 Alright, so we're now going to look at the last geometric shape for change in boundary conditions with transient conduction and so we're looking at convective boundary conditions where we're changing them and this we're going to look at a sphere in this segment. Okay, so geometry of the sphere we have r0 which is the outer radius and then we're interested in what is going on at some radial location r and we'll begin just like we did for the slab and the infinite cylinder we'll begin with the approximate solutions and we begin with the centerline temperature. So here we're evaluating theta not star which is theta not divided by theta i and those are defined as being the centerline temperature minus the free stream convective environment and then ti our initial temperature minus the free stream convective environment temperature and just like for the slab and for the infinite cylinder we have the expression c1 exp minus zeta 1 squared and then we have the Fourier number Fourier number alpha the thermal diffusivity times the time divided by the outer radius of our sphere squared bio number we need the bio number and that is h r0 that's your convective environment convective heat transfer coefficient r0 is the outer radius divided by the thermal conductivity of the solid and the reason why you need bio is because c1 and zeta come from tables and those tables are functions of the bio number so you would look those up you compute your bio number first and then you look them up so that is our centerline temperature function spatial distribution we have a function for the spatial temperature and this one's a little simpler I guess you could say it doesn't involve bezel functions like we saw for the cylinder however in order to solve it you do need to know what happens at the centerline which we have in the theta not star term and so here we have a trig function remember to be careful that needs to be evaluated in radians and our star just like we saw for the cylinder that is basically just the radius non-dimensionalized by the outer radius so that's how we get the spatial temperature you solve first of all for that using the centerline temperature and then you using the values that you looked up for the bio number and for the spatial location you calculate our star and it's pretty straightforward and remember what we're after here is to get that because that gives us temperature at a given spatial location at a specific time and then finally heat loss how do you calculate heat loss from this sphere so you get that that's a little bit more complex that we've seen for the slab and the cylinder and then q naught that is just like we had for all the other ones total amount of energy assuming that your sphere goes all the way to t infinity with time so that is heat loss spatial temperature centerline temperature just like before we also have the Heisler charts that we can use and so that's the graphical solution I'm starting with the centerline temperature we have this plotted as a function of the Fourier number that's going to be theta naught divided by theta I and here we're going to have these curves and they are functions of 1 over the bio number spatial distribution you have to determine what goes on at the centerline first before you can get to this one because you'll notice theta naught is in there and that's what you're getting so this theta naught that you determine here goes there that's what's going on and this is as a function of 1 over the bio number and we have a number of different curves that change depending upon the radial location that we're interested in and so that would be r over r naught that's our r star value and then finally heat loss we have curves for that and those are going to be q over q naught Fourier bio squared the bio number and then we have this here that's a function of the bio number increasing in that direction so those are the Heisler charts we will be looking at those in the next lecture because we work an example problem we won't look at the heat loss one but look at the central line of the spatial Heisler chart so anyways those are the different techniques for calculating transient heat conduction in an infinite slab in a infinite cylinder and in a sphere when you change the external convective heat transfer boundary conditions so in the next lecture we'll go on and we're going to use one example but we're going to work it twice once we'll use the approximate solution and then in the second segment we'll use the Heisler charts and we'll compare them to see how well they compare to one another so that is where we're going with transient conduction