 okay so we're taking a look at fins conductive convective systems and in the last segment what we did is we came up with an expression for the temperature distribution along a fin we had made a substitution of variables and we were expressing it in this theta of x but what we said is the temperature distribution could be expressed as a linear combination of this solution that we came up with and in order to solve for C1 and C2 we need boundary conditions that's what we're going to look at now and just to refresh your memory we had this as being the base temperature of the fin that we were looking at and we made a bit of a simplifying assumption we assumed that the cross sectional area of the fin did not change with x and so we were dealing with what we call a fin that has no taper or it has uniform area and the things that we were specifying we specified the perimeter so there would be some perimeter at a given location x so let's say that is x and what we're after here one of the things that we're after is to know the temperature at that location and then we're also going to try to come up with a heat flux and and the heat flux is going to be at the base because we want to know how much heat the fin is removing from the base so we need the boundary conditions what we're going to do we're going to assume three idealized cases and from that I won't go through all the math but I'll give you the results of each of the cases and the results being the solution basically you know determining what the boundary conditions would be and then determining what the constants of integration would be the C1 and C2 so we're going to take a look at three cases case one that would be the case of a very long fin so if you imagine we have a very very long fin not Fing fin and what will happen as the fin gets very very long eventually the temperature of the fin is going to become the same as the free stream temperature so we can say temperature at this L L being some very long distance eventually we'll get to the free stream temperature and with that with our variable theta at that L is then going to be equal to zero so that's boundary condition scenario or case one case two is a more realistic one because you'd never have a fin that long that it gets to the ambient temperature while you could but usually you wouldn't case two is finite length and we're going to lose heat from the tip via convection so this is the most realistic scenario because that is what actually happens and if you imagine here we have our fin and let's assume that it's round in cross section so AC is the cross section as we come out along the length what we're assuming is that we have q coming in here and then that's going to go into q convective heat transfer so let's try to express that and giving us a mathematical representation so we have Newton's law of cooling on the end HAC because the area is not changing as we go along that length of the fin and it will be the temperature at the end of the fin minus the fluid temperature that the fin is exposed to and then on the right hand side we'll put Fourier's law and we'll apply Fourier's law right at the end of the fin tip so that is going to be dT dx evaluated at x equals l so essentially it's equating the slope because what's going to happen here is that cross sectional area is going to cancel out and what we end up with is the following at x equals 0 okay so that's finite length we lose via convection at the tip and case three is another one that's a little bit of an idealization that would be the case where instead of having free convection at the tip you put insulation there and so there is no heat transfer from the tip and if the tip is insulated we know that when we look at the boundary conditions we looked at this when we came up with the heat diffusion equation if we have the case of insulation that means that through Fourier's law the slope of the temperature profile at that point is equal to zero and therefore writing that in terms of our variable theta we get that so those are the three different cases that we have case one case two and case three so what you can do is you can take these and plug them into the solution that we had from the fin equation and there is one other boundary condition that I forgot to mention before we go to determining C1 and C2 now let me mention the other boundary condition the other boundary condition is what is happening at the base of the fin and if you recall from our schematic we said the temperature at x equals zero is equal to Tb for the base temperature so we can write out a theta the base at x equals zero is Tb minus T infinity or the free stream fluid temperature so with that what I'm now going to do I'm not going to go through the math but I'm going to give you the results in a table for all three cases that of an infinitely long fin that of convection from the tip and that with an insulated tip so let me write out all of those and when I do that I'm going to give you two values one is going to be the temperature distribution and the other is going to be the fin heat transfer rate which will be evaluated at the base it basically tells us Q leaving the base and that gives us amount of heat being removed from the surface okay so those are the results that you get when you put in the boundary conditions and you solve for C1 and C2 and in here we have a lot of hyperbolic signs cosines and tans but recall theta was T minus T infinity we said m squared was HP divided by k ac so the cross-sectional area convective heat transfer coefficient perimeter and thermal conductivity of the fin theta B was equal to theta at zero which is then the temperature of the base minus the free stream temperature and the last thing I haven't mentioned it yet but you'll see in the table we have this m term that appears in the heat flux coming out of the base m is defined in the following manner theta so those are the different terms this here on the left gives us the temperature profile in the fin and this gives us the heat transfer that the fin is removing from the surface and when you look at these if you recall the long fin case that wasn't really a physically realistic application the convecting tip was the one that was very accurate but when you look at the mathematical expression it's rather complex although in a computer it's not a big deal and then finally case three the insulating tip one that is a rather simple solution and so what we'll be doing in the next segment is comparing these three different solutions and seeing how well they compare for for different types of applications for a given problem so that is the fin equation and solutions to three different cases and like I said in the next segment we'll be plugging some numbers into these and taking a look at what the temperature profile looks like