 Okay, so what we've been looking at is this alternative method. We've come up with the equations for the heat flux in one-dimensional conduction that is steady state, the thermal conductivity being constant, not a function of temperature, and no internal generation. We looked at a plain wall, a cylinder, and a sphere. What we're going to do now is we're going to take those results, we're going to put them together into this concept called thermal resistances. And this part of the motive for this is the fact that heat transfer in a way is very similar to the flow of electricity through a conductor. And if you recall, when we looked at heat transfer mechanisms, conductive heat transfer mechanisms in solids, we said that the conduction was through two ways. It was through lattice vibration and through electron motion. And consequently, it's natural to expect that there's some sort of relationship or analogy between electrical conduction and heat transfer. So let me begin by making a comment about that. So if you look in the tables of thermal conductivity in the back of any heat transfer book, you'll find things like copper, aluminum, aluminum is not as good, but it's still pretty good. But copper, gold, platinum, silver, those are all really good heat conductors. And they're also very good electrical conductors. And consequently, we have this relationship here. And the reason is because conduction in solids is via both lattice vibration and electron motion. And the electron motion is the one that enables us to have a good electrical conductor. So this is what brings up the idea of a thermal resistance. So this brings up the idea of a thermal resistance. And we can use this when we have all of the restrictions that we had when we were using that alternative method. So you probably have them memorized by now, steady state, one dimensional, no heat generation. And the last one was that the thermal conductivity was a constant. So it's not a function of temperature. So with that, what I'm going to do, I'm going to summarize what we found when we looked at the plane wall, the cylinder and the sphere. And then we're going to look functionally for commonalities here. So let's look at all three of them. All right. So we have that for the three different geometries that we looked at. And what I'm going to do, I'm going to rearrange each of these. And I'm going to rearrange them with the temperature differential on the left hand side of the equation and q in the numerator on the right and the other remaining items in the denominator on the right. So let's rewrite those. Okay. So we get those three different expressions. And when we look at these, what we can do, and this borrows from electrical conductors and Ohm's law, these relationships, this looks a lot like v1 minus v2 equals i times r, if you recall from a course, if you've taken a course in electrical circuits. And what we have here, this is the basis of our analogy. And if we look at electrical conductor or resistor, I should say, we know that we have voltage one and then we have a voltage drop across our resistor or resistive element. We have current flowing in and we have current and continuity. So the current continues flowing and flows out. But there is a voltage drop. Well, temperature drop is similar to the voltage drop. The current is similar to the heat flux and resistance is similar to the thing that is in the denominator on the right hand side. So with this electrical resistance or thermal resistance analogy, we can write t1 minus t2 is similar to v1 minus v2. And heat flux is similar to current, the flow of current. And so with that, what we can do is we can rewrite our temperature difference and our heat flux and we introduce this new term here, this RT. We haven't seen this before, but what RT is, is everything that was in the denominator on the right hand side of our equation. So coming back here, we had this, we had this, and we had this one here. So that's what our t comes out to be. So what we're going to do now, let's take a look at all of these. I will go to a new slide just that we have enough space. So looking at a plane wall, we have a thermal resistance, and that's what we're going to call this, and that is equal to the thickness of the wall divided by the area multiplied by the thermal conductivity. And then repeating that for a cylinder and a sphere. Okay, so that is for a plane wall, a cylinder, and a sphere. And you'll see we have RT, and that stands for thermal resistance, and then we show conduction. Now, we haven't seen anything about convection, but it turns out that convection, we can express it in a very similar manner as well. And if we look at Newton's law of cooling, again we have a temperature difference or temperature differential. And here, we have a thermal resistance for convection. I put CO and V for convection. And that then is equal to 1 over HA. So that would be the thermal resistance for convection. So there we go, we have thermal resistances for conduction in a plane wall, a cylinder, a sphere, as well as for convection. And with this, what we can do, we can put these together. And first of all, we can say that high thermal resistance, if we have high thermal resistance, that would be analogous to being a good insulator. So RT equals good insulator. And we can do similar things as we do in electrical circuits. We can put the thermal resistances in series, and we'll take a look at an example problem in the next segment that demonstrates this. So what you do is you sum up the thermal resistances for all of the individual components. So you'd have one thermal resistance, another one, another one. And that might be a case where you have heat conduction through a wall that might look like that. And you have one material here, you have another material there. It would have a different thermal resistance and then a third material here with a third thermal resistance. So that's an example. And we also can do parallel, parallel circuit analysis. And for here, very much similar to what we see with electrical circuits, it's 1 over the sum of 1 over all of the individual thermal resistance components. And so that would be the case where if you have a parallel circuit and you have another thermal resistance, that would be a parallel circuit. And what might that look like in terms of an object, that would be, let me erase that. So let's say you have an object, you have a plain wall, but then half of it is one material. And the other half is a different material. And your heat flux is going in this direction. And same up here, the heat flux is going in that direction. So that is thermal resistance. And what we'll do in the next segment, we're going to take a look at applying this to some example problems. And then we'll talk about a couple of other things with thermal resistances, our values and contact resistance.