 Okay, in this segment we're continuing on looking at transient solutions, convective boundary and we're going to be looking at an infinite cylinder and so we're looking at approximate solutions as well as a graphical solution using the Heisler charts. So let's begin with the approximate solutions. So this is the infinite cylinder. Sketching out the geometry, remember we had r naught is the outer radius and we're interested in what's going on at some radial location and we're assuming that this cylinder goes off to infinity. Beginning with the approximate solutions. So we have the centerline temperature and whenever you're working transient problems you always got to start with the centerline. If you're wanting something spatially you start with the centerline you get that and then you move to your spatial temperature as well as for heat loss you need to know the centerline temperature. So let's take a look at that using our theta designation it's theta naught star and that is going to be theta naught divided by theta i and defining what those are. Centerline temperature minus T infinity divided by the initial temperature of the entire cylinder divided by minus T infinity and the solution here just like we saw for the slab is going to look like this. That is zeta and then the Fourier number very important non-dimensional number in transient conduction analysis. Now our length scale you'll notice before when we had this slab it was l squared now we're dividing by r naught squared and the bio number h times the characteristic length scale which is the outer radius for the cylinder divided by the thermal conductivity of the cylinder. Now c1 and zeta where are you going to get those from? Well you get them from a table and they will be as a function of the bio number and those are listed in a table. Okay so that is how you can evaluate the centerline temperature. Let's take a look at spatial temperature variability and you'll notice that this is quite similar to what we saw for the slab but there are slight differences in the equations so just be careful with that. So spatial temperature theta star theta over theta i and that's going to be the temperature at our radial location and specific time the radial location that we're interested in that's off the centerline minus T infinity divided by the initial temperature minus T infinity and that is going to equal the centerline temperature so the theta naught star that we just calculated in the previous part or I showed you the equation and then this is where you get the fun Bezel functions so that's a Bezel function of a first kind times zeta r star where do you get the Bezel functions well there should be a table in your book if not search it out on the internet I'm sure you'll find them. Bezel functions of the first kind and then our star what is our star that's our non-dimensionalized radius so that's going to be r divided by r naught so that's spatial temperature you'll notice you have to evaluate what's going on at the centerline first before you can get what is going on there and this is what you'd be looking for in that equation. Let's take a look at heat loss. Heat loss is q divided by q naught and I'll give you q naught in a second. Again this is a Bezel function. Evaluated at zeta 1 and q naught that is the total heat loss that would occur if your solid your cylinder was to go all the way from t and original ti all the way to t infinity t free stream. Now we see heat loss you can have cases where the object is getting hotter so just be aware of that sometimes t infinity can be larger than the initial temperature that we're looking at so it doesn't always have to be heat loss it could sometimes be heat gain so that is the approximate let's take a look at the Heisler Heisler charts that's the graphical technique and this is quite similar to what we saw for the slab. First of all we start with our centerline and you'll have a plot of theta naught that is at the centerline divided by theta initial and that's going to be plotted as a function of the Fourier number and you'll have these interesting curves that have breakpoints and that is one over the bio I'll show you these in an example problem in the next lecture so if you're wondering what do these look like spatial you have to do your centerline first and once you've done your centerline then you can do the spatial and that's plotted as a function of one over the bio number and here we're going to have curves like that and they are in order of increasing radial location so as you go out towards the outer radius of your sphere and then finally heat loss there'll be curves for a heat loss and these are q over q naught again plotted as a function of Fourier bio squared and these curves are plotted for different bio numbers so those are the curves that pertain to the cylinder for the Heisler charts and very similar to what we saw for this slab the only difference is that you're using r naught for your characteristic dimension was before we're using L okay so that is the cylinder the next segment we'll look at the equations for a sphere and that will give us all the equations that we can use for collective boundary conditions