 In the last segment we looked at a correlation for determining the convective heat transfer coefficient across a two bundle and you recall we had a Reynolds number that was evaluated based on the diameter and a thing called Umax. So what we're going to do in this segment is figure out how to determine that maximum velocity. Now in determining Umax going through a two bundle it's going to depend upon the configuration that we have. So what we're going to do we're going to begin with the simpler and that is for the inline arrangement. So let's assume that we have this configuration here where we have three cylinders inline arrangement. The spacing between tubes is Sn in the direction normal to the flow. We have free stream coming from the left to the right and so the free stream velocity is at U infinity and if you go back and look at the flow visualization from the first segment of this lecture you'll notice that there is fairly significant jetting going on or accelerated flow between the tubes and we have this region of high velocity in the middle there. That is what we are going to refer to as being Umax for the inline configuration. So what we want to be able to do is this is where we have the lowest area, oops sorry about that, the lowest area and consequently that is where we will have the maximum velocity. So what we want to be able to do is evaluate the velocity at that plane, basically the point between the two cylinders and so the way that we're going to go about evaluating Umax for the inline configuration is we're going to rely on fluid mechanics and the continuity equation. So if you're called from your fluid mechanics courses continuity here we would look at the mass flow coming in and what I'm going to do is I am going to look at a section of flow that extends out in this direction and so on the inlet what we would have let me erase those that and that and on the inlet what we would have is we would have this here and these are all at U free stream. So coming in what we have is the density multiplied by the free stream velocity Sn is our spacing and we're going to assume unit width and that is going to be equal to the mass flux coming through this region here where we have the constriction. So again it is going to be the density of the fluid multiplied by this new maximum velocity. We do not know what that is yet and the size of that opening we can compute that from Sn as well as knowing the diameter of our cylinders and so what we have here on this side we have d over 2 that would be this distance and then here we have another d over 2 that would be that distance so I'm looking at half of the cylinder so what we can say is that we have Sn minus d over 2 minus d over 2 which just turns out to be Sn minus d we can now do some rearranging first of all density is going to cancel from the left and the right hand side and what I want to do I want to evaluate this that I can isolate Umax because that's what we are interested in so we then obtain an expression for Umax is equal to the free stream velocity multiplied by Sn divided by Sn minus d and that is the way that we can evaluate Umax when we have the inline arrangement the next thing that we want to do let's consider the case that is slightly more complicated where we have a staggered tube bundle okay so there is our staggered tube bundle what I want to do is I want to put the dimensions on here so you recall from the first segment in this lecture we talked about the direction parallel to the flow that was sp or the spacing in that direction and we also talked about spacing normal to the flow that was Sn and again like before we will have d as being the diameter of each of our cylinders and what we are now going to have again we have u infinity out here and if you go back and review the video for when we have the staggered tube bundle you'll find that the flow comes along and and it goes through a fairly severe change in direction and and so what is happening we have two locations where we could say that we have a fairly significant constriction in in space between the two bundles one of them would be what we just saw where we called u max that was for when we had the inline tube bundle but I'm going to call this u1 because at this point we don't know if that is the maximum velocity or not and then another location where we have a constriction is although in this particular drawing it may not look like a constriction but if you go back and watch the video you will see that between these two tubes here we have velocity going in this direction and I'm going to call that u2 and one of those two is the one that would give us the maximum velocity and what we're going to do we're going to again use geometry and continuity and come up with an expression for u1 and u2 and whichever one is larger that would be u max that we would use in our Reynolds number so what we're going to do we want to find which is greater u1 or u2 and we know from the analysis with the inline arrangement we've been able to determine what u1 is so that was the expression for u1 so that one we can put off to the side and now what we want to do we want to work on u2 and determine what it is so let's take a look at determining u2 and looking back at our diagram we notice that sn which we have here is the distance from this center to that center and so if we look at the distance from this tube center that distance there is going to be sn divided by 2 so we will use this in our drawing here and then in the direction of the flow or parallel to the flow direction we know that this is sp so what we have here we can construct a right angle triangle right here and with that we can write the following and i am going to solve for l so we now have an expression for l which is telling us the distance from center to center of our tubes that are in the tube bundle and just like before what we'll be able to do we know that this distance here is d over 2 and this is d over 2 as well going from the center line to the surface of the tube and so we will be able to correct for that we're now going to apply the continuity equation and what i am going to do is i am going to say u infinity was up here looking back at our larger picture what we're doing is we're zooming in on this section of the flow here so we're looking at that to there so we're making the assumption that a streamline coming along here it would hit a stagnation point on that cylinder so half of the flow needs to go this way and half is going to go up that way and that's how we're able to do this and apply it at sn over 2 so with that on the inlet side we will be looking at rho u infinity sn divided by 2 and on the right hand side that is going to be equal to rho u2 which we don't know yet l minus d because i'm minusing d over 2 i'll put it out explicitly so we're doing minus d over 2 minus d over 2 and so with that we can go through canceling out the density and rearranging our terms we end up with the following expression okay so we obtain that for u2 a little bit more complex than what we saw for u1 but what we can do we can come up with a method and essentially what we can say is the following so if this condition is met then what we can say is that u max equals u1 and if that condition is met therefore u max is equal to u2 and all i'm doing here is i'm taking the the term that is multiplying u infinity and comparing it so that term or looking back this term so i'm taking those and doing a comparison between them seeing which is greater and that enables us to determine u max and once you have u max once you have that then you can evaluate reynolds's number based on diameter at the max velocity which is required in order to determine the convective heat transfer coefficient within the two bundle so that is the way that we determine u max in the two bundle