 Okay, so when we're dealing with convective heat transfer within pipe flow, we have to come up with a way to express the temperature of the pipe flow. And the temperature depends upon the radial location. So there really is no one fixed temperature. And so that brings up this concept of a mean or a bulk temperature. Now, if you recall Fourier's law when we looked at the flat plate, we had HAT wall minus T infinity. Well, for pipe flow, we can have a wall temperature. That's quite straightforward. But the thing that we don't have here is we don't really have a free stream temperature like we did with the flat plate boundary layer flow, where we had the boundary layer growing. But out here, we had U infinity, T infinity. And so we could define. And then that would be our wall temperature. For pipe flow, what we have we've seen is we're going to have different temperature profiles dependent upon whether or not it's a constant wall temperature, or if it's a constant heat flux boundary condition. And consequently, we need an expression so that this is supposed to be temperature that we're looking at here and there. Now, so we need a way to be able to define some form of temperature that we can use in our calculations. And that brings up this concept of a mean or bulk temperature. Okay, so with that, what we're going to do, we're going to consider pipe flow. And we're going to consider the total amount of thermal energy in transport. And so we will refer to that as being energy in transport. So this is going to be in units of joules per second. And the way that we're going to compute that is we're going to integrate across the area of the pipe. And what we're going to integrate is row U. And there will be an area that we have in there. And that area will be a differential area as we integrate around it. And that will give us a mass flux, row U area is a mass flux. And we're going to multiply that by the specific heat. So m dot Cp times t, t at a given location. And then this brings in our area to close out the integral. So looking at the units here, we have kilograms per meter cubed. We have meters per second. We have joules or kilojoules per kilogram kelvin. We have kelvin and then meters squared. And so with that, we have meters squared with a meter. So that and that castles with a meter cubed. Kilograms goes with kilograms. Kelvin goes with Kelvin. And so looking at the units, we're left with units of joules per second, which is what it should be because we're looking at energy per unit time being transported through our fluid. Now what we're going to do, we're going to equate this with some bulk temperature. Sometimes they call this a mixing cup temperature because if you were to put a cup at the end of the pipe and let the flow come out into that cup and then mix it up, that would be your bulk temperature. But we're going to equate it with a bulk temperature. And again, so this is thermal energy in transport. And this time what we're going to do, we're going to say that it's equal to m dot. So the total mass flux in our pipe, specific heat of the fluid times Tb. Tb is what we're trying to determine. We don't know what that is. That is our bulk temperature. So I'm going to equate those two. So when we equate them, and what I will do, I will, I'll write it out first, make it more explicit. Okay, so that's the integral over our pipe. And this DAC, this is going to be a 2 pi r dr. So that's what we will be substituting. But what I want to do, this is what we're interested in. So that's what we're going to try to solve for. So let's solve for that on the left hand side. And then the right hand side becomes the following. Now what I'm going to do in the denominator, I am going to put in a value for m dot Cp. And I'll pull the row and the Cp out of the integral in the numerator. And so that is our m dot term in the denominator. Notice here, we have a 2 pi and a 2 pi, those cancel. We have a row and a row, a Cp and a Cp. So all those cancel. And what we end up with is, you get that in the numerator. And then in the denominator, we have that term. Now if you look back a couple of segments, we had, when we were evaluating, let's see, I have it here, we were evaluating the mean velocity in the pipe. And when we were doing that for the mean velocity, this is what we said was the mean velocity. And when we look at this, we can see that this integral here is exactly what we have in the denominator. And so consequently, what we can do is we can re-express the denominator as um r naught squared over 2. And so making that substitution, we can then rewrite our bulk temperature in the following manner. And so this becomes an expression for the bulk temperature. Sometimes also called the mixing cup temperature. But essentially, this is a temperature that we can then use once we've evaluated it. We can then use it in Newton's law of cooling. Because recall, at the beginning, we said that we did not have a free stream temperature. Well, this is kind of the equivalent of a free stream temperature. But with that, we would then have heat flux. And we would write it this way, T s, that is going to be the wall temperature. And T b, that is our bulk temperature. So that is the mean or bulk temperature and how we would then use it in Newton's law of cooling. What we'll be doing in the next segment is starting to take a look at new salt numbers for pipe flow. And those then become the way by which we can estimate the convective heat transfer coefficient. And then enabling us to determine heat flux for pipe flow.