 We're now going to take a look at the methods that you can use to determine the convective heat transfer coefficient on two bundles. So the equation that we're going to be looking at is one that enables us to determine the average heat transfer over the entire bundle. So this is an equation that is derived from empirical data, so from experiments, and there are a couple of things to note. First of all, this equation applies when you have 20 or more tubes that are in the parallel to flow direction. If you have fewer than those, if you have fewer than 20, then you have to apply a correction factor. Another one is that it applies over a range of prantle numbers going from 0.7 up to about 500, and the last one is the Reynolds number applies over a certain range, and in a moment we'll talk about what RED max refers to, but it's from 10 up to 2 times 10 to the 6 is the range of Reynolds numbers that this correlation applies for. Okay, so looking at this equation, we have C and N, and these we can obtain from tables, and the values are going to depend upon a number of different things. One, it's going to be on the configuration, whether or not we have staggered or an aligned tube bundle, which we saw in the previous segment. It will depend upon the spacing between the tubes in either the normal to the flow direction or parallel to the flow direction, and it will also depend upon the diameter of the tubes themselves. And in the equation, we can also see we have Reynolds number based on diameter at max, and what that is referring to is Reynolds number based on diameter evaluated at the maximum velocity, and I'll be going through that in the next segment. I won't go into details about what the maximum velocity is right now, but essentially we'll come up with a method by which we can determine the maximum velocity going through our tube bundles. So with this equation, there are a couple of other things. We'll note that here the Prandtl number is being evaluated at the wall temperature, that's what the subscript W denotes. The other properties in this equation are evaluated at the average between the inlet and the outlet to the tube bundle. Okay, so with that, let's take a look at what is going on within our tube bundle, because you might be wondering if you don't know the outlet temperature for example, how are you going to evaluate the properties? So let's take a look at geometrically what is going on when there is flow coming across our tube bundle. So what we're showing here is a tube bundle that is in the align configuration, and I'm just going to say there's some arbitrary number of rows in the streamwise direction. One thing that we do know here, we'll say that we know the velocity coming in, and we will call that U infinity. The other thing we're going to say is that the temperature of the flow coming in, the inlet flow to the tube bundle, is equal to T infinity or T free stream. The wall temperature, what we're going to do is we're going to make an assumption that the wall temperature of all of the tubes is the same temperature. Now that's a bit of an approximation, however, and sometimes you may have something where you have fluid going through a phase change, such as a condensing unit. And in that case, that would be a fair approximation to say that the fluid on the inside of the tube bundle was all at the same temperature. So that's the wall temperature. Another thing that we will say, and we saw this in the previous segment, the spacing between the tubes themselves, going from centerline to centerline, will be S subscript N, denoting normal direction to the flow. And the final thing is the diameter of each of our tubes is a little D. And then what we have on the back side of the tube bundle, where the flow is emerging, it will have gone through a change in temperature, either a higher or a lower temperature. And two more things that we will denote are capital N with the subscript N, that denotes the number of tubes in a row. And capital N is the total number of tubes in the bundle. Okay. So if you recall from the previous slide, what we said is that we want to evaluate the properties at this temperature, T i plus T outer divided by two. Well, usually you're going to know the temperature coming in. But quite often you're not going to know the temperature of the fluid leaving the tube bundle. And consequently, what we need to do, we need to go through an iterative process. And so solving problems with two bundles can be a little bit on the laborious side, because you have to go through a couple of iterations. But the first place to start, if you do not know the temperature of the fluid leaving, what I would recommend is you begin by evaluating properties at the inlet temperature, if you know that. And then what you can do is you can go back to the correlation equation, assuming you know how to determine umax, which we will be doing in the next segment. You use this equation here to come up with an expression for h bar. So that will be the average convective heat transfer coefficient within our two bundle. And then I'm going to give you an equation that will enable you to estimate the fluid temperature leaving the two bundle. Okay, so using this equation here, what we can do, we know most of the information on the right. Well, it's going to depend upon the problem. But what we're after, we're after this. And so we can determine the temperature of the fluid leaving. And then what we're going to do, we're going to use a thing called the log mean temperature difference. And we will see this when we look at heat exchangers later on in the course. But for now, I'll just give you the equation and we'll work with that. So that there is the log mean temperature difference. And what we then do is we use the LMTD, log mean temperature difference. Sometimes you'll see this expressed as LMTD, depending upon the book. But then what we do is we use that to determine the total heat transfer. And so the total heat transfer, I'm going to do heat transfer per unit length, is going to be using Newton's Law of Cooling, that is the average convective heat transfer coefficient for the tube bundle, all of the tubes, times the area. Now it's going to be area divided by the length, because we're doing heat transfer per unit length. That's going to be the total number of tubes times pi times d. So the circumference, then you'd normally multiply it by the length of the tube bundle itself. But we've brought that over to the left hand side of the equation to give us heat transfer per unit length, or watts per meter. And then we will have delta Tlm for the log mean temperature difference. And so with these equations here, the first one is to estimate T0. And once we have T0 or the outlet temperature, we can evaluate the LMTD. And then we can get the heat transfer. And so with this, what we can do is we can go through an iterative procedure. And we would then be able to go through a step by step. And once you determine a T0, at this point, you could go back and you could reevaluate the properties, again at the average of Ti plus T0 divided by 2. And if you go through a couple of iterations, it should converge relatively quickly. Other thing to notice in the denominator here, I'm evaluating these properties at the free stream temperature conditions, so the inlet temperature that we have coming into our tube bundle. And that, the C essentially is just mass flux. On the bottom here, that's m dot C sub p. It's not quite m dot because I don't have the length in there. And then on the top here, that is basically Newton's law of cooling. We have the area times H. So that's H bar times A. And that just relates to the way that this approximation is formulated. But anyways, those are the equations that you can use to solve for heat transfer in a tube bundle. Another thing to note is there are different correlations. I've shown you this one here. But there are different correlations that exist. And you're free to use whatever you can find for this. The one that we're looking at here does work, but there are restrictions. And also remember that if you have fewer than 20 tubes going in the flow direction, you have to apply a correction. And you would get that from a table that would be in your heat transfer textbook.