 In this lecture what we'll be doing is taking a look at some of the correlations that exist for various shapes dealing with natural or free convection. So what we'll do, we'll begin by looking at the case of an isothermal surface. So that would be either a plate or a vertical cylinder that is at a fixed temperature. And if you recall from the last lecture what we said is the analytical techniques can only take us so far. And quite often when we're dealing with natural convection and we're dealing with either turbulent or complicated geometries we end up having to use empirical relations. So that's pretty much what we'll be looking at in today's lecture. But the relationship that we often use is the following. So what we have here, we have the new salt number expressing the convective heat transfer in any of these natural convection cases. We have a length scale L and then finally what we have in the denominator with our new salt number is the thermal conductivity for the gas evaluated at F. So wherever you see the subscript F that is referring to properties at the film temperature and the film temperature is the wall temperature plus the ambient temperature outside of our heated object. So outside the thermal boundary layer divided by 2. And that's a common approximation that we saw for force convection boundary layers as well. Now typically what we have here in this part of the equation we have the Grashof number, Prandtl number. That combination if you recall from the last segment we said that that could be clumped into what we call the Rayleigh number. Now typically what we have here, we have this coefficient M in the equation as well. And we have this value of C and I'll talk about that shortly. But to begin with let's take a look at the coefficient or the power M that is in this equation. So typically M will be one-quarter for laminar boundary layers on our heated vertical surface. So for free convection the boundary layer if it's laminar M would be one-fourth. Now that is typical and remember we said that the way to determine whether or not we had laminar or turbulent depending upon the Rayleigh number or the Grashof times the Prandtl number. And if it was less than 10 to the 9 that is what we said would be a laminar boundary layer. Now if you have a turbulent boundary layer typically M would be on the order of one-third. And you may find correlations that are using the same sort of relationship but M does not meet that one-quarter or one-third for being a laminar or turbulent boundary layer. So just be aware that this is typically the case but sometimes there will be correlations that don't exactly match up to this. Now for turbulent remember we said that the Grashof Prandtl or the Rayleigh number that would be greater than 10 to the 9. And I should put an approximate because we're on the order quite close to that but whenever you're dealing with transition that number could be plus or minus a little bit. So that is a relationship that we sometimes use for natural convection. I should get rid of that because that might confuse you. It was Grashof Prandtl that we had there for the and then that is expressed as we said Rayleigh. Okay, so that is an expression that we will quite often find for free convection on a vertical surface. Now I had mentioned that you can use this for cylinders. It's under certain approximations that you can use this relationship for vertical cylinder. So we can treat a vertical cylinder as a flat isothermal plate if the ratio of the diameter to the length of that cylinder satisfies this relationship here. And so what this is getting at it's basically saying if you have a very very large cylinder and the length would be the vertical here, the diameter would be there. But if you have a very very large cylinder with a large diameter and you zoom in on what is happening in the boundary layer, if we have this requirement here satisfied then we can basically approximate the boundary layer as being that just over a plain flat plate and consequently we can use the flat plate relationship. So essentially what we're saying is that the curvature would have to be not that significant for the cylinder that we're looking at and would have to satisfy the requirement that we have there. So what we're going to do now, let's take a look at some of the correlations. So we'll begin by looking at the values of C and M for the isothermal vertical flat plate. So we'll begin by looking at the case of a laminar boundary layer. And remember what we're looking at here is Nusselt number which is Grashof, oops I forgot the C, it would be C Grashof-Prantel to the power M. So the value of C if we have laminar is 0.59 and again this is going to vary, you can look at any heat transfer book and you might find numbers that are a little different from this. And because what has happened, people are fitting their experimental data to this functional fit or this functional form and so there could be slight variability. If you have turbulent flow, our coefficients are going to change and we already said that M would be one third but C would be 0.10. So those would be the values of C and M for a vertical flat plate in the case of it being isothermal, so constant temperature and there are a couple of other relationships that exist and I will provide those now. So it's a one correlation that this would be a single correlation and then this would be laminar and turbulent. So that is one single correlation that would cover both laminar and turbulent and here you can see I have replaced the Grashof-Prantel with the Rayleigh number in the expression and the properties would be evaluated at the film temperature. Now taking a look at a correlation that works only for the laminar flow regime and so we have that correlation there. Now which one would you use? Well it depends on if you have a turbulent Grashof-Prantel number, Rayleigh number greater than 10 to the 9, if it's less than 10 to the 9 then obviously use the laminar relationship because that will give you a more accurate result and it is interesting to note in the first equation what we have is Rayleigh number to the 2, 6 which is Rayleigh to the one third which was the value M equals one third that we said for turbulent flow and then in this relationship here we have Rayleigh number to the one quarter and that would correspond with our M equals one quarter. So it's consistent with the relationship that we saw earlier, the one with C we had Grashof-Prantel to the M which is C Rayleigh to the M. Essentially it's the same thing but you can see there's a bit of a correction here for Prantel number effects and that is what is in the denominator in these two relationships. But those are expressions that you can use for isothermal vertical flat plates and it also works for a cylinder provided that the curvature is not that strong in the cylinder and we saw a relationship that enabled us to determine that. In the next segment what we'll do we'll move on to not isothermal but we'll be looking at constant heat flux relationships for the vertical flat plate.