 Alright, in this segment, what we are going to do, we are going to derive the equation for the log mean temperature difference and this will be kind of a long lecture or a long segment. The derivation is not that difficult but we will step through the process of coming up with this equation that is used quite often for heat exchanger analysis. So if we are dealing with the heat exchanger where we know our inlet and exit conditions, what we can do, we can write the heat transfer, we saw this equation in the last lecture and U is the overall heat transfer coefficient. A is the surface area corresponding to the way that U is defined, remember it could either be the inner or the outer area. And finally delta Tm, so delta Tm is an appropriate, now what does appropriate mean? It is an appropriate mean temperature difference and that is one that characterizes the temperature difference. Remember the two fluid streams are changing temperature as they flow through our exchanger and what we are going to do, we are going to model it as being a double pipe parallel flow arrangement heat exchanger and we will come up with a value for delta Tm and then if you want to apply it to different types of heat exchangers, there is a correction factor F that is applied and we will get to that later on in this lecture. But what we are going to begin with, we are going to begin with deriving delta Tm. Okay, so let's begin by drawing a schematic of a double pipe and what I am doing here with the double pipe heat exchanger, if you recall, I am zooming in on one of the walls. So this is what we are talking about in terms of the geometry, what it looks like for the double pipe. So we have fluid going through here and we have fluid going through here. Now what I am doing, I am zooming in on one of these wall sections and that is what we are looking at here. Okay, so that is the schematic that we are going to be using for the derivation and what you can see in the upper part, we have zoomed in on the interface wall between our two fluid streams and what is happening is the fluid, the cold fluid is coming along and its temperature is changing, we are assuming that it is going to be going up. The hot fluid is coming along and its temperature is also changing. We don't know what DTH is right yet, but we do know that the heat transfer is taking place across some differential element, we will call that DA and the amount of heat transfer is DQ. Now looking at the temperature diagram, remember this is a parallel flow double pipe heat exchanger and what we have here, we have our delta T and this is actually what we are going to be trying to solve for and again we can see we have heat exchange going from the hot fluid stream down to the cool fluid stream and in the process the hot fluid is changing and the cold fluid is changing, this is acting over some differential element DA, so consistency between physically what is going on as you zoom in on a section of the heat exchanger as well as the temperature diagram. So what we are now going to do, we are going to take this and we are going to write out a number of equations that we are going to work with that are modeling the processes that we are looking at for this particular schematic, so let's begin doing that and we will begin by looking at equations for the entire heat exchanger, so looking at the hot side and these are equations that we use over and over and over again when we are looking at heat transfer and heat exchangers, it is the mass flow rate of the hot fluid times the specific heat capacity of the hot fluid, now usually you have to average that because the fluid is going to change between the inlet and the exit, so we have that one and then looking at the cold side we can write out a similar equation, I made a little error here, I was wondering about that because the sine of Q would be wrong if I wrote it that way, so this should be TC2 minus TC1, so be careful of that when you are working with heat exchangers, because cold is going, TC2 is greater than TC1, so in order for Q to be positive we have to write it that way, now looking at the differential element, we can write out similar equations but for the differential amount of heat transfer and notice here for the hot side I introduced the minus sign because the hot temperature is going down whereas the cold is going up and so in terms of DT, the way that DT is defined we have to introduce that negative sign in order to make those consistent, now what I'm going to do, I'm going to label these equations, I'm going to call this equation 1 and this is going to be 2, this is 3 and this is 4 and we have a few more equations so let's keep writing those out, now we have Newton's law of cooling that we can use and this is looking at the differential amount of heat transfer using our overall heat transfer coefficient, so I'm going to do the overall times delta T times DA, the differential element area and in this expression delta T is the hot fluid temperature minus the cold fluid temperature and ultimately that's what we're trying to solve for, we're trying to solve for this delta T, so I'm going to call this equation 5 and we can also rewrite the differential of the delta T, so that's another equation that will enable us to further our analysis, so now what we're going to do, we have all of our equations, we're going to start subbing different ones into different equations and we will begin by subbing 3 and 4 into 6 and so we get that, now what we want to be able to do, we want to be able to replace this, so we're going to sub equation 5 and for DQ, so we get this, now what are we going to do with that, let's bring the delta T over to the left-hand side, so we get this equation, we have D delta T by delta T on the left, you can see that this is going to start turning into a logarithm when we integrate and that is what we're going to do in the next step, we're going to integrate this equation, so we get this equation here, now when we integrate the left-hand side, that's where we get the natural logarithm, hence log mean temperature difference, that's where that's going to come from, so we have this, now what we're going to want to do, we want to get rid of the mass flow rate on the left-hand side of the equation because we want to be able to express this in terms of temperatures and in order to do that, we're going to use equations 1 and 2, so we're going to go through and we're going to sub in for that, we have that now, usually we don't look at an equation like that, we want Q on the left-hand side, so let's rearrange and get Q on the left, so we get this big long expression here and this is our delta Tm, so that is the way to calculate the temperature between the two fluids and this is referred to as being the log mean temperature difference and sometimes you'll see it written with the acronym LMTD, now with this we assumed a parallel flow double-pipe heat exchanger, so if you want to apply this to other types of heat exchangers and we'll see later a factor, a correction factor comes in that you would get that out of figures, but what we're going to do in the next segment we're going to take a look at the temperatures, how these temperatures map to different types of configurations be it parallel or counter flow and then we'll solve some problems using the LMTD method and it works quite well with the exception of you don't know all four temperatures of your fluid streams and so we'll see later on in this lecture that this technique does break down, you have to do an integration which can be a bit of a pain and then there's another technique the effectiveness NTU that we'll be looking at, but for now we're going to play with LMTD for a little bit and explore where it can take us so that's what we're going to be doing in the next segments.