 In this segment, what we'll be doing is taking a look at free convection from horizontal plates. And in what we're going to be looking at, we're going to be looking at plates that could either be round in cross-section or they could be square or rectangular. And so with that, we first of all need to begin with a length scale that we would use to characterize a horizontal plate. And so what I'll do is I'll give you a number that enables us to determine, or it's an equation, that we can determine the characteristic length scale to use in our non-dimensional numbers. So this is an equation that we can use in determining the length scale that we'll use in our non-dimensional numbers. So that would be in the Nusselt number as well as in the Grouchhoff number, they both have length scales. And so that would be the value of L that you would use in both of those equations. And notice that the area, that would just be for one side of whatever shape that we're looking at. And then it would be divided by the perimeter. And this works for square, rectangular, and circular discs. So that is the characteristic length scale that we can use in the correlations. Now what we need to do, very similar to the incline plate, it will depend upon where the surface is either heated or if it's cooled and if it's an upper hot or a lower hot. So what we're going to be doing is we're going to look at a couple of different conditions. And we'll begin by looking at the case of the upper surface heated or the lower surface cooled. And it turns out that both of these scenarios result in similar fluid mechanics although they're reversed. And I'll try to sketch that here. So let's begin with a sketch of what the flow looks like around these plates. So what we have here on the left hand side we have the case of upper hot. And we can see as the fluid on top of the plate heats it moves up and in the process through the continuity equation we know that we're going to have entrainments of cooler fluid around the outside. And that's what I've sketched there. And then in the case of a lower cold surface we have that on the right. What we have the fluid on the bottom is cooling and therefore it becomes more dense and it will descend. And in the process we will have again entrainments of fluid. And I've shown these columns just to basically show that we have a separated flow or clumps. It's a three-dimensional flow field and that would be the scenario of the flow that you would have if it's either an upper heated or a lower cooled. And with that we have correlations that pertain to these types of flows dependent upon whether the Rayleigh number would be in the laminar regime or if it would be in a more turbulent regime. Actually there's a range. It's not the exact same number that we've had before for 10 to the 9. It's actually 10 to the 7. But let me sketch out or write out what those relationships are, the correlations between the new salt and the Rayleigh number. So there we have our two different correlations. The first one here, you'll notice that we have the Rayleigh number raised to the power of one quarter. And before when we were looking at the correlations we said that that would be an indication of laminar flow. And although the Rayleigh number, it's not 10 to the 9 like we saw before. It's a little lower. But still that would be an indication of more of a laminar like flow. And then as the Rayleigh number increases the exponents on the Rayleigh number goes to one third. So that again could be construed as being more of a turbulent type of a regime for the flow. So that is the case of upper hot, lower cold. Now let's look at the flip side of that where if we have the lower surface hot and the upper side cold. So again what we have here on the left hand side, this would be a case where the lower surface was hot. So looking at it, and I think I drew this in the wrong location, let me flip that around. So the lower is hot. And then on the right hand side that would be a case where the upper is cold. And so what we can see is when the lower is hot what is happening is the fluid is moving up due to the fact that it's becoming heated on the underside and then that entrains cooler fluid from below and then we would get kind of a jetting of fluid going up on the left and right hand side. And when the upper surface is cool, again what it's going to do is it's going to entrain fluid down and it's going to cool and then it's going to come along and that will be more dense fluid and it will move downwards and so that would be upper cold on the right. And with that, when you look fluid mechanically, these do look similar, they just flip from one another. And consequently with that we can write out the new salt number relationship, it would pertain to either of these and it would be the following. And here this would apply from 10 to the 4 to 10 to the 9 so that we have to the power of one-fifth but I would assume that that would be more of a laminar type flow that we're looking at. And we don't have the clumps of three dimensionality like we saw in the other one and that's probably why this would be viewed more as a laminar type flow. So anyways that is the correlations for free convection from horizontal plates for different conditions, either heated, oh you know what, I think I goofed that up. This here, Ts, should be on this side and then over here we said that lower was cold and consequently Ts should be there so the lower surface was the part that was cooled so we would assume that that was a cooled surface and then here we would assume that that was a heated surface and then for this other one the hot surface was down here and the cold surface was up here and so I apologize about that, hopefully it's not too confusing but you would have either the heating either being on the upper or the lower surface and you get different fluid mechanic and heat transfer correlations coming out of that. So anyways that is how you hand all the case of free convection from horizontal plates.