 In this segment, what we'll be doing is taking a look at free convection, free or natural convection on inclined surfaces. So thus far we've looked at the vertical flap plate and the vertical cylinder with large diameter. We're now going to be looking at inclined surfaces over variable angles. And this is for the case of a constant wall temperature. So when we have an inclined surface, we could have two different scenarios. We can have the case where we have a plate that is cooler than the ambient temperature and we can have a case where the plate is warmer. So we'll begin by looking at the case where the plate is cooler than the ambient temperature. So what we have here is a schematic of a plate that is cooler than the ambient temperature. And what will happen in this case is the fluid cools on the surface and it becomes more dense. When it becomes more dense, the difference in buoyancy or the difference in density, I should say, causes the fluid to descend and move downwards. As we saw on one of the video clips in the previous lecture, I showed you the case of a vertical flap plate where the wall is cooled and we saw the fluid moving down. Now, when you incline the surface, the top of the plate is actually okay. And what happens is our boundary layer forms as we would expect. And then outside here, we would have our ambient or a free stream temperature, but on the lower side, what happens, depending upon the angle, the fluid will start descending, but then we get three dimensionality starting to creep in and we get clumps of fluid kind of falling away as the fluid is descending. And consequently, the correlations on the lower surface are not reliable. The ones on the upper, the boundary layer looks very similar to what we've seen before and consequently, we can use our correlations from before, but for the lower surface, we cannot. So that is a schematic for the cool plate. Before jumping into the correlations, what I want to do, let's take a look at what the flow looks like for a heated or a hot plate. So in the case of a heated plate, what is happening is the density of the heated fluid is going down and consequently, it rises and so we get the buoyancy force going and then we get the current and we have the fluid moving up on the lower surface is very much like what we would see for the vertical heated plate. Now on the upper surface, however, what happens is again, just like for the cool plate on the lower surface here, we get the boundary layer forming, but then we get clumps of fluid breaking away and when that process occurs what happens is our correlation breaks down and consequently on the upper side of the heated plate, the correlation does not work, but on the lower side, the correlation will work and so depending upon what surface you're looking and if the plate is heated or cooled, you have to be a little careful in terms of the application, but there is a slight modification to the correlations and I'll introduce those in a moment and you'll notice that I have indicated the coordinate system both for the cool plate. I put the coordinate surface on the upper side of the plate because that is where the coordinate system would apply given that our correlation works on the upper side of the plate and for the heated plate, you'll notice that I put it on the bottom because that is where the correlation would work for the heated plate. So what we can say is for these two different scenarios of an inclined plate, the correlations apply for the following. So the correlations that we've seen thus far will apply and this is for the constant temperature plate. The cool plate, it will work only for the upper surface so okay and then down here not okay in terms of the correlation and for the heated plate, correlation here would be okay and then up here not okay. So just keep that in mind and then the other thing is that it will only act over restricted angles. So what we're looking at here, theta is the angle that we're looking at and it turns out that the correlation works obviously from zero degrees because that would be a vertical plate that we've looked at previously but it goes from zero degrees and theta will go up to a max of 60 degrees and when you get both beyond that, you start moving into what we would call a horizontal heated or cool plate and the fluid mechanics becomes different and their correlations break down. Another thing that we do though, when we use the correlations that we've looked at for the heated vertical plate, we make a substitution for the gravity constant or the gravitational constant and that substitution is g cos theta under the either the top surface or the lower surface. So in the Rayleigh number or the Grashof number, we have the gravitational constant. We make that substitution. So with that, which correlations will apply? Well, any of the ones that we looked at for the vertical heated plates, I'll just write those out. So I guess there was the one that we had with the Nusselt number was equal to C times that was Grashof-Prandtl and then we had it to M with the power there. So that correlation would apply but also we had the more detailed and more accurate ones and so I'll write those out now. So that's the correlation that applies if we have laminar flow and remember within the Rayleigh number we have the Grashof number where the gravitational constant as we make that substitution and then if you have the other correlation that extended to laminar and turbulent so this was the correlation that would pertain to a wider range of Rayleigh number and this is with again g equals g cos theta and theta ranging from zero up to 60 degrees. So that is how you handle an inclined surface over this angle range and if you want to determine what is happening in terms of the connective heat transfer coefficient on the surfaces that we were not able to do so essentially on the hot plate on the upper surface and on the cool plate on the lower surface if you're looking for correlations for that what you're going to have to do is you have to go into the literature and look in journal articles to see if people have done experiments investigating those types of flow fields because these correlations will break down they won't work there the flow gets three dimensional and a little bit more complex than what we can do with the equations that we have here. So anyways that that is how we handle inclined surfaces with free convection.