have our spreadsheet. And what we're going to begin by doing is selecting a cell. And so we'll select that one there. And then we will go across 13. So 12345678910111213. Now you'll notice I've already adjusted the width of these columns to be narrower. And that way, it enables us to visualize what is going on within the grid that we're investigating. And I always like to color these. So let's color that so that we know where our grid is. And then we said that there would be 11 cells in the vertical direction. So let me select 1234567891011. And I'll color that orange. I just wanted to make sure that there were 11. And then what I'm going to do, I'm going to populate the rest of the grid. And I'll make that orange as well. And we'll begin by putting in some arbitrary number. I like to put 25 degrees C as being the starting point. And we may not necessarily need that for the current problem. But if you're doing radiation conditions, sometimes the copying and pasting of the boundary conditions can cause problems. And so by putting in numbers at the beginning, it makes it a little easier and you avoid those problems. So what I'm now going to do, I'm going to modify the perimeter boundary conditions. And if we want, we can make those different colors. So let's do, we'll just make all the perimeters a different color. Now they're going to be different temperatures, because remember, we said that the four walls are at different temperatures. But this is kind of just a way to visualize it a little quicker. And now the upper wall we said was 100 degrees C. So what I'm going to do is I'm going to copy and paste all the interior or the upper wall. The corners, I'm not going to treat yet. The corners will have to do the average of each of the walls. Looking at the left wall, it was 75 degrees C. So copy. Boy, that reddish are hard to see. Maybe I should change that. I probably will in a moment. 25 degrees C here. Well, that's already done, because that was the one that we started with. And then 90 here. And I'll fill in the rest. Let me get rid of that red. It's kind of hurt on the eyes. And we'll put another color there for that boundary condition. What looks good. You can't see anything with the blue, yellow. Yellow is not bad. Let's do that. And finally, we'll do that one there. Okay. So what we have now, we have the boundaries. With the exception of the corners, the corner there, the average of 175, that's 87.5 degrees C down here. Now the average of 75, 25 is 50. And then this corner, it's 90 and 25, 57.5. And then finally up here, 190, the average is 95. Okay. So there we have all the boundaries specified. And this is quite a simple one because we have no internal generation. We don't have any kind of boundary conditions other than specifying the temperatures on the perimeter of the object. So all we need to do now is copy and paste in the interior node cells. So clicking up in the formula bar that shows us that the interior node is being computed from the upper, lower, left and right, which is what we found in our finite difference formulation of the energy equation, or the heat equation, I should say, the heat diffusion equation. So what I'm going to do, I'm going to take that, I'm going to do control C, and then I'm going to drag in the middle, and I'm going to do control V. And there you can see it does a calculation. I'll push F9 a number of times. And we'll see that it slowly converges. And what it's doing is it's doing the calculation over and over and over again. And there we go. Looks like it's pretty much converged. So that is the solution of the square plate that we're looking at with the boundary conditions as we've imposed. Now it's sometimes nice to be able to graph this up in a contour plot. So let's select that data, and then we'll go up on to insert. And depending on the Excel version you're using, you might have different ways of doing the contour plots. You'll have to figure that out on your own. But I want the contours. I'm going to select that. It puts it there. Let me put it right below. Now one of the things that Excel does, and I still don't know why, I'm sure some of you out there know why, but what it does is it inverts the plot. So let me show, if I change this temperature at the top, let me put it 50 degrees, you'll see that it's actually at the bottom of the contour. And we don't want that. What we want, we want to actually represent where we think the object is with our grid. So the way to correct that, if you come up under chart tools, not design layout, then you go under axis, depth axis, and then show reverse axis. And there now we can see the little anomaly that I introduced this up at the top, which is where it should be. Let me change that to 100. And I'll hit F9 a couple of times that it converges. And there we go. What we can see here, this is a contour plot showing us the temperature distribution in our plate. We had temperature scales as shown over on the right hand side in the legend. So the bottom of the plate was at around 25 degrees C. And when we look here on the contour plot, 20 to 40 is red. And that's what we see at the bottom of the plate. And then the blue, light blue is actually the hottest section of the plate. And when we look at our boundary conditions, the hottest is there. And it's also quite high temperature over here on the right hand side, a lower temperature over here at 75. And we can see that in the purple. So that would be from 60 to 70 degrees C. So anyways, that's a very, very quick demonstration of the Excel model. And there are many, many other boundary conditions that exist within the Excel model. But you can see it's pretty easy to do calculations that we're taking quite a while when we were trying to do the separation of variables technique and solving the heat diffusion equation for a plate with different boundary conditions. And the problem that we looked at earlier in the course was one with, I think we had 100 degrees C on three of the walls and then 110 on the fourth wall or something like that. It wasn't as different as what we're seeing here. And so anyways, that shows a demonstration of the Excel model. We'll take a look at another example in the next segment. And we'll be introducing slightly more complex boundary conditions for that one.