 In this lecture we're going to take a look at the topic of numerical methods. And numerical methods represents one method by which we can solve heat transfer problems. So when we're solving heat transfer problems, we basically have three different techniques or methods that we can use and numerical methods, which applies to topics other than heat transfer. But we'll be talking about applied to heat transfer in this lecture. Numerical methods is one of those methods, but the other methods, we've looked at some of them thus far. So if you look back at a couple of the lectures that we've been going through looking at conduction, we looked at Fourier's law, and that led to what we referred to as being the alternative method. And from that, we then came up with expressions for the thermal resistance. And so we looked at that method of being able to solve for heat flux as well as temperature at interfaces within certain systems. And we also looked at the heat diffusion equation, that was a complex PDE. And we looked at one solution for that, that was heat distribution in a plate with different boundary conditions. And we saw that if the boundary conditions were more lifelike, the solution turned out to be an infinite series. So fairly complex solutions, and you can get them for some certain basic shape. So when we look at the analytical methods that we have studied thus far, we can say that they apply to relatively simple geometries with fairly simple boundary conditions. So those were the shapes that we looked at, thermal resistances. We had them for one dimensional conduction. I call that the slab. We had it for cylindrical conditions as well as for a spherical. So that was what we were able to get to with analytical methods. We weren't able to get very far. We were able to solve some problems. Another method of solving heat transfer problems involves just doing physical experiments. So setting up a system and setting the boundary conditions correctly and putting thermocouples throughout the object, measuring the temperature distribution. And so through experimental methods, we're able to obtain certain physical quantities, mainly temperature and perhaps heat flux. And in some cases, you're forced to only be able to use experiments, especially if you're dealing with convective boundary conditions. But we'll get to that later on in the course. But one comment that we can make about experiments is that they're fairly expensive to set up and conduct. So we have analytical methods. We have experimental methods. The last technique that we can use for solving heat transfer problems are the topic of this lecture, numerical methods. And so that's what we're going to spend a little bit of time looking at. And when you look at numerical methods applied to heat transfer, there are two main methods that people use. There is the finite difference method. And there is the finite element method. Now we're going to be looking at the finite difference method in the next lecture or two. And that will be the technique that I will be presenting. Finite element method is a slightly different technique, mainly used in solid mechanics, strength materials, things like that, although not exclusively. But I'll be talking mostly about the finite difference technique or method. So the nice thing about numerical methods applied to heat transfer is you can handle complicated geometry. And so whatever you can put into the computer and grid, which we'll talk about, you can solve. And they're relatively inexpensive as computational power has come down and down, they become less and less expensive to run these models. And so they all run quicker on computers, even small laptops today. You can do computational heat transfer fairly easily. Now there is a bit of a catch, and that's what I'll talk about in this next little bit. And so the catch or the downside is that the boundary conditions are sometimes difficult to estimate, especially if you have a convective boundary condition, either free or forced convection, which we'll be looking at later on in the course, or with radiation. If you remember the lecture segment that we looked at Lesley's cube, and we had a cube that had different surface finishes, and we looked at it with the IR camera, we found that the copper surface that was slightly polished had a very, very different emissivity from the other surfaces. And so you need to know that emissivity in order to get accurate results. And consequently, the boundary conditions are the place that they sometimes challenge engineers when they're using numerical methods to solve heat transfer problems. So that is an overview of the three different techniques. We have analytical methods that we've looked at, experimental methods and numerical methods. And what we're now going to do, we're going to spend the next couple of lectures looking at the finite difference method applied to heat transfer problems. OK, so that is where we are going with numerical methods applied to heat transfer. We'll be looking at two-dimensional steady conduction problems. So we looked at the flat plate earlier, the PDE, and we use separation of variables technique to solve. With numerical methods, the method that we'll be developing, that problem is quite easy to solve. And so we'll be taking a look at that as we go through the next couple of lectures.