 Okay, the last thing that we're going to look at in this introductory lecture to convective heat transfer is going to be an analogy between heat transfer and fluid friction. So this analogy, it turns out, it will apply to flat plate flows be it laminar or turbulent, so it is quite flexible in that regard. Okay, and so what this analogy is going to enable us to do, it's going to enable us to take friction measurements on flat plate flows, so a frictional drag, and relate it directly to the convective heat transfer coefficient H, which is what we're after here. So let's begin by taking a look at wall friction or shear stress along a flat plate boundary layer flow, and the friction coefficient, or the shear stress I should say, along the wall is related to the dynamic viscosity and the gradient of velocity in the y direction, and this can be determined in a number of different ways, but typically the most common ways are either indirectly measuring the velocity profile and then taking the derivative and you get the wall shear stress, and another method is actually measuring the drag force, and with MEMS sensors that's becoming more and more possible, but this can be measured by, and in measuring the velocity profile what is quite commonly used is a pitostatic tube called the Preston tube, which is basically a squashed pitostatic tube that enables you to go right along the wall, and so the probe body itself would come up and away from the wall. Let's see, I don't know if I can draw that in 3D, I probably can't, but let's give it a try. So you would have something like this where your probe body is coming up and then eventually it goes away from the wall, and so your flow is coming in this direction here, and that would enable you to get the velocity right very close to the wall. You could also do that with different optical techniques as well, although it would be difficult to get seeds particles very close if you're using laser Doppler velocimetry for example, but anyways it would involve some indirect measure where we get the velocity profile. A second technique is to directly, physically measure the drag or the friction on the wall, so conceptually what this might look like is you may have a section of the wall removed where you have some element. Now this is going to be a very very small element, I've drawn it as being a little bit larger, but with MEM sensors microelectromechanical systems, we can make these very very small now, and so you would have that above, you have your shear, so you have your velocity profile coming over. You'd want to ensure that the gap here is very very small so that you don't disturb the flow, but out here you would have your velocity coming along, and by measuring then a drag force or the restoring force on this little element that would be area A. By measuring that we can say drag is then going to be the wall shear stress multiplied by that area A, and from that we would then be able to determine what the wall shear stress would be, so we would measure that, we would know that, we would get the wall shear stress, and once you have the wall shear stress, if we look back here, and that is what we're after here, once you have the wall shear stress, you can then determine the friction coefficient, and so the friction coefficient, and I'm going to put it here as being a function of location on the plate. If we divide the friction coefficient by two, we have our wall shear stress divided by the dynamic pressure, but I've divided by two, so it's going to be rho u infinity squared, and for Blasius's boundary layer solution, he got 0.664 for the skin friction times Reynolds number to the minus one-half, I've divided by two, so that turns out to be 0.332 Reynolds number x to the minus one-half, and it'll become apparent in a moment when I'm writing this out, so that is for the Blasius solution for a laminar boundary layer, and what we're going to see in the next few lectures is for the laminar boundary layer, we'll find that the, you can determine the new salt number, and the new salt number for the laminar boundary layer is 0.332, parental number to the one-third, Reynolds number to the one-half, and you'll see we're starting to look kind of similar, we have the 0.332 there, that's interesting, and that this would also be basically Blasius and energy using the energy equation temperature profile, you can get the convective heat transfer coefficient from that, like we've seen in this lecture. Now, what I'm going to do, I'm going to divide both sides of that by parental Reynolds number, so if we go new salt number x divided by parental Reynolds number x, and here, oops, sorry, that's not one-half, that was one-third. Okay, so what we have here is this is going to result in parental number to the two-thirds in the denominator, and I'm going to take that over to the left-hand side of the equation, and with that we get new salt number x, okay, so I brought that over to the left-hand side, and then the other thing that we have here, we have Reynolds number to the one-half divided by Reynolds number, that results in being Reynolds number to the minus one-half, and so what we end up with on the right hand side, and what is interesting about this is this here, and this, well, actually that are the same, and that is where the analogy starts to come from, and it turns out that the supplies for turbulent flow as well, but what can we do with that? Well, let's take it a little further, and so what we're going to do, we're going to play with this term over here, and we're going to expand everything out, so expanding that term out, we have the new salt number, we have new salt, we have parental and Reynolds in the denominator, so I'm going to invert them, that is the parental number, and then the Reynolds number, so with this here, we can cancel some things, k goes with k, mu goes with mu, we have an x and an x, and what we're left with, I think I've canceled everything out that I needed to, we're left with the convective heat transfer coefficient divided by rho c sub p u infinity, and that, I forgot something, I forgot the parental number to the two-thirds, so let me pull that in, this should have been multiplied by parental number to the two-thirds, so we will keep it here, and then on the right hand side, we had 0.332 rex to the minus one-half, so what we can do, we're going to introduce a new non-dimensional number, and that is referred to as being the Stanton number, and that is right here, and it is given the symbol s t x, do not confuse it with the Struel number, we always have different numbers with, hopefully, different letters, but in this case, it looks like they're the same, anyways, okay, so what do we get here, we get s t x, so that is the definition of the Stanton number, and then plugging it into the above relationship, what we get is Stanton number, parental number to the two-thirds is related to the local skin friction coefficient divided by two, so what's so significant about that, well, and let's see, before I get into that, I should say that this is restricted over a range of parental number, some textbooks say 60, but essentially 0.6 up to about 50 or 60, what's significant about this is it applies for both laminar and turbulent, and this is referred to as being the Colburn analogy, and a fellow named Colburn came up with this, I think it was in 1933, and it applies for laminar and turbulent flow over a plate, and so what is nice about this is that there already is a great deal of data characterizing the skin friction coefficient over flat plate and boundary layer flows, laminar or turbulent, and so all of that data can be used in this relationship to determine the Stanton number, and from the Stanton number, we get the convective heat transfer coefficient, so that's the beauty of this, it opens up a lot of data that we can use for heat transfer calculations, and so anyways that is the Colburn analogy, the Stanton number, and we will use it from time to time and coming up with relationships that we will be using to determine the convective heat transfer coefficient, so that concludes this introduction, what we'll be doing in the next lecture, we're going to start going into further detail with laminar boundary layer relationships, and then we'll start looking at different empirical relationships for determining the convective heat transfer coefficient.