 Okay, in the last segment we introduced the idea of the boundary layer and the fact that fluid mechanics is very important in terms of determining the connective heat transfer coefficient. What we're now going to do now we're going to look at the flat plate again but we're going to consider only the first part of the flat plate boundary layer flow and that is where we have a laminar boundary layer. So in analyzing the laminar boundary layer some of the earliest development came with Ludwig Prandtl and what he did is he took the Navier-Stokes equations and these are the governing equations for fluid mechanics basically f equals ma and he took the Navier-Stokes equations and he simplified them for a boundary layer flow and then what happened is one of Prandtl's students, Blasius, he solved these equations through numerical hand calculation and so the Blasius solution is what will become the basis of a lot of the results that we're going to be using when we look at the laminar boundary layer. One thing that I should say is that no exact closed form analytical solution exists. Blasius solution only comes about by doing numerical integration and consequently you get a table of values that you can use for the velocity. Now Theodor von Karman a little later played around with different types of profiles quadratic cubic and he came up with approximations for the velocity profile but it was not an exact velocity profile like Blasius's came out to be and both von Karman and Blasius were students of Prandtl at Göttingen in Germany and so what von Karman came up with was a velocity profile that looks something like this you divide by the free stream and this is looking at a quadratic there's also cubics that that he came up with different functional profiles and so if you want you can look at my fluid mechanics lectures and I go through the use of this profile for coming up with things like the skin friction coefficient and the momentum thickness we're not going to worry about that now in this lecture because we're looking at heat transfer but looking at the velocity profile it would be something like this here's our flat plate remember x is going in the direction of the flow y when we're looking at boundary layers is always normal to the plate and then we have our delta back center velocity profile and so here is the velocity profile once we get out into the free stream we get to you infinity and here we have you and I'll show it of sorry not why it's you the velocity you of wine really implicitly it's also of x and and that results given the fact that delta of x is in here and and consequently it is scaling with you with y and x and delta the boundary layer thickness is delta of x remember we said that's where we get the 99% the free stream now that is a velocity profile that von karmann came up with now with blaze yes he was able to using his numerical integration come up with the value for delta of x so let's take a look at what that value was obtained as being and he got delta of x the boundary layer thickness was 5.0 x divided by Reynolds number of x to the one half and and so this was blaze yes solution okay so that is only one half of the problem and I'm not going to go through and come up with the blaze yes profile you can look in textbooks of fluid mechanics and and find that some heat transfer books may have it as well but having the velocity profile is only one half of the problem because really remember we're interested in heat transfer in the convective heat transfer coefficient so we have to go a little bit further than that okay so having the velocity profile that's only taking us halfway now what we need to do we need to determine the convective heat transfer coefficient and so how can we go about doing that well it turns out that you need to be able to get the temperature profile in the boundary layer in order to estimate the convective heat transfer coefficient now let's take a look at how we may go about doing that so here we have a the flat plate again assume that's our boundary layer and here we're assuming that the flat plate oops not T infinity that should be T wall and then we have some flow field and this is not the velocity boundary layer that I'm drawing that this would be a profile of the temperature distribution let's say the wall is hotter than the fluid you may have something that looks like that so that would be your temperature profile and then eventually you get to the point where you are at T infinity which is the free stream temperature profile and what we're going to do we're going to introduce a thermal boundary layer thickness which is going to be a function of x as well and so that would be the point where we get out to the free stream temperature and you do not see the presence of the wall any longer so T of y I remember y was normal to the wall and the result of this is we have heat transfer taking place and so there's q and we know through Newton's law of cooling we have the convective heat transfer coefficient which is what we're ultimately after here so looking at the equations what we can do we know through Fourier's law and what I'm doing is I'm taking advantage of a thing called the no-slip boundary condition and what that means is right along the wall if we look at the velocity profile that's not a very good plate let me do this okay so here is our flat plate and and if we look at the velocity profile right along the wall if you're to go microscopically into the wall at the wall the velocity is zero u at y equals zero is equal to zero and and so that is what we call the no-slip boundary can condition and with that the only mechanism of heat transfer when there is zero velocity is going to be via conduction and consequently we can use Fourier's law and so that's what we're doing with this expression up here and we also know that that through Newton's law of cooling is going to be equal to h times T wall minus T free stream and we have that expression so what I'm now going to do is I'm going to work to isolate h so let's isolate h and we obtain that expression there and so this is going to be the basis by which we're able to determine the convective heat transfer coefficient for the laminar boundary layer flow that we're looking at but in order to get this what we need to do we need to know temperature as a function of y and and that is part of the solution technique that is required so we need to know the temperature profile okay so in order to get the convective heat transfer coefficient we need to know the temperature profile and then once we obtain h what we do and typically in fluid mechanics is we embed that within a non-dimensional number and and that number is the new salt number so there is the new salt number what it is it's the convective heat transfer coefficient times some characteristic length scale in this case acts the distance from the start of the plate divided by the thermal conductivity of the fluid and so let me write those out okay so a new salt number is the number that we will use quite often and one thing I should say is notice we have new salt number x that denotes that this is a local new salt number not an average other times you'll see new salt number with an over bar that denotes average for an entire plate so just be careful this new salt number refers to convective heat transfer coefficient evaluated at some specific x location other times this would have h bar and that would be the average convective heat transfer coefficient over an entire object but anyways that is the new salt number and that is what we will use just like the Reynolds number but we'll use it for characterizing the amount of convective heat transfer coefficient on some object that we're studying so that's the laminar boundary layer the new salt number and what we're after we need to get the temperature profile so what we are going to do in the next segment we're going to take a look at the thermal boundary layer in relation to the viscous boundary layer the velocity boundary layer and and they are related to one another but we'll be looking at that as we move along looking at an introduction to convective heat transfer