 okay so what we're doing is we're taking a look at the boundary conditions in the last segment we looked at the boundary condition with heat flux on an external surface what we're going to do in this segment is we're going to take a look at a convective boundary condition and so what I'll do again is I'm going to begin by drawing out a schematic that we'll use when we come up with the equation that will enable us to determine the boundary condition and finite difference form and so we're assuming a convective environment with convective heat transfer coefficient H and free stream temperature T infinity so let's draw out the schematic so again I'm putting an interior node m minus 1 n and just like before what we are going to do we are going to prescribe a control surface before I do that let me denote our convective environment so out here we have a fluid could either be forced or natural doesn't really matter T infinity and H is the convective heat transfer coefficient so what I'm going to do just like before we're going to prescribe a control surface here and that is going to be what we're going to use in coming up with the equation for our boundary condition and just like before delta X over 2 is the width and the vertical dimension or the height is delta Y so we are going to perform an energy balance on that control surface and that will be the basis for coming up with the equation so let's start with that and you'll notice when I'm writing out this equation I'm always treating heat flowing into the control volume as being positive and so we start off with heat in via surface convection plus heat in via conduction and looking back at our schematic we can have conduction coming in let me put that I'll use red conduction can be coming in from this surface or from that node I should say through there and through there as well convective heat transfer obviously is going to be coming in this way and then finally we have to consider the fact that we may have internal generation within our little control volume within the control surface and given that we're operating at steady state all of this has to equal zero so what we can do we can go through and we can sub in values and that will be the basis by which we will come up with our boundary condition equation so let me go ahead and do that beginning with convection and I'm multiplying by delta y just say before let's assume unit width so the width is equal to one and with that the area remember we have Newton's law of cooling h a delta t so that would be h delta y times one I'm not going to draw out or write the one and given that the energy is flowing from the fluid into the wall we'll assume that the fluid is hotter and then going through and applying Fourier's law for conduction and that last term that this is essentially delta v the volume of our control surface or control volume so the goal that we have now is to isolate for TMN so we're going to try to isolate for TMN everywhere in this equation and we're going to bring that to the left hand side and so I'm not going to do it I'm just going to show you the result of that exercise and this is under the assumption again of a uniform grid spacing so delta x is equal to delta y and if we can make that assumption then TMN turns out to be the following so this becomes the equation that enables us to handle the boundary condition where we have convection convective heat transfer through the surface and we're doing this in a manner where we can have internal generation that's why we have the q dot term but that would then become the equation that you would put in for that boundary within your finite difference formulation for the heat diffusion equation so that is convection at a boundary the last segment what we're going to do is we're going to take a look at the most complicated and that is where we have radiation and convection on a surface and so that's what we're going to do in the next and last segment for this lecture.