 In this segment what we're going to do, we're going to take a look at an equation that enables us to determine the bulk temperature when we have a constant temperature boundary condition for pipe flow. So if you recall in the last segment what we did is we looked at a constant heat flux boundary condition. Well it turns out that this boundary condition is going to be a little bit more complex than what we saw for constant heat flux, which was just a linear variation in the bulk temperature with position. So going back to the differential equation that we have that enables us to investigate how the bulk temperature changes as a function of position in the pipe. And I'm going to write out the flow equation this time. Okay, so what we're doing here, we're saying that our wall temperature, the wall temperature is right here, we're saying that that is a constant. So how would you achieve that type of boundary condition? Your pipe might be going through an ice flow, ice and water, it might be in a condensing unit where you have fluid, for example, steam going through a phase change on the outside. So those are common places where you might have a constant temperature boundary condition on the outside of a pipe. But what we're going to do in order to work with this differential equation, we're going to introduce a new variable and that will enable it to become a homogeneous differential equation. So we're taking this term in brackets here and we're replacing it with this new term delta t. And so let's work on the left hand side and re-express the derivative term. And with that, we know that the temperature, the wall temperature is constant and therefore it cannot change with x. And we're left with this expression for the left. So basically this left side of the equation becomes minus d delta t by dx. So what I'm going to do, I'm going to rewrite our differential equation with this substitution and I'm going to rearrange this equation. Okay, so we now have this new expression and what I'm going to do is I'm going to integrate it from the inlet of a pipe to the exit. So remember we have inlet and we use i for that. And then outlet, we use o and x is going in this direction here. And the length of the pipe, let's see, we will use that here. x equals l, x equals zero. Okay, so let's integrate that expression and integrating. So upon integrating, we obtain this expression here and h refers to the average convective heat transfer coefficient in our pipe. Now, if it's fully developed flow, h will not change. But if we have an entrance region, then we may have to worry about entrance region effects. And what I am going to do, I want to recast this equation so that we can look at what this term looks like on its own. And so in order to do that, we take the exponential of both sides so we can rewrite it. And so this becomes an expression that helps us look at how the temperature is changing within the pipe. Now, it might look a little abstract at this point. So in order to assist what we're going to do, let's plot the temperature as a function of position. And remember we have our constant surface temperature. This is x equals zero. It goes up to x equals l. And on the inlet to the pipe, we had our bulk or our mean temperature on inlet. And looking at the relationship that we have here. So this is our relationship. We have the change in temperature related to an exponential function. And consequently, when you plot that, you end up with something that looks like this. And then this here would be our bulk temperature on outlet. But what we can see here is the temperature difference between the wall or surface temperature, constant surface wall, and the bulk temperature of the fluid in the pipe, delta t, this is a function of position x. And consequently, if we want to determine the difference between the two fluid streams in order to evaluate the total connective heat transfer coefficient, it's a little bit of a challenge because delta t is constantly changing. If you recall when we looked at the flat plate, it was quite simple because we had T wall minus T infinity. And T infinity was not changing. That was a constant here. However, we don't have that case because we have T wall minus T mean or T bulk. And that is changing as a function of position, which makes it a little confusing, a little challenging. So what we need to do, we need to find a way to be able to handle the fact that our temperature differential or temperature difference is a function of position. And so what we're now going to do, we're going to try to relate this back to the convective heat transfer, which is taking place between the wall and the fluid flowing through our pipe. So this was an expression that we put down on the very first segment of this lecture. And what we can do, we can rewrite this in terms of our delta t. And this was where delta t was defined as being T wall minus T mean or bulk temperature. So we can re-express it in that way. And what I'm now going to do, I'm going to solve for m dot c sub p from this equation. And we obtained that expression. And the other thing I'm going to do, so I'm going to park this now for a moment. And I'm going to go back to the expression to this expression here, the one with the natural logarithm. So let's pull that expression. And now I'm going to isolate for m dot c sub p in this equation. And that will enable me to compare the two equations that we have here. Okay. So here we have an expression for m dot c sub p. And here we have an expression for m dot c sub p. So what I'm going to do, I'm going to combine or unite equations one and two. And I'm going to, in the process, what I'm going to do, this is what we're interested in. We're interested in the convective heat transfer taking place between our fluid streams. So that's what I'm going to try to isolate. Okay. And then we obtain this expression here. Now as here is the surface area, the wetted surface area along the pipe wall, the inside. And this last term here, the one that has the natural logarithm in the denominator, we call this the log mean temperature difference. And sometimes you'll see it with the acronym LMTD. But it's basically a temperature difference that enables us to determine the convective heat transfer between the wall at a constant temperature and the fluid stream. So if we look back, essentially what LMTD is doing is it's taking this changing difference and coming up with an average temperature difference, which is the LMTD. But it's taking into account the fact that we have this nonlinear function here. It's not a linear function. If it was linear, we could just do the average difference from the inlet to the outlet. But we can't do that. It's a nonlinear function. So what LMTD does is just an expression that computes the difference, given the fact that we have a constant function and an exponential function. And that is what LMTD does. And with that, we're equipped with a way to be able to compute the convective heat transfer from a constant wall to a fluid that is flowing through a pipe. And so that is how we handle the constant surface boundary condition in pipe flow. And we will be looking at this equation again when we look at heat exchangers. And there's even a method called the LMTD method of analyzing the flow through heat exchangers and the amount of heat transfer. But we'll be getting that to later in the course. For now, we're just interested in pipe flow. So this is the expression that tells us if we're dealing with pipe flow with a constant wall surface temperature. And we've come up with an expression for the convective heat transfer. But it also gives us an indication as to how the bulk temperature within the pipe changes as a function opposition.