 In the last lecture, what we did is we took a look at the introduction to convection to begin with, and then we looked at the concept of the boundary layer, both laminar and turbulent. What we're going to do in this lecture, we're going to focus in a little bit more on the equations that enable us to determine the convective heat transfer, and we'll be looking at both the laminar as well as the turbulent boundary layer. In this segment, we're going to begin with the laminar boundary layer. What we're going to do, we're going to consider the laminar boundary layer with an isothermal plate. The entire plate is at one fixed temperature that is different from that of the free stream. In looking at this, this is one that there is a solution. It's not an analytic solution, it's a numerical solution. It's one, the velocity profile was solved by Blasius, I think it was in 1908, and if you recall, he was a student of Ludwig Prantl, and then later on, it was coupled with the energy equation to give the temperature profile, which provided then the convective heat transfer coefficient and the new salt numbers. That's what we're going to look at in this segment. So Blasius was able to solve, he did this numerically, hand integration. He used the momentum and the continuity equation, so momentum is Navier-Stokes equation that has been reduced for boundary layer flow, so it makes it a little bit easier. He was able to solve it using what we call a similarity solution, and so he converted the partial differential equation into a non-linear differential equation, ordinary differential equation, and then he integrated that by hand. But when you get that solution for the velocity profile, you can then take it with the energy equation and come up with a temperature profile above the wall. So we have our flow, we have it coming in this direction, and if you recall, we said that our temperature profile, if the wall was heated, might be something like this, and we're going from the temperature of the wall would be here, and then out here would be T infinity, and this is T of y. That's what we're after in order to get the convective heat transfer coefficient. I am not going to go through that solution procedure. You can look at many different textbooks and find it, be it a fluid mechanics book or some heat transfer books have it as well. But what we're going to do, we're going to move into taking this temperature profile and determining the convective heat transfer coefficient from it, which is what we are interested in in heat transfers. So if you recall for the boundary layer, we said in an earlier segment, that was actually in the last lecture, we did basically a coupling of Newton's law with Fourier's law, and we applied Fourier's law in the fluid right at the wall. And this approximation is made assuming that there is no slip or no fluid velocity right along the wall, which is true because there would be no velocity in a boundary layer right at the wall. And so we can use Fourier's law right at the wall, and then we equate that with Newton's law of cooling. And so we know the temperature profile from the Blasius solution coupled with the energy equation. We can take the derivative and everything else in this equation is known. It's an isothermal plate. We know the freeze stream. We know the thermal conductivity of the fluid above the plate. And so with that, we can determine the convective heat transfer coefficient H. And writing out the equation, it would be expressed this way. Okay, so that's the equation that we can use to determine the convective heat transfer coefficient in the laminar boundary layer. And so when you take the solution with Blasius's solution coupled with the energy equation, this gives the following, okay. So we get this expression here and notice what we're looking at is H of X. So what this is, this is the convective heat transfer coefficient at some position from the leading edge of the plate. So X would be from here. So what we're looking at would be H at X. That's the convective heat transfer coefficient at this little element. Let's say I should draw that being X. Let me redo that. So that would be location X and that's where we're evaluating H of X. So H of X is changing along the length of the plate. And we'll look at how to compute the average in a moment. But knowing this, we can then go ahead and with the definition of new salt number. Remember new salt number is the convective heat transfer coefficient times some characteristic length scale divided by the thermal conductivity of the fluid above the wall and the characteristic length scale in this case is going to be distanced from the leading edge. And so with that, we can write out that the new salt number for the laminar boundary layer is as follows. And notice here, I have Reynolds number X that denotes that the Reynolds number is evaluated at that X location. And so this equation here, this is valid for a laminar boundary layer. And there is restriction on the Prandtl number. So 0.6, that would be something like air up to 50. That's the range of the Prandtl number. So what we're going to do with this, this is giving us the new salt number at given location along our plate. What we're interested in, and typically in engineering calculations, we're usually interested in average heat transfer from a surface. So let's take a look at how to compute the average heat transfer from the plate. So if we're looking for that, we can evaluate it using an integral. And so plugging in the value that we have for the convective heat transfer coefficient. Okay, now this is a peculiarity or something that's interesting with the laminar boundary layer. We have an integral of X to the minus one-half. And if you look at the process of integrating that, we have X to the minus one-half. When we integrate that, we're going to get 2X to the one-half. And when we sub in the limits of integration, that is going to be 2L to the one-half, so the overall length. And with that, that essentially enables us to take the L to the one-half and pull it into this term here. And if you look at that, that is nothing else but the Reynolds number. And so it's kind of a convenient little thing that occurs. So let's take a look at the average or mean. And sometimes it is expressed as being H over bar. I did HM there, but sometimes you'll see it with H over bar. But with that, and the 2 goes to the front. So we obtain that for the average convective heat transfer coefficient over the plate of length L. And looking back at the expression that we had for the convective heat transfer coefficient itself, we have basically this. What that tells us is the average convective heat transfer coefficient for the laminar boundary layer is equal to 2 times the convective heat transfer coefficient at the end of the plate. And so when I say end of the plate, remember that we have a plate like this. X is starting here. And then this would be the end of the plate, and that would be X equals L. So that would be the length of the plate. But the average convective heat transfer coefficient is nothing more than 2 times the value of the convective heat transfer coefficient at the end of the plate. So that's kind of an interesting thing. It's kind of an easy thing to remember. Remember, it only applies for the laminar boundary layer with an isothermal plate. But we can then write an expression for the new salt number. And I'll put an over bar here to denote average. And finally, Reynolds number at L, that is nothing more than the Reynolds number evaluated with the length scale being the length of the plate. Mu is the viscosity of the fluid. Now, you'll notice in here in these equations, we have a number of things going on. We have the Prandtl number. We have the Reynolds number. We have our thermal conductivity. Within the Reynolds number, we know that we have the density and we have our viscosity. And within Prandtl number, we have C sub P. Again, we have viscosity and we have thermal conductivity. So the question arises at what temperature do we evaluate the properties within these equations? And this is always something with fluid mechanics, depending upon the equations. Sometimes they'll evaluate properties at different temperatures. But for these equations, what we do is we evaluate the properties at a temperature that we call the film temperature. So what are we talking about for the film temperature? What that is, it's nothing more than the wall temperature. Remember, we're talking about an isothermal wall plus the free stream temperature divided by two. It's just the average between those two temperatures. And this is going to have an impact on the following properties. And sometimes you'll denote it with K with a little f and that denotes film properties evaluated at the film temperature. So we have there for the thermal conductivity of the fluid over the plate, the viscosity with a little f denoting the f for film means that it's evaluated at the film temperature. And finally, for the density, same thing. And you can combine, oops, not all of them. You combine these two and you go to the kinematic viscosity instead of the dynamic. And so you might see this. And that is essentially just the ratio of mu over rho. Okay, so you will sometimes see that. Sometimes you'll also see r, e, x with a little f or r, e, l with a little f. The little f is denoting film temperature, meaning that the properties are evaluated at the film temperature. So that is how we are able to compute the heat transfer coefficient, as well as the new salt number for a flat plate isothermal flat plate with a laminar boundary layer.