 We're now going to take a look at solutions for convective boundary conditions under transient conduction for three different shapes. We're going to look at a slab, an infinite cylinder, and a sphere. In the previous segment what we did is we looked at the semi-infinite plate for three different boundary conditions and we came up with solutions. It was using a similarity variable and a similarity solution. But in this case there are other solutions. We'll be looking at approximations and it's actually a series solution that we look at the first few terms and that converges relatively quickly. And so the solutions we're going to look at will be in the forms of charts and in terms of an approximate solution using the equations. Before we can get to the solutions however what we need to do we need to introduce the nomenclature that we're going to be using for these solutions. So that's what we're going to be doing in this segment. And then in the subsequent segments we'll look step by step we'll begin with this slab then we'll look at the cylinder and the sphere. So the geometry that I refer to as the slab which is probably not technically correct but it's an infinite plate 2L thick and what we're going to be doing we're going to be solving for temperature first of all at the midplane as a function of time. And from that we will be able to get the spatial temperature. So temperature at some point off of the midplane at a specific time. And the last thing we'll be able to calculate is the heat loss from this slab up to a certain point in time whatever point of time we are examining. Now that's the infinite plate 2L thick and as I mentioned we're also going to be looking at a cylinder. And just like for this slab we will start by being able to determine the centerline temperature. And from that we'll be able to estimate the spatial temperature distribution as a function of radial location. And then just like for this slab we are going to be able to evaluate the heat loss. And the last shape that we will look at is that of the sphere. And just like for the previous two we will have the exact same three things that we will be able to estimate. Okay so those are the shapes that we're going to be looking at. And what I will do now is let's go through and take a look at the nomenclature. The nomenclature is very important when solving these problems. Beginning with the infinite plate. So the infinite plate is 2L thick and x is denoted from the centerline of our infinite plate. And in this t0 is going to be the centerline temperature. And we will introduce a variable x star. And that is going to be our spatial location non-dimensionalized divided by the length scale L. And the length scale is just L. And so that is the infinite plate looking at the cylinder. Okay so the cylinder is going to have an outer radius r0. And we will be interested when we look at the spatial temperature at location r, radial location r. And just like for the infinite plate we had x star here we will have r star which is going to be r divided by the outer radius. And the length scale that we deal with here is just the outer radius of our cylinder r0. And finally for the sphere the geometry that we will be using is as follows. Okay so we're interested in what is going on at some radial location r. The outer radius is r0 just like we had for the cylinder. T0 is the centerline temperature. r star is the non-dimensionalized radial location non-dimensionalized by the characteristic length scale which in this case is r0. The outer radius of our sphere. Okay so those are the some of the temperatures and spatial variables that we will have in our solution. Now a couple of other things that we need to be aware of and those are temperatures. We've already looked at the centerline temperature T0 but we have other temperatures the initial temperature. And that's going to be the temperature at tau or T less than zero. So before we change the convective boundary condition T infinity and h that describes the convective environment. So we assume that there is no convection at the beginning and then when we start we expose our either the slab the cylinder or the sphere to this new convective environment T infinity h. T0 we already saw and that is the centerline temperature. And then we introduce some values of theta which are differences in temperature and theta could either be the temperature at a given location so spatial and time minus the free stream that's if we're dealing with the slab or if we're dealing with either the cylinder or the sphere it would be expressed in the following way where we'd be evaluating temperature at a radial location. Theta i that is going to be our initial temperature minus the free stream. Theta 0 is going to be the centerline temperature minus the free stream. And we can also have a theta star and that is going to be theta divided by theta i or the initial temperature difference. Other numbers that we will be working with we have a couple of non-dimensional numbers that are very important in in heat transfer and especially transient heat transfer. And those are the Fourier number and it is given the symbol capital F o and it is the thermal diffusivity times time divided by our characteristic length scale squared. So it could either be l or r naught and another number that we have we saw this earlier when we were looking at the lump capacitance method. It is the bio number and that is bi and is the convective heat transfer coefficient times some characteristic length scale divided by the thermal conductivity of the solid that we are examining. So those are two other numbers that we'll be using. And then finally we'll be evaluating heat loss in the solids. And heat loss is referenced to some value of q naught and what that is rho cv so that is basically mc times delta t and the delta t is going to be the initial minus the free stream. So that's basically the total change potential change in energy possible change I should say going from the initial state to the new free stream state and ultimately in time the entire solid will eventually go to the value of t infinity but that's after all the transients have gone away. So those are some of the values that we're going to be using for the analysis here and what we're going to do in the next three segments we're going to look starting with the infinite plate and I will give you the approximate equations as well as the charts which are called the Heisler charts and then it will do the same for the cylinder and then for the sphere. And then in the next lecture what we're going to do is we're going to solve a problem with a sphere using both the approximate technique that we'll look at as well as the Heisler charts that's where we're going.