 In the previous segment, we looked at the hydrodynamic entrance region for pipe flow. And so what we're going to do now, we're going to take a look at the thermal effects. We're going to look at the thermal entrance region in pipe flow. And so just like before, what I'm going to do is I'm going to sketch out a section of pipe. And we can consider that we have a fluid flowing into that pipe. And we're going to begin with a uniform temperature profile. So let's assume that we have the fluid coming in. And this would be at location x equals zero. Now what is going to happen is just like before, depending upon the wall conditions. So remember here, we can either have two different boundary conditions. One we can have a constant surface temperature. Or we can have a constant heat flux on the surface. And so those would be two different types of boundary conditions we could have upon our wall. But with that, we have either heating or cooling taking place. And consequently, the temperature along the wall will begin to change. And as the fluid flows, that temperature will move in. And consequently, we will eventually get to a point where our thermal boundary layer has moved from the walls in towards the metal, just like we saw for the flat plate. So if we go a little bit further into the pipe, our temperature profile may look something like this. And so we have this heating effect. This is the case where we would have heating on the wall, is what I'm sketching here. And that should be symmetric. So let me clean that up. So we would have a temperature profile that may look something like this. And you'll notice that in the center, we still have the same original temperature that we did on the inlet. And that's because our thermal boundary layer is growing. And so if I draw a dotted line for the thermal boundary layer, eventually what's going to happen in the pipe flow is that boundary layer will close. And then that is when we get to the case where the effects of the wall have been felt throughout the entire pipe flow. And when that occurs, we define that as being at the location where we're at what we would call fully developed flow with the thermal boundary layer. Now the nature of the profile that we will have is going to depend upon the boundary conditions around the pipe, be it a constant temperature boundary condition or a constant heat flux. So I'm going to begin by sketching the case of a constant temperature boundary condition. So this would be one where the wall surface temperature is different from that of the fluid. And the temperature profile for a constant temperature boundary condition will look something like this. So that would be a constant temperature boundary condition. And then the case of constant heat flux, a constant heat flux boundary condition would look something like that for the temperature profile. Now the thing about the thermal boundary layer and the temperature profile, if we continue to add energy, be it either through a constant temperature boundary condition or a constant heat flux, this temperature profile will continue to change as we continue to move in the x direction. So unlike fully developed velocity profiles where the velocity profile doesn't change, this thermal energy or the temperature profile will continue to change. And so that's a little bit of a difference. But when we say fully developed, we're referring to the fact that the uniform temperature no longer exists and the effects of the wall have been felt throughout the entire pipe. So across all the radii or radius. So with that, what we can do just like before, we defined the length for before we got to the fully developed flow for the hydrodynamic boundary layer. We can do the same thing for the thermal boundary layer. And so beginning if we have a laminar flow, the growth of our thermal boundary layer non-dimensionalizing by pipe diameter, it's approximately 0.05 Reynolds number based on diameter, which is what we saw earlier for the hydrodynamic. But then we multiply it by the Prandtl number. And similarly, if we have turbulent pipe flow, and so fully developed thermal, that's what the T stands for, non-dimensionalized by pipe diameters for turbulent approximately 10 pipe diameters. So it takes about 10 pipe diameters before you get to the case where the discontinuity or whatever your surface boundary condition has been felt across the entire pipe. And that's if you have a turbulent flow field. So that is the thermal entrance region. What we'll be doing in the next segment, we'll be looking at a way to be able to compute the bulk temperature or the mean temperature. Remember we looked at the mean velocity in the previous segment when we're looking at the the hydrodynamic growth or entrance region to the point where we had fully developed flow. What we'll be doing is we'll be looking at an equivalent thing to the mean velocity in a pipe flow. We'll be looking at the bulk temperature and within pipe flow.