 okay so what we're going to do now we're going to look at the last geometric shape that is a sphere for this alternative method of doing conduction analysis so we're going to look at a sphere and we are going to assume that it is radial conduction only and the other assumption steady state 1d no heat generation and the last one is k is equal to constant so it's not a function of temperature so those were all the requirements that we had for this alternative method I'll begin by drawing out a sphere and then we'll work through using Fourier's law okay not a great sphere but it is a sphere nonetheless so what we have we have some hollow sphere and we're going to say we have internal radius r i we have outer radius r not and we're interested in what's going on at some arbitrary radius r and we're going to then try to solve for t at r and temperatures those are the boundary conditions that we're going to have let's say that we know t i and we know t outer all right so that's the information we have the area of a sphere you better know the area of a sphere the volume of a sphere is four-thirds pi r cubed and the area would be the derivative of that with respect to r so area is four pi r squared all right that's the area of a sphere you have no idea how many times during exams I have students ask me what's the area of a sphere and what is the volume of a sphere those are things you should have memorized that in the quadratic equation remember memorize those things okay here we go Fourier's law we have Fourier's law we're going to rearrange that we put in the area of the sphere and we're going to integrate this to our arbitrary r location okay so we get that equation there now what we want to do is we want to integrate that and looking at this equation here remember this is the thing that we're after we want to have a way to determine the temperature distribution within our hollow sphere so we rearrange this and we can come up with the following expression and we get this relationship now like before we have not finally solved everything's we still have that heat flux in there how are we going to get that while we have our boundary conditions coming back here we have we've used this boundary condition we haven't used that one so let's apply the boundary condition okay there we go so that gives us the heat loss for this spherical object where we know the inner and outer temperature as well it gives us the temperature distribution so those are three different things that we can do we've looked at spherical cylindrical and we looked at one dimensional that was for the conical section as long as you know the area as a function of position and you can assume it to be 1d steady no internal generation and the thermal conductivity being a constant as a function not a function of temperature if you have all of those you can come up with these equations so we've done a lot with this but what we're going to do in the next lecture engineers are lazy not quite we're not lazy where you fish so what we're going to do we're going to package all of this stuff together and come up with a more compact way of being able to apply these ideas and apply them to calculations in a fairly quick manner and so that's what we'll be looking at in the next lecture as we continue on using the alternative method for conduction analysis