 okay at the end of the last lecture what we did is we looked at an example problem for a rod fan and we looked at the temperature distribution in the rod fan and what we found let's just go back and take a look at that we found that when we had the case of the convecting tip or the insulated tip the solutions were really quite similar and the solution for the insulated tip that this was case three was a lot simpler than the solution for the convecting tip which is actually the more accurate and approximating the physical reality better but it's much more complex solution so what we're going to do we're going to come up with a method here that enables us to correct for the length of the fin and it approximates the solution for case two but we use the results of case three which was just the hyperbolic tan and so functionally mathematically a lot simpler so what we're going to do now and we're not looking at the rod fin we're going to look at fins and correcting the length but let me begin by saying what we're really after when we're looking at fins okay so what we can say is that the main thing that we're after we want to know how much heat is being removed from the fin and it's either cooling application or your heating space using these fins and what we do know is that all of the heat that goes through the fin flows through the base and so by applying Fourier's law at the base we can then determine the total amount of heat leaving the fin so with that we can write the following and so that's one thing now another thing that we're going to do we are going to introduce a concept or a term called the fin efficiency and this makes it a little simpler when we're solving problems efficiency for engineers it's always eta with the subscript f for fin and that is going to equal the actual heat being withdrawn by a fin or that is going through the base of a fin divided by some hypothetical theoretical maximum and and so qf is going to be in the numerator and this maximum is going to be if the entire fin was at the base temperature so if that was the case then we would have haf that would be the area of the fin multiplied by t base minus t infinity so here we're making the approximation that the entire fin is at the base temperature and that is an idealization so realize that that is an idealization so that is the fin efficiency and that is something that we'll use you can also look it up in charts if you have more complex fins that are not the type that we've been able to solve analytically in the last few segments here now the other thing we're going to do is this approximation between case one and case two by correcting the length so let me talk about that now okay so what we found when we did the example problem with the rod fin is that the case two and case three solutions were very very similar uh case two is a more accurate solution because it's modeling convection from the tip uh but what we're going to be doing is we're going to correct for the length and we're going to use the simpler case three solution in order to approximate a case two solution and so the correction depends on if we're dealing with a rectangular or a round fin but if we have a rectangular fin the corrected length and we will denote that with capital L little c for corrected is going to be the length plus the thickness divided by two and if you have a round fin again we'll correct the length and that will be corrected by taking the diameter and dividing by four and and so that is the way that we will correct for the length and then we can use the simpler case three solution which if you recall was just a hyperbolic tan and with that we can introduce the fin efficiency so this would be the fin efficiency uh if we had the insulated tip solution and then the fin efficiency if we have the corrected length and if you recall we defined m i'll give us m squared here so that is the corrected convecting tip so essentially what we're doing here let's say we have a fin and what we're doing is we are saying uh we're not going to have convection at the tip but we're going to add this small length here and that is going to be whatever the corrected amount is going to be so here we would have l and then this is going to be lc so we're adding a slight increase oops a slight increase to the surface area here but then we're saying that we're going to have insulation on the tip so what we're doing to approximate for the convection we just add a little bit of an area there and there to increase the convective heat transfer and then we use the insulated tip solution by saying that we have insulation here which was simpler and with that we can then come up with an expression for the fin efficiency and then compute the amount of heat being withdrawn by the fin and if you have a more complex fin such as an annular fin uh as we saw in the video that would be if you have a fin that is something like this where you have here it's more complex because if you were to model this what you'll find is the area the cross-sectional area is going to change as you move further and further out and so consequently it's more complex but there would be tables that you can look at tables or charts and so that would be the way to deal with more complex fins or if you have even a fin perhaps with taper or anything like that where the area is changing and it does not satisfy the simplifying assumptions that we've used thus far in order to come up with these relationships that we're using so that is the fin efficiency and what we'll do we'll close out this lecture in the next segment by solving an example problem where we will be computing the heat being withdrawn from a base by a fin