 okay so what we're doing we're doing conduction analysis we're looking at this thing called an alternative method but recall methods of conduction analysis there are several we have the heat diffusion equation that's the big complex partial differential equation you need to have boundary conditions you can solve for the temperature inside of an object and sometimes for certain shapes you can do analysis and solve that analytically which we will be looking at later in the course numerical analysis I've already talked about what we're going to be doing there and we will look at numerical analysis I'll give you an excel spreadsheet tool that enables you to do two-dimensional numerical analysis but there are other techniques as well and then finally what we've been focusing on for the last we did this last lecture is what we call this alternate or alternative method and essentially it's using Fourier's law under some rather severe assumptions but turns out these are pretty good assumptions because for many of the systems that we study these assumptions do apply so let me just briefly overview what that alternate method was and then what the results were and then we'll move on looking at how to apply it for engineering problems so if you recall the alternate method we started with Fourier's law minus KADT by DX and if you know the area as a function of X or it could be radial location as we saw for cylindrical and spherical coordinates but if you know that then you can make a substitution into Fourier's law and then what we did is we rearranged that and we set it up to enable us to be able to integrate but typically what we were doing is we were pulling the area over to the left-hand side in the denominator and then we were left with minus KDT this is where we said that we're going to assume that the thermal conductivity is not a function of temperature so that enabled us to pull it out of the integral which I'll show you in a moment so we have QX and we integrate from some boundary condition that we know what's going on to some X location that we're trying to figure out what the temperature is and you divide that by A as a function of X you pull K out of the integral from T1 that's the temperature at the boundary condition at 1 and then we have the integral DT now this works it works subject to a couple of constraints one of them we had to have steady state the second constraint that we had was one-dimensional so that means that all the heat is flowing only in one dimension not in two or three dimensions which we'll look at later on in the course third one no heat generation inside of the material that we're studying and the fourth was that K is equal to a constant so the thermal conductivity is not a function of temperature and what do we get out of this well you get T of X and you get Q of X so we were able to get the heat transfer and the temperature profile in terms of engineering this is usually the one that we are most interested in because we want to figure out what the heat loss is so what I'm going to do now I'm going to summarize the three different systems that we looked at and we looked at the plane wall well it wasn't quite a plane wall we looked at that example problem with a conical section but this would be what you would get if you were to look at the plane wall so I'll draw out a schematic so that's a schematic now what we were able to determine was the temperature distribution we were able to determine the heat flux and so what we did for all three cases we plugged Q of X into T of X in order to give us the temperature distribution in the solid okay so that was for a plane wall we also looked at a cylinder so that was the geometry for the cylinder R i R outer and what we obtained here was the temperature distribution and if you recall for the cylinder it resulted in a natural logarithm we were able to solve for the heat flux and just like for the plane wall if you plug the heat flux into the temperature distribution we were able to come up with the temperature distribution function in the cylinder okay so that's what we got for a cylinder pipe flow and many many engineering applications where we would have cylindrical coordinate systems and we're trying to determine heat loss but that's what we got for the cylinder and then finally spherical that this might be something like a storage tank you could have spherical coordinates okay so that's a scenario for the sphere temperature distribution and just like for the other two you plug Q of R into T of R and you get the temperature distribution in the wall of this sphere okay so those are the three different geometries that we've considered and the reason why I put them all out is because we're going to summarize them in the next segment and what we'll do we're going to look at the form of them and it turns out that there is some sort of commonality to all three of the forms that we have here for both the heat flux as well as a temperature distribution and so what we'll do we'll come up with a shortcut method that enables us to analyze what is going on for heat transfer either through a plane wall through a cylinder or a sphere and we'll be able to do more than just what we're looking at here we'll be able to do scenarios where you might have a sphere with insulation on the outside of this sphere so you might have some insulating layer and then there might be convective heat transfer going on out here so you have wind blowing and that has an impact we're going to develop a technique that will enable us to look at that and that is referred to as being thermal resistances so that's where we're going with all this in case you're wondering we don't want to have to solve these equations all the time and write out all these different terms we'll come up with shortcut methods and that's why I said engineers are efficient I said engineers are lazy but that's not true engineers are very efficient we like to find shortcuts some ways to be able to do things more effectively and efficiently so that's where we're going in the next segment we're going to work towards thermal resistances based on the results that we got thus far with this alternative method using Fourier's law