 In the last segment what we did is we talked about an alternative method using Fourier's law for determining conduction problems and what we're going to do we're going to begin by solving an example problem and this is for the alternative method and what this will do it will show what types of things we can calculate using this technique and if you recall it was for the case of study so we don't have time derivative no generation and one-dimensional conduction so let me begin by writing out the problem statement and then we'll work through it okay so there is our problem statement what we have we have a conical section which is very similar to what we looked at in the last segment we're told the thermal conductivity and we're given information about the diameter and recall we needed to know the area as a function of position so knowing the diameter we can get the area we are told the temperatures at the two surfaces of this conical section and then we're asked to derive two things one is the temperature distribution is a function of position throughout this conical section and the second one is the heat transfer rate so let's begin by writing out what we know what we're looking for and then a schematic okay so that's what we know and what we're looking for our schematic and I won't put it in three dimensions this time okay so that is the schematic the assumptions that we're going to make well they're going to be the assumptions that we used for the alternative method so those are steady state two is that we're dealing with one-dimensional conduction so that means that the surfaces are well insulated as we've shown here three is that there is no internal heat generation and the last one in which enabled us to pull the thermal conductivity out of the integral sign when we derive the alternative method was k is equal to a constant so k is not a function of temperature nor position within the object so for the analysis what we're going to do let's begin with Fourier's law and we're going to follow in the same steps that we did before what we need to do we need to figure out the area as a function of position position being x so it is a conical section so it has round cross section so the cross sectional area at any point is pi d squared divided by four subbing in the value for the diameter so what we can do we can take this area that we have here and we'll sub it into Fourier's law and then rewrite and rearrange so we get that equation and what we're going to do looking back at our schematic we're going to integrate between this position and any particular x location so we're going to come up with an expression for t as a function of x and consequently we'll be evaluating the temperature at an arbitrary x so that's what we're going to do now so we get this equation here let's go ahead and integrate that and recall the first thing that we're after we want to be able to find t of x so we have t of x in this equation here let's isolate for it so we get that and that gives us an equation for t of x which was one of the things we were after however you notice in this equation we have the q of x and we still don't know what that is so we need to solve for q of x and how are we going to get that well we're going to use the boundary conditions and so we've already used t at x1 we do know t at x 2 and that is t2 and so let's go ahead and do that in order to find what q is so we have this equation here what we want to do we want to be able to isolate for the heat flux and going through this conical object so let's solve for q of x so we have that let's plug in all the values when we plug them in we get minus 2.12 watts so we have a minus why do we have a minus well if we go back let's take a look at our schematic which was right here we said that x positive was going in this direction and notice we're going from a low temperature 400 up to a higher temperature 600 consequently he's going to flow from the hotter temperature to the lower so really the heat flux is in that direction opposite the direction that we've shown it and so consequently what that is telling us is that the heat flux is going from x2 to x1 and so it's consistent it makes sense and that's why it's a negative so that is the alternative method for one-dimensional conduction in a conical section what we're going to do in the next segment we're going to look at applying same technique to cylindrical coordinates and then after that we'll have another segment looking at applying it to spherical and then we're going to start to consolidate things and make this into a technique that we can use for general 1d conduction analysis so that's where we're going