 We're now going to work an example problem for an enclosed vertical space and the problem that we're going to look at is that of a double pane window. Okay. And so there's our problem statement. What we have is we have a window that is 0.8 meters high, is double pane, and we have two temperatures that we know, we know a temperature on one side and the other, and we're looking to determine the heat transfer through the window, and we're also looking for the R value. So in order to solve this problem, and we're told that the fluid in the window pane is air in the gap between the two window panes is air and atmospheric pressure. Sometimes you can reduce the pressure within the gap to reduce the density, which then reduces the amount of natural convection taking place, but in this case we're told that it is at atmospheric pressure. So let's begin this problem by writing out what we know. So that is what is known, and then what we're looking for, we want to find the heat transfer through the window, and we're also looking for the R value. Okay. So to begin this problem, what we're going to do, we're going to go and we're going to get the properties, and the properties for this, we're going to be using one of the correlations for natural convection and vertical cavities, and for those the properties were evaluated at the average temperature. So the properties here at Tav, so 280 Kelvin. So going into the property table and looking up the values for air. So those are all the properties that we may be using for this problem. The next thing that we're going to do is we are going to write out a schematic. The schematic is not too complex for this. Actually it's just the schematic that we looked at earlier for vertical cavities. We're told T1, T2, T1 is greater than T2, and the gap here is L, and that was two centimeters, and we know that there is air in here. So for the analysis, what we're going to do, we're going to begin by determining the Grashof number, and then from that we'll get the Rayleigh number, and then we'll determine which correlation we can use based on the Rayleigh number. So let's begin that. So it turns out the aspect ratio is 40, and that's to be expected because we're dealing with a very, very tall window, and the gap between the two panes is quite minimal. That's a horrible looking window. Let me just not even do that. OK, so we know that it's a window. It's very tall, very narrow aspect ratio of 40. The Rayleigh number here, Grashof times the Prandtl number, we have one times 10 to the 4. So looking back at the previous lecture, you'll be able to find the correlation number two that we can use. I'm going to use one that's a little bit more complex, and so with that, we have the Nusselt number. So we can plug in the values, and what I'm going to do, I'm going to take the K over and the L over, so I'm going to isolate for H on the right-hand side, I'll write out explicitly what all the values are. Now the Prandtl number that we're using here, I believe the correlation said that it would apply for Prandtl numbers greater than one. Let me double check that. Yeah, we did see the Prandtl number would be greater than one, but 0.7 is close enough. This is heat transfer, and we'll use that. That's probably fine. And that's all that we have at our disposal to solve this, so that's what we're going to do. So with that, what do we get? We get H is equal to 1.73 watts per meter squared kelvin. And Q, we want to find out the heat transfer coming through this. We use a lot of cooling. Temperature difference is the hot minus the cool. Plug in the values. So 27.68 watts is what is flowing through this window. Now let's look at that. Q double prime or Q over A, 17.3 watts per meter squared. And we are now going to use that in determining the R value. So this is the first part of the problem, and getting Q. The second part is the R value. So let's go on and take a look at that. And if you recall, we looked at this a long time ago in the course. Much earlier lecture. But we had said that the R value in SI is delta T divided by Q over A. So here we know those values. So you get an R value of 0.578. And now let's put this into context of the R value that you would see if you went to a hardware store. And if you recall, we said you take the value, the R value SI, and you multiply it by 5.679. And so with that, we get an R value of 3.28, which is pretty low. Because typically, typical walls, if you have an insulated wall, you could have R12, R16, R20. And here we have R3.28. And so therefore, windows, and no real surprise here. Windows are not great insulators. So what do we do about that? Well, there are a couple of things you can do. Different gases in the gap of the window. So sometimes argon gas they'll use or different things like that. You can lower the pressure, which I said earlier, although your window might implode, probably not something you want to do. Although they're incredibly heavy, you can get triple pane windows. And so the triple pane windows, you would have one convective cell here. It'll be going like that. And then you'd have another convective cell on the other. But it would be reduced. And so you're significantly reducing the amount of heat transfer that goes through. They're very expensive, very thick, very heavy. The triple pane windows is another way of bumping up the R value for a window. So anyways, that is looking at the case of enclosed vertical space. And we looked at an example of a window. We used the correlation. We determined the heat loss and the R value. And that will conclude our coverage of natural convection flows.