 Okay, in this segment we're going to look at a couple more expressions for the convective heat transfer coefficient, our new salt number for laminar boundary layer flows over flat plates. And in the last segment we looked at the case of an isothermal plate. We're now going to begin by looking at a plate with constant heat flux. Okay, so the expression, the new salt number for a plate with constant heat flux, is as follows. And that would enable us to determine the new salt number at a particular location along this plate. So recall that this would be our plate. X is coming from the leading edge. And so we're looking at the situation at some locations. So we're assuming that we have some U infinity upstream of the flat plate. And then we have a boundary layer that's going to grow above the plate as the flow moves along. So we have the boundary layer growth. Now what we have here is a case of constant heat flux. So we're pumping in heat at a constant watts per square meter. And that would be Qw. So maybe you have an electric resistance heater or something like that that is along the wall and pumping in heat. Now the thing about this is the temperature will not be uniform like it was in the previous case where we were looking at an isothermal plate temperature. So here the plate temperature is going to vary as a function of position. And from the new salt number, what we can do using Newton's law of cooling, we can write the following. So we have that expression there. Now what I'm going to do is I am going to use our expression up here for new salt number as well as the convective heat transfer coefficient to rewrite this and express the temperature difference between the wall and the free stream as a function of X. Okay, so all I've done there is I've plugged in the value that we had for the new salt number above. Now what we're going to do, we're going to try to compute the average wall temperature. So the average wall temperature would be the average difference between the wall and the free stream because what I should have done is said out here that is the free stream temperature. And that is far away from the plate where it has not been impacted by the plate. And in this case, T wall is a function of X. So let's look at the average wall temperature. And we're going to be doing a little bit of mathematical manipulation here in order to get this expression. Okay, so we have that integral. And what I've done here is I pulled X to the one half out of the Reynolds number term. So we have X over X to the one half. And it will kind of become apparent why I have done that in a moment. So let me integrate that and rewrite the expression. Okay, so after integrating what we have, we get L to the three halves in the numerator. And what I've done is I've taken the L here and I've split it into two L's, L to the one half and L to the one half. And the reason why I've done that, I want to pull this L in here to form the Reynolds number. So let's rewrite that. So I get that expression. And what I can do, looking back at the expression that we have for the Nusselt number, so let's go back here. What I'm going to do is I'm going to use this expression here. And I'm going to pull it into the equation that we have here, enabling us to rewrite it. And in the process, what I'm going to do essentially is isolating for this H value here. And you can go through the algebra as an exercise. But what you get out of this is the following. And so with that, what we can say is that the heat flux in the plate is the following. So the average temperature is related to the heat flux multiplied or divided I should say by 1.5 times H at x equals L. And so that is the expression that we get from doing the calculation for a laminar boundary layer flat plate constant heat flux condition. So that's another expression that you might use if you have constant heat flux. There is another expression that if you recall when we looked at the solution with Blasius and the energy equation, it was limited over a Prandtl number from 0.6 to I think it was 50. There is another expression that exists that is based on empirical data. So based on experiments but gives us a wider range of Prandtl numbers. So let's take a look at that now. And so again, this is still laminar flow. But it is wide range Prandtl number or wide range of Prandtl number. And it is empirical. So what that means, I'll talk about that in the next segment, but that means that they've done experiments and they would have used the original functional fit from the solution that Blasius came up with and the coupling Blasius with the energy equation. But the Prandtl number, the Prandtl number will vary depending upon the fluid that we are looking at. So we can have Prandtl numbers that are very low. And things like liquid metals would have very low Prandtl numbers. And typically in this course, we're going to be dealing with things like water and air. If we look at the Prandtl number for air, it's around 0.7. Prandtl number for water has about 7.0. I do realize it's going to change as a function of temperature, but in the ballpark, it's something around there. And then if you have oil, oil is going to have a much larger Prandtl number. And it would be up on the order of maybe 2,000. And looking at the definition of Prandtl number, remember we talked about this in an earlier lecture. It is, if we write out our kinematic viscosity divided by the thermal diffusivity, it's essentially quantifying viscous diffusion over thermal diffusion. And so that is why, if we look at a liquid metal, liquid metals, they are going to be highly conductive and consequently K, which is in the denominator, is going to be really, really large. And consequently our Prandtl number is very, very small. Oil on the other hand, the viscosity for oils tends to be quite high. And consequently we can have very high Prandtl numbers. So that gives you a little bit of a feel for how the Prandtl numbers vary depending upon what liquid we're looking at. But with this expression, it's a wider range and it's for laminar bounduaries. I'll write it out. Okay, so that's an expression that you can use for the new salt number. Now what we're finding, even here, we're relying on an empirical fit. So we're relying on experimental data to be able to give us a relationship for a laminar flow. And we haven't even started to talk about turbulent flow yet, which we will be in the next segment. So what we're starting to see is that the analysis capabilities for determining the connected heat transfer coefficient are really quite limited. And so with that, what we'll be doing as we move into more complex shapes, we're really going to be moving into this area where we are doing empirical functional fits of data. Now we use the theory in order to give us an idea as to what the functional relationship would look like, like we have here. But then there will be corrections that result due to the experimental data that is being collected. So that's where we're going in the next couple of segments. We're continuing looking at convective heat transfer over flat plates, but we're moving into the cases where we would have turbulent boundary layers. And that's what we'll be looking at in the next segment.