 All right, in the last segment, what we did, we spent quite a bit of time coming up and deriving the heat diffusion equation, and we concluded by saying that the heat diffusion equation enables us to determine temperature inside of an object provided that we know the boundary conditions on the surface. So what we're going to do in this segment, we're going to take a look at the typical boundary conditions that you can come across when you're solving heat transfer problems. And so it's with these boundary conditions that we're then able to solve with the heat diffusion equation for the temperature inside of an object. So what we're going to do in all of the cases here that we're going to look at, we're going to consider a one-dimensional object or surface. So we're going to look at a 1D object and we're going to assume that the surface takes place at x equals zero. And we're going to begin with the simplest type of boundary condition, and that is where we have a constant temperature on the surface, so a constant surface temperature. And so drawing out our object, so this would be the solid that we're looking at. So there we have our solid, this would be x equals zero on the surface. And let's assume that we know the surface temperature, we're trying to solve for the temperature within that object, and so that is an unknown, but we know the surface temperature. So then this would be constant surface temperature, the way that we can write this out would be T at x equals zero, and for time is equal to the surface temperature. So how might you create that type of a boundary condition? Well, one way that you could do that is with an ice bath, for example, or boiling water, although that would vary depending upon the elevation of where you are located. But a good example here would be that of an ice bath. So that's the most simple boundary condition that you can encounter. Then what we can do, we can have conditions where we have constant heat flux at the surface. So let's take a look at heat flux boundary conditions. So drawing out our object again. Now if we know the heat flux on the surface, so that would be qx, what that is saying is that we know the slope of the temperature on the surface. So what we would do is we would use Fourier's law right on the surface here, and that would then provide us with an indication of the temperature, the slope of the temperature at that point. So when you say constant surface heat flux, essentially what you're saying is that you know what the slope is. And we call Fourier's law qx equals minus KADT by DX. So essentially what we're saying was that if we know this, we know that. And that is what we get the slope from. So that would tell us what the slope is right on the surface. And then as we go further into the object, it can deviate in some other way. And that would be our X of T for the temperature then. And so writing this out mathematically, we would say qx over A is minus K DT by DX at X equals zero. And that comes directly out of Fourier's law. An example, how could you create a constant heat flux condition? One thing that you could do, there could be a radiation from some source, or an electrical resistance heater would be another example. Provided that you insulate the outside of it, so that you don't lose heat to the outside. So that is constant heat flux. And there is another form of constant heat flux surface that of no heat flux. And we refer to that. So let's call this here, sorry, I should have put an A, this is finite heat flux. And we can have B, which would be adiabatic or insulated surface or no heat flux. And if you have an insulated surface adiabatic, we're then assuming that there is no heat flux coming through the surface. In reality, you will, any kind of insulation, you will have heat flux, but this is something we assume in heat transfer. Whenever it says insulated surface, you usually assume that to mean zero heat flux going through it, or if it's adiabatic. And with that, we then get DT by DX at X equals zero is equal to zero. And so looking at that in terms of a schematic, I'll draw this a little more compact here. So what that is saying is that the slope of the temperature is going to be zero at the wall. And so we would have a condition that might look like this, and then it can deviate a lot. TX of T would then be our temperature distribution. And X, again, is going in that direction. So that's the case of an adiabatic or insulated surface as well as one with a finite heat flux. The last boundary condition that we commonly look at is that where you have a convective boundary condition. And so you have fluid outside of the wall. And so we call that a convective surface condition. And if you have convection on the surface from Newton's law of cooling and from Fourier's law, from Fourier's, we know this. And then from Newton's law of cooling, we can write this. And so writing out a schematic of what is going on here. We have our wall. X is going in that direction. And then out here, we have some fluid. And so what this is then telling us is it's essentially giving us relationship between the slope and the convective heat transfer. And with that, it is specifying the slope on the surface. And it is assuming that we have the temperature at X equals zero and time there. Because that can change as a function of time, but you would need that surface temperature for the convective surface condition. So that is the third boundary condition that we can have. We can have constant heat flux, constant temperature, or convective boundary condition on the surface. So with those boundary conditions, we use those to solve the heat diffusion equation. Okay. So we use the heat diffusion equation in these boundary conditions. And the boundary conditions are applied on the surface of the object. And what that does, that enables us then to determine the internal temperature distribution in the object. And so the way that we do that, there are a couple of different techniques that we use. We can do this analytically. And that means that you solve a closed form mathematically for temperature distribution would be a function. Then there are a number of different types of objects that you can actually do that for. And sometimes it turns out into being an infinite series. And we'll look at that when we look at multi dimensional conduction later on in the course. And the other way that we can do it is numerically. And there are different solution techniques for solving the heat diffusion equation numerically. We will look at one in this course as well that uses Excel. It's kind of a very basic one, but it's actually not bad. It solves for things in two dimensions. And so we'll take a look at that. And so that is how you would go about applying these boundary conditions. So that provides an overview of the heat diffusion equation and the boundary conditions that you would apply. And we're going to be looking at these a little later on in the course. What we're going to do in the next lecture, we're going to take a look at kind of an alternative mode of doing analysis. And that is for the case where you're only looking at one dimensional problems. It is steady state and no heat generation on the inside of the object. And we'll be able to do some analysis for those. And but then if you want to do more complex, then you got to go to the heat diffusion equation and use these techniques that we have here, which we will look at later on in the course. So anyways, that is heat diffusion equation and the boundary conditions.