 In the last segment we took a look at an energy balance for pipe flow and what we came up with was we came to a conclusion that we weren't sure how to be able to calculate the bulk temperature as the fluid was flowing through the pipe, be it a constant surface boundary condition or a constant heat flux. So what we're going to do now, we're going to explore in a little more detail the bulk temperature with position and what we'll do, we'll derive a differential equation that enables us to explore the two different boundary conditions, either constant surface temperature or constant heat flux. So let's begin by pulling a couple of equations from our energy balance segment that we just looked at. So this is one of the equations that we came up with from our energy balance. And if we consider a section of pipe and what we're going to do, let's consider a little differential element in the middle of that pipe, which is what we did when we came up with perform the energy balance. dx is the length, so x is going in this direction as we go along the length of the pipe. And we have some form of heat transfer taking place. We'll quantify that as qs, double prime, giving us watts per square meter. Now what we can write is knowing this is one equation that we have, but we can also do a balance based on the amount of convective heat transfer taking place. So what it'll do, let me go to the next slide. So dq convection can also be expressed as our surface heat flux multiplied by p, the perimeter of the pipe times dx, that would be the area. And then that can be equated to our m dot cp dt, the bulk or the mean temperature change with distance. And so what we're going to do here, we're going to play with this equation beginning by rearranging. So what I've done here is I've taken these two and I've done a little bit of rearranging. And we end up with a differential equation that gives us an expression for how the bulk or mean temperature is changing as a function of position within the pipe. What we're going to do, we're going to take this and let's take a look again at Fourier's law or the way that we quantify convective heat transfer in the pipe. And so we have this expression here for qs double prime. And that would be the heat transfer, the convective heat transfer, due to convection on the wall. There we have our wall temperature and then the bulk temperature. And what I'm going to do is I am going to use this expression and I am going to bring it up into here for the qs double prime term. So let's go through that process. So this provides us with a differential equation where we have the slope or the gradient of the bulk temperature with position on the left. And on the right we have an expression that contains the bulk or the mean temperature. And also embedded is the convective heat transfer coefficient. Now if we're outside of the entry region of the pipe, so we're in the region that we call fully developed flow, the convective heat transfer coefficient in pipe flow does not change with position. So that is a constant outside of the entry region. And these other two terms, if we can assume that the specific heat is remaining a constant, mass flux is going to be a constant. And if it's a constant area duct or pipe, perimeter does not change. And so those are constants as well. But that gives us a relationship. And what we are going to do, we're going to explore this equation. And we're going to explore the equation to examine how tm varies with x. And we'll do it with two boundary conditions. One is a constant temperature boundary condition. The other one is constant heat flux. And so that's what we're going to be doing in the next couple of segments is looking at this equation and seeing what it can tell us about the bulk temperature for those two different boundary conditions.