 Alright, so we're now on to the last boundary condition that we are going to consider by applying the finite difference technique to the heat diffusion equation, and it's the most complex one, but it is radiation and convection. Okay, so just like for the other boundary conditions, we're going to begin by drawing out a schematic of an internal node, or a node on the surface, and from that, that is what we're going to use to come up with the equation for the boundary condition. So let's begin by doing that, and just like before, we are going to prescribe a control surface, and we will use that in coming up with our formulation for the boundary condition, and we have our convective environment, and for this one, we have radiation. So let's draw a radiation. I'll give it a big red symbol. So we have radiative heat transfer coming in, and in order to formulate that, I need to specify the emissivity and the surrounding temperature, and that leads to radiative heat transfer coming into our surface, and we will be looking at it acting on the external surface at node location m, n. And so with that, we can now go ahead and we're going to write an energy balance on the control surface, and just like before, what we're going to have is heat in through convective heat transfer. We have heat in through radiation heat transfer, and then we're also going to assume that we have conduction coming into our little control volume here across the control surface. So we have heat in through convection, radiation, conduction, plus the rate of energy generation within our control volume, and all of that has to equal zero. So what I'm going to do now, I'm going to write out the equation, and from that we're going to try to solve for what's happening on the boundary at location m, n. And just like before, what we're doing, we're assuming a unit depth into the page. So I could write here unit depth, and that would equal one. And so we begin with Newton's law of cooling. We then have our radiation term, the Stefan-Boltzmann constant. And with this, we're implicitly assuming that our temperatures are all in Kelvin. And then we have Fourier's law handling the conduction coming in. And then finally, we have our internal generation term. Sorry, that's a delta, not a very good delta, delta x over two looks about the same as it did before. And all of this has to equal zero. And so what we want to do now is we want to find a way to be able to determine tm, n. And the reason why I said radiation was the most complex is because we have this fourth power term here. So we're not going to be able to find an explicit relationship for tm, n. But what I'll do is I will rewrite this. And I'm going to assume that we have a uniform grid again, just like we did before. So delta x is equal to delta y. And if we have that condition, then we can write out kind of doing a little bit of isolation, but not really because we still have a tm, n in our radiative heat transfer term, which we'll see right here. Okay, so that's what we get. They got a little crunched over on the right hand side. I apologize about that. But this then becomes the basis for the equation that we can use to determine what is going on on the surface when we have both radiative heat transfer and convection. So with that, that concludes setting up the finite difference formulation of the heat diffusion equation. What do we do with all of this? Well, what you do, you put together all these equations for whatever object you're looking at, and you'll then get a series of equations that you can then solve and set up a matrix. And you solve by doing a matrix inversion technique. And what we'll be doing in the next lecture segment, I put together a tool using Excel, and I'll show you how to use that tool. And essentially what that tool has done is it's taken all of these boundary conditions and the finite difference formulation for the two-dimensional heat generation equation or heat diffusion equation with internal generation. And we're going to apply it to solve some problems. And we'll look at problems that perhaps we'll look at the one that we looked at earlier when we did the heat diffusion equation using separation of variables. But you can do much more complex things than that. But we'll play around with that in the next segment. I'll show that. And that will give you an idea as to what happens when you put all of this together. And that's what we'll be doing in the next lecture. So I've gone through a lot of derivations here with the finite difference technique. Don't get too hung up on it. Basically, it's just showing you the method. But once you put it all together, you just got to be careful. And you can accomplish some pretty neat things when you put all of this together. So numerical methods and finite difference methods applied to the heat diffusion equation can be very, very powerful. And we'll see that when we start playing around with some of the solutions in the next lecture.