 What we're going to do in this lecture, we're going to take a look at a method of conduction analysis called the alternative method and essentially it's a method that relies on Fourier's law. It applies to one-dimensional conduction without heat generation and for steady state scenarios. But in general when you're looking at conduction analysis there are a number of different methods that exist and we'll be looking at in this course. One method is the heat diffusion equation which we looked at in the last lecture. We haven't solved it yet but we derived the equation. We'll be solving it later when we look at two-dimensional conduction for very simplified cases. So the heat diffusion equation, numerical analysis, and we'll be looking at a technique later on in the course as well that uses Excel. And then finally what we call the alternative method. Essentially it's a method that relies on Fourier's law and so there are some restrictions in terms of where this will apply but it kind of gives us a shortcut way of doing conduction analysis and so that's what we'll be working on in this lecture. But looking at this alternative method there are some heavy restrictions on this alternative method that we're going to be looking at. We have to have steady state conduction so that means no transience. The d by dt term goes away. One-dimensional although we will extend this to problems dealing with cylindrical and spherical coordinates but for those you consider them to be one-dimensional and you'll see what we're talking about when we get there. So one-dimensional conduction basically what happens is the only change is in the radial direction for cylindrical or spherical and the final restriction is no heat generation. So if we can make those three restrictions then we can use this alternative method. So what I'm going to do I'm going to kind of set it up and then we're going to look at it for three different applications one being just one-dimensional conduction and a solid then we'll look at radial cylindrical coordinate and spherical. But let's begin in a generic manner looking what we're talking about here. So let's assume that we have some chunk of material a very technical word for it but it's some piece of material. We have insulated walls and what I'm going to do I'm going to assume that this object that we're looking at is aligned about the x-axis. So that is our x-axis there and with that x will be going in that direction and so this would be x equals x naught at this location. And what I'm going to assume is that at each of the positions along the x-axis and that the temperature is constant across that entire plane so we'll assume that we have t naught there. And the other thing I'm going to assume is that I know the area as a function of distance in the x direction and then on the back surface here this here I'll say I know the temperature there and we'll call that t1. So what we're going to do we're going to draw out a little differential element here so let me do that now. And so there is our differential element and what we're going to do let's take a look at what that differential element might look like. Okay so there's our differential element we're going to say the thickness in the x direction is dx and then what I'm going to assume is we have heat flux in this direction qx and then on the back surface of the differential element it will be qx plus dx. So that is the heat transfer in and the heat transfer out. Let's write out Fourier's law so this is one-dimensional heat transfer so I have an ordinary differential here instead of a partial differential and what I'm going to do I'm going to rearrange Fourier's law and essentially what I'm going to do I'm going to bring the area over to the right hand side and the dx over so let's rearrange this so we have that if we rearrange Fourier's law and now what I'm going to do I'm going to integrate this so the integral on this side is going to go from x0 to x and over here we're going to integrate from t0 to t. Now notice what I've done is I've written that thermal conductivity is being a function of temperature. Thermal conductivity is a function of temperature. Quite often for our analysis however we will consider it not to be a function of temperature so sometimes you can pull that out and we will be making that assumption as we go on with this analysis. So let's take Fourier's law as written and if we can assume thermal conductivity is a constant and in this case I'm seeing it's a constant it means it's independent of temperature. If we can make that approximation then we end up with this equation here. Okay so this is an equation that if we know the change in area as a function of x we should be able to determine the temperature distribution in an object. Now we do have heat flux in there as well and we'll show that that comes out of the boundary conditions but we're able to figure that out but what we're going to do we're going to take this equation we're going to apply it to a number of different scenarios one is just the one-dimensional conduction problem and then we'll look at applying it to cylindrical and spherical coordinates but that kind of sets up the alternative method technique.