 In the last segment we took a look at the idea of the shape factor and what we're going to do in this segment is we're going to solve an example problem involving the shape factor. If you recall the shape factor is kind of a quick way of being able to compute conduction in a problem that cannot be approximated as only being one-dimensional. This would be for two-dimensional problems and so what I'll do I'll begin by writing out the problem statement and then we'll work through towards the solution. Okay so there's a problem statement kind of a long one but what we have we have the whole is drilled through the center of a solid block of square cross section and the dimensions of the square cross section we're told are one meter by one meter on a side. The hole is drilled along the length so the length of this square cross section is two meters long and we're told that a warm fluid passing through the hole maintains the inner surface at T1 equal to 75 degrees C so a very simplified approximation for a convective heat transfer due to the fluid and as a result what we'll do is we'll assume the inner wall temperature of this hole is at 75 degrees C and the other thing we can assume is the outer block temperature is at 25 degrees C and then we want to find the heat transfer so let's go through the steps that we do for all the problems we begin with what we know and then we'll draw a schematic and work through the assumptions so our known items are okay so that's what is known and what we're looking for let's draw the schematic and that will help visualize what is going on in this problem okay so there is the schematic for our problem we have a block it is dimensions w by w and length inner temperature T1 and the outer temperature T2 and so when you look at this it becomes quite apparent that we can't a lot assume this to be one-dimensional conduction as we were able to do and we looked at thermal resistances where we could have a pipe a very poor looking pipe but if you recall when we had thermal resistances we could assume that the conduction was in one dimension basically going in the radial direction here given the fact that we have a square geometry around a circle we're going to have edge effects going on and consequently this becomes a two dimensional conduction problem and that's why we have to go to the shape factor in order to solve it so let's take a look at the assumptions and then we'll work through the theory basically using the shape factor okay so we're assuming this is happening in steady state and that we have 2D conduction and the other thing that we're assuming is that the ends of the block are well insulated because if we look at the schematic again we have something like this and this is one meter that's one meter and this is two meters so the block is really not that much longer than it is wide and consequently in order to ensure that we do not have conduction going in the axial direction what we'll assume is that the edges of the block are well insulated preventing any kind of heat transfer going in that direction and consequently we're assuming that all of the heat transfer is going in this direction well it's two directions because we have two dimensions that we're looking at but then we'll have edge effects the fact that we have these corners here which I just mentioned okay so analysis how are we going to solve this what we're going to do we're going to use the shape factor so you have to find a table that has a shape factor with this object in it that's usually step one for solving these problems and I just happen to have a table that has this value in it so we look up the conduction shape factor and this is for a cylinder centered in a square of length l and for that the shape factor we're told is the following okay so we're given that r would be half of the diameter of the hole within the object it's interesting to look at this because we have the natural logarithm in the denominator and that was what we found when we looked at conduction through a pipe we always found the natural logarithm in the denominator but anyways when we put in the values we know all of the dimensions here so we can determine this and we get 8.588 meters is the value for the shape factor and then it's a pretty straightforward and simple calculation with the shape factor really the biggest trick is to ensure that you have that shape factor for the shape that you're looking at and then we plug in the values we know the shape factor so we can do that thermal conductivity was given and then the temperature difference it was 75 on the inside and 25 on the outer wall and with that we can then evaluate this and that tells us that the heat loss for this particular object turns out to be 64.4 kilowatts so that is an application of the shape factor and it enables us to solve in kind of a quick and efficient manner problems that involve two-dimensional conduction but again the biggest trick is to ensure that you have a shape factor for the particular shape or configuration that you're looking at and and one thing to say is that shape factors only exist for a limited number of scenarios okay so what we can say is that shape factors only work for a limited number of scenarios you need to have a shape factor for it so if we have a problem and we cannot assume it to be 1d and there are no shape factors then what happens is you end up going to numerical methods and so that's where we're going in the next couple of lectures we're going to be taking a look at two-dimensional conduction problems using numerical solutions and and we'll see the power of numerical solutions it's not as quick as you would get with hand calculations like what we've been doing thus far but it does prove to be a very efficient method especially for heat transfer being able to calculate temperature distribution in objects and what we're going to find is the biggest challenge with numerical methods is going to be estimating the boundary conditions so what goes on in the solid is pretty easy but what's happening on the surface is usually the biggest challenge but that's where we're going in the course we're moving into numerical methods for two-dimensional conduction