 Alright, in the last segment what we did is we took a look at the laminar boundary layer and we said that in order to determine the convective heat transfer coefficient we needed to be able to determine the thermal boundary layer or the temperature distribution normal to the wall. So what we're going to do in this segment we're going to take a brief look at what the thermal boundary layer development may look like and we're going to begin by looking at the thermal boundary layer thickness. Remember we saw the boundary layer thickness itself that was delta of x. Now here we will have delta t of x and just like we said for the hydrodynamic or velocity boundary layer this is defined as the point where we reach 99% of the free stream temperature and there may be other definitions for it but that's one that we'll use here. Now what I'm going to do I'm going to begin by looking at just a generic flat plate and we're going to make it very generic from the perspective that not all of the plate is going to be heated. So this is the leading edge of the plate up here and this is our plate. And what we're going to do we're going to assume that the first little bit of the plate is not heated and consequently we can say that in this region up here the temperature of the plate is equal to the free stream temperature and the free stream we have coming in this direction so that is u infinity and t infinity and coordinate system normal to the wall and then along the wall is x and we will call this distance here x naught and what we are going to assume is that heating is starting there and consequently this here is equal to t wall and how you go about obtaining that that would depend but you'd have to have some form of heat flux and if you want to maintain an isothermal wall temperature at t w you would probably have to vary the amount of heat flux as a function of position along the plate it would not be a constant q it would have to vary but anyways you'd have to have some sort of regulation process a controller or something like that but nonetheless what we have is heat flux coming in and that would maintain the wall temperature at t w but what we're interested here in this segment is looking at what happens and how the boundary layers grow so I said boundary layers we do have two and so I'll begin by drawing the hydrodynamic and so I'm assuming that we're dealing with a laminar boundary layer and the thickness we've defined that as being delta now with the thermal boundary layer it is going to begin at the point where we have heating starting and consequently its growth is going to be slightly different and we don't know exactly what it is yet it depends upon the fluid that we're looking at but the growth of that and the symbol for it is going to be delta t denoting thermal boundary layer so that is a case where we could have a plate where we do not have heating at the beginning but then it eventually starts now if x not was zero well then obviously the heating would be at the beginning but the boundary layer and the thermal boundary layer may not grow at the same rate and so they could grow at different rates so just be aware of that now quite often when we look at these types of flows we put the temperature profile into a an expression that would be non-dimensional and we've seen this quite often as we've studied heat transfer we just saw it in the Heisler charts when we were looking at transient heat transfer convective heat transfer but what we do we take t minus t wall divided by t free stream minus t wall and t here would be at a given spatial location in the boundary layer so that is one thing another one is our thermal boundary layer thickness and delta that is our hydrodynamic or velocity boundary layer thickness divided by delta t now we haven't seen this yet but we will it is can be expressed as being a relationship involving prantle number raised to some power n and we haven't determined what that power is yet and the prantle number that this is a number that is used quite often in heat transfer pr it is new over alpha being our thermal diffusivity so it is the kinematic viscosity over thermal diffusivity expanding these two so another way of expressing it is cp mu over k but essentially what the prantle number is quantifying is the kinematic viscosity over thermal diffusion so basically viscous diffusion over thermal diffusion and so depending upon the fluid that you're looking at the prantle number will vary quite significantly if you're looking at oils or liquid metals they they would be very very different prantle numbers we typically look at water and air but we could look at many of these other substances as well and and so if you're looking at liquid metal well their thermal diffusion would be very very high because high thermal conductivity consequently the prantle number would be very very low if you're looking at an oil the viscosity might be very high so the numerator in that term could be high and consequently you could have a very high prantle number but those are different prantle numbers that would depend upon the fluid that we're looking at so that is a bit of a brief introduction to some aspects about the thermal boundary layer and we'll look at it a little more closely when we start looking at some of the solutions that come out for the laminar boundary layer we'll look at the new salt number and the relationships the last thing we're going to do in this lecture we'll look at in the next segment is a relationship between heat transfer and and skin friction and there's an analogy that that works really quite nice and and it's one that enables us to get a lot of heat transfer data using fluid friction measurements and many of those have been collected so we'll be talking with that in the last segment