 Okay, so we're looking at transient conduction and what we're going to do, we're going to begin by looking at the heat diffusion equation. And what we'll be doing is considering a case with the simplest possible boundary conditions that we could encounter and we'll come up with a solution for this technique. Well, I'll give you the solution, we won't go through the math, but it essentially uses the separation of variable techniques. So what we're going to do, we're going to begin, we're looking at a slab or what we call an infinite plate and the boundary conditions are a fixed temperature. So these are probably the simplest boundary conditions that you could have. And then the boundary conditions are suddenly changed to another temperature. So let's take a look at that. Okay, so there is the problem statement that we have. What we have is an infinite plate and when we say infinite plate and heat transfer, we're not assuming that it's infinite thickness. What we're doing is we're assuming that the plate goes in this direction and that direction to infinity and into and out of the plane of the screen going towards infinity. And the plate itself actually does have a finite thickness and so here we will define our x-axis and the width of the plate we will assign it to be 2L. So that is the width of the plate. So this here is our plate and we say it's infinite because it goes up and down into and out of the plane of the screen to infinity. And what we were told is this plate is initially at temperature Ti. So the entire plate is at a fixed temperature at time zero. And then suddenly what we do is we change the boundaries at time T greater than zero. We change the boundaries here. So this is T1 and this becomes T1. And so with that, what's happening is we're changing our boundary conditions and that change in temperature is going to move inwards towards the center of the plate with time. And so what we've looked at thus far in the course has been looking at things steady state. Finally we're dealing with the problem dealing with transients. And so let's take a look at how we would approach the solution to this problem. Now what we're going to do for the solution, we're going to begin with the heat diffusion equation. Okay, so we've worked with this equation quite a few times already in the course. And what we can say off the bat, looking back at our schematic here, X is one of the variables and T times T is another one. So let's look at the heat diffusion. So this term and this term go away because we said that they go to infinity into and out of the screen and up and down. And so there's no change in those directions and there's no internal generation in this problem. So consequently that is neglected. And what we're left with for the first time, we get to keep this term on the right hand side. We've always neglected that thus far but we're dealing with transient conduction and consequently that is going to remain. So what we can do, we can rewrite the heat diffusion equation and it is then recast to look something like this. And you'll notice that we still have temperature. We cannot convert this into an ODE like we did for some of the other solution techniques because temperature is a function of X and time in this solution. And so that's the equation that we're going to be working with. Another thing, what I've done, I've made a substitution here. You might be wondering what is alpha? Well alpha is a very common parameter that we use when we're doing transient conduction analysis and that is referred to as being the thermal diffusivity. And what that is, it's the thermal conductivity divided by the density and the specific heat capacity of the solid that we are considering. And so that is the thermal diffusivity. And you can find that in tables. If you have a heat transfer book, look in the back. I'm sure you'll find thermal diffusivity there with thermal conductivity, density and all the other variables that we're using for the different substances that we're solving. So that is the equation that we're going to work with. We have the thermal diffusivity. If you were to go through a solution technique to this, the solution technique is similar to what we looked at for 2D conduction. And if you're called for 2D conduction, temperature was a function of X and Y. Here we have temperature as a function of X and time. And so that's what we're looking for. That is the nature of the solution that we're looking for. We know the boundary conditions for this current problem. The boundary conditions are fixed temperatures and they just change from the initial temperature. So the solution to this can be determined using the separation of variables method. And just like before when we looked at 2D conduction, I am not going to go through and give you the method of solving that. If you get any textbook in heat transfer, I'm sure they will go through it and you can look at it there. But what comes out of this is going to be the temperature as a function of position in the solid and then as a function of time. And we'll denote time by tau. And so I'll write out the rest of the solution. Okay. And so that is the solution to our problem. And we were told that it was initially at Ti. And then the boundaries go to T1. So those are the temperatures on the boundary. Other things that we notice here, this is an infinite series. And so it is going from n equals 1 to infinity. And there is a restriction on n. And it's for odd values. So we have 1, 3, 5, and so on. So we only use every second value of n in that infinite series. So very similar to what we saw earlier when we looked at 2D conduction analysis. And we looked at the square plate with different boundary conditions. So what I'm going to do now, I'm going to show you the solution to this equation. And I'm going to come up with the solution for different numbers of terms in the series. So we're going to look at a solution with one term. And we'll look for n equals 3. That would be a solution with two terms. And then n equals 5. That would be three terms. And so on. So we're going to look at solutions for a different number of terms. The parameters, alpha, are thermal diffusivity. L, I'm going to set equal to 2. And that is going to be in meters. And then time, t, or actually in this equation here we had tau. But I used time. So tau is, I'm going to go from 0.1. And then I am going to go in steps up to 50 seconds. So we're going to look at the evolution of temperature within a substance. And essentially what we're going to have is a plot like this. This is going to be x. And this is going to be temperature. And then what you're going to see, we're going to go from the initial condition. The initial condition here would be something like that. We have a step change. And then what happens is we, initially we are at ti. And then what we do is we drop the temperature on the wall down to t1. And so what we're going to be doing is looking at the evolution of that. And for the particular example that we're looking at, let's see here, what do I put t? I put ti is equal to 100. And t1 is 15 degrees C. So those are the conditions that we're going to have. We drop the temperature down. And then we're going to see this system move in with time. And so that's what we'll be looking at in the solution. So let's take a look at that now. And we'll look at what the solution looks like with different numbers of terms. So here we have one term, two terms, three terms, four, five, six. And I've really sped this simulation up. We're going to look at it quickly. And you can see down in the bottom right hand side where we have six terms, we'll look at it slower now, there's a bit of ripple at the beginning when we have the square wave. Because remember, we're filling in using sine functions. But as you get further on in time, that ripple goes away. And then slowly you can see the step change moving to the right. And that is as conduction is taking place within the solid object that we're looking at. So that is what that solution looks like. You can see that you need obviously more than six terms because you have quite a bit of ripple. And even I didn't show you really, really small time steps. I just showed you that the first time was 0.1 seconds. Had I done 0.001 seconds, you would have even needed more terms to be able to resolve that step change, discontinuity, right at the beginning of that simulation. And so even with 50 terms, you still get quite a bit of ringing going on. And we refer to that as being Gibbs phenomena when we take sine waves and we try to approximate a step, discontinuity. And we saw that as well when we looked at the solution for the heat diffusion equation for 2D conduction analysis. So that is the simplest approximation to a solution to the heat diffusion equation for transient conduction. What we're going to do in the next segment, we're going to move on to a different technique. And that is called the lump capacitance method. And that's where you assume that there is no temperature distribution within the object. You're assuming that the entire object is at the same temperature. And then we just model the temperature as a function of time. So that's what we'll be looking at next in the lump capacitance method.