 Okay, in this lecture what we're going to be doing is taking a look at solutions to the transient heat diffusion equation for certain geometrical configurations and we're going to begin with the semi-infinite solid. Now for the semi-infinite solid it turns out that the heat diffusion equation, if you cast it in a form 1D transient heat conduction, you can come up with solutions for semi-infinite solids by the introduction of a similarity variable. And the similarity variable eta is introduced and essentially what it is, it's a combination of spatial location x and time. And with that we collapse the two variables into one and cleanly then what that does it turns the partial differential equation into an ordinary differential equation. This is a technique that is used in mathematical physics problems, it's used in fluid mechanics, heat transfer, but it's just a way to solve partial differential equations. That's the best way to think about it. And what we're going to do, we're going to take a look at three different solutions. We're going to begin with the simplest boundary condition for a semi-infinite solid, so let's take a look at that one. Okay, so we have a semi-infinite solid with a sudden change in external temperature, so let's draw out what our solid might look like. We will introduce spatial location x coming from the surface of the solid. And what we're going to say is what we're after is we want to determine the temperature as a function of spatial location x time tau, and that's what we're looking for as for the solution for this particular problem. And what we do is we change the external boundary condition by changing the temperature on the surface. And so I'm going to give that the symbol t naught, and that's going to be for time greater than zero. So as we start the time, we'll change the surface condition. And the other conditions that we have at time zero for all x, so for the entire solid, we will say that the temperature is ti, and the surface condition that I just wrote was zero. And for time greater than zero, we specify the surface to be t zero. And that is for time or tau greater than zero. So when you start the, you change the surface condition, and then what happens is that change propagates inward into the solid. So I'm not going to go through the similarity solution for this. You can look at pretty much any textbook on heat transfer, and you can find it. I'm just going to give you the final result. And we're going to do that for all three cases that we look at in this segment. So that is the temperature distribution. And notice what we have here, we have an error function. And the error function inside of it, that is our similarity variable, the adoth, that we referred to. We've brought the four outside of the square root. And that's why it's two square root alpha tau. But you can see the similarity variable. And you're going to see that in all the solutions we look at in this segment. And the other thing that can be calculated with that is the heat flux at any x location in the solid. So that's heat flux at any specific location within the solid. And then finally, if we want to find heat flux on the surface of the solid, you would obtain that by setting x equals to zero. All right, so there we go. So that is the solution if you change the surface boundary condition. And it is the simplest case that we'll be looking at for the semi-infinite solid. The next one, increasing complexity of the surface boundary condition is one where we change surface heat flux. So let's take a look at that. Okay, so for this one, what we're doing is we're changing the surface heat flux on the surface of the solid, so right here. And what we know is that the heat flux, it could be an electric resistance heater, maybe a shine laser on it, radiation, anything like that, that you have a heat flux on the surface, that is going to be equal to, we can use Fourier's law right inside of the surface. And that represents Fourier's law with conduction inside of the surface. And so with that, we can rewrite out what the boundaries are for this problem. Again, we start at t initial throughout the entire solid. So for all x at t zero, it's t initial. And then we change the heat flux on the surface. And we can say that that is then going to be related through Fourier's law. Essentially, it specifies the slope of the temperature right inside of the surface. And that is for tau greater than zero. And the solution for this one, it gets a little bit more complex. So let me write that on the next slide. Again, we have temperature as a function of spatial location and time. Okay, so that is the solution that we get for the change in surface boundary condition. And one thing that we're starting to see, we saw the error function earlier. And the error function is typically tabulated. You look it up for whatever value that you are assessing. And the other thing that we're seeing here is this one minus error function. That is sometimes represented as being the complementary error function. So the complementary error function, you may see sometimes something like this. And I'm going to write it in terms of some generic variable w. But that is one minus the error function of w. And so sometimes that will be re expressed. And again, you can see the two or four alpha tau, that appears in a couple of places in this solution. It's there. And it's there. That was the similarity variable that we used for this particular problem. And consequently, it's starting to appear in the solution for the temperature distribution. So that is if you change the external heat flux. The last one that we're going to look at is going to be where we change the convective boundary. And so assume we have a semi infinite solid, and we have a sudden change in the convective boundary condition. So just like for the other ones, what we're interested in is determining the temperature distribution as a function of spatial location and time. We're told the initial condition for all x at time zero is ti. And for the boundary condition here, we have convection. And so again, what we're going to do, we're going to kind of do something similar to what we did just previously for the constant heat flux. We're going to equate that using Fourier's law. And I've canceled the area out here, if you're wondering what happened to area. But this is going to be right on the surface. So right when you come inside of the solid, the slope of the temperature is going to be related to the convective heat transfer. And we have to account for the fact that the surface temperature can change with time. And so we do t at zero, x equals zero, and for time as time evolves. And let's see, should that be tau? You know what, that probably should be tau to be consistent with what I have in the other thing. So I'm going to put tau. Okay, so those are our boundary conditions looking now at the solution. And this solution will be the most complex one that we look at. Okay, so that is the solution that we obtain for the surface convective condition for a semi-infinite solid. Again, you can see some resemblance to the similarity variable embedded within there. You probably don't want to calculate that all the time. And consequently, that is something that you'd probably want to put either into a computer program. Or sometimes what we do is we plot this data. And in the plots, I'll show you one in a moment here, but typically what we do is we plot our temperature. So what we have up here is plotted on the vertical axis. On the horizontal axis, what we do is we plot the similarity variable itself, x, or it could be four alpha t if I pull the two inside of the square root symbol. And then what we do is we plot this as a function of different values of time, the convective heat transfer coefficient, thermal diffusivity, and the thermal conductivity of the solid. And so that is represented with this variable. So that is how we can plot it. And then depending upon what particular location you're looking at, you find the appropriate curve, you go to the x location, you get the value. So you would go there and you go over. It's quite obvious you read the value there. Oops. Sorry about that. Disappears on me. And then we get the value from there. So let's take a look at what some of those curves might look like for this last solution. And this becomes a trend in transient heat conduction, just because the equations start to get quite complex. And so what we do is we plot the equations and curves. And so we'll begin with this one. So there you can see on the bottom we have the similarity variable, and then our temperature on the vertical axis. And then you're going to see curves of increasing the time, thermal diffusivity, convective heat transfer coefficient, and the thermal conductivity. And going all the way from 0.05 up to infinity. And all you would do is you just go up into that curve, whatever the particular parameters you're looking at. And you would read off the specific value. So those are three different solutions for the case of different boundary conditions for semi infinite solid. And what we're going to do next is we're going to now look at three specific geometries. And we're going to stay on the convective idea. So for this last one, we looked at a convective boundary condition. What we're going to do in the next segment is we're going to move into three other geometries, one being a slab, a cylinder, and then a sphere. And we're going to look at the temperature within the center line and as a function of position within those objects for transient convective boundary conditions. And those will be in the form of Heisler charts or tables and charts that we'll be looking at. So that's where we're going.