 Okay, in this segment what we're going to do, we're going to work an example problem involving the lumped capacitance method. So I'll begin the example problem by writing out the problem statement. Okay, so there's our problem statement. What we have, we have a can of soda pop, and let's say it's sitting in the refrigerator cooling, and then what we do is we take it out and we put it into a room at a given natural convective or small convective environment, 7.5 value for H, that would be kind of a low convective heat transfer environment. The room temperature is 19.5 degrees C and we're told H is 7.5, and what we want to do, we want to find how long, so the time for the temperature to go from 1 degrees C all the way up to 15 degrees C. So how long is it going to take for this soda pop just sitting in a room to go from 1 degrees C to 15? So you could assume you're taking it out of the refrigerator, you put it on a table, and you monitor how long it takes for that temperature to change. So let's begin by what we know for this problem. So the following is known. So we know that 355 milliliter can, we'll assume it's an aluminum can, and although that's not critically important for what we're doing here, because we're not going to consider that. And we're looking for the time T when temperature is equal to 15 degrees C. So let's begin by drawing out a schematic for what this looks like. And what I'm going to do is I'm going to assume that this can is placed on an insulating material at the base. Okay, so that is our can of soda pop. We take it out of the fridge, put it on a table with an insulating base. So I'll say insulation here, so we assume there's no heat transfer. So with that, another thing that we're going to need for the lump capacitance, we need to know the area, the surface area. So we're going to take the top area, pi D squared divided by 4, plus the perimeter area of the cylinder, so pi DH. And when we do that, and when we plug in the values, we get 2.82 times 10 to the minus 2 meters squared. Okay, so that is the different parameters that we need to solve this. What we're going to do, we're going to use the lump capacitance technique to analyze the problem. So let's go forward and do that. And writing out the equation for lump capacitance, and the expression for the thermal time constant. So what I'm going to do now, I'm going to plug in all of the values. We know everything and we can evaluate the thermal time constant. So let's go ahead and do that. So when we do that, we get the thermal time constant to be 7,016.3 seconds. So we can take that and plug it back up into this equation here. We know everything on the left-hand side, and we can solve for small t, and that would be how long it takes for the soda pop to reach 15 degrees C. So let's put in the values for that. And so when we solve that equation for time, we get very close to about 10,000 seconds. And so I'm going to divide that by 60, and then I'm going to convert it into hours. So that translates into 2 hours 45 minutes. So that's the amount of time that it's going to take this can of pop, starting at 1 degrees C, and a room at 19.5 degrees C to get up to 15 degrees C. So it takes quite a time when you look at that. Now with the lump capacitance techniques, that is our answer. There is the thing that we have to be careful with, the applicability. There are certain limitations to when we can use lump capacitance when we can't. So the lump capacitance technique is only valid if our bio number, and in heat transfer the bio number is given the symbol Bi, and what that is, that is the external convective heat transfer coefficient, some characteristic length scale, and we determine that by taking the volume divided by the area divided by the thermal conductivity. If that is less than 0.1, then we are okay with our approximation. If we look at the above example, the bio number there turns out to be 0.159. So really we were pushing things, we were above the 0.1, and consequently the approximation of saying that we could use the lump capacitance technique is not entirely valid. Another thing is in the case of the can that we just looked at, this is our can and it's sitting there. We have a liquid on the inside and so it's not really a solid, and consequently you can have fluid motion going on, be it natural convection or other things, and consequently that could also kind of stretch the results that we just used. But we're pretty close, but that is something that you have to watch for. Usually when you're dealing with solids, you're okay, but when you have a liquid in the system like we just did, you're kind of pushing things in order to be able to try to use the lump capacitance technique. Anyways, that is an example showing how we can use the lump capacitance technique. What we're going to do in the next segment, and that will conclude this lecture, is we're actually going to do this experiment. We're going to take a can and pop out of a fridge. We're going to do two different ways, one where we open it up, another one where we don't open it and look at it with the IR camera as well as we're going to stick a thermal couple in it, and we're going to see whether or not this solution actually makes sense. So that's what we're going to do in the next segment. We're doing an experiment involving lump capacitance technique.