 We're now going to take a look at a couple of the internal combustion gas cycle, gas cycles that we discussed in the previous lecture, specifically what we will do is we will begin with the spark ignition cycle, that is the auto cycle, and then we'll take a look at the compression ignition cycle, which was the diesel. So let's begin with the ideal or the auto cycle. We call it an ideal auto cycle because in reality, the real auto cycle, we aren't able to make the approximations and assumptions that we will throughout the analysis that I'll present here. So a couple of things to point out here. First of all, the mechanism itself of the engine, we're looking here at a four stroke cycle is what we'll be talking about, but the mechanism is going through a cycle, but the working fluid itself does not. That is, remember the working fluid comes in, it gets compressed, the combust that goes down on the power stroke, and then it exhausts, and then it gets purged out into the atmosphere. What we're assuming, however, through our approximations is that the working fluid stays within the cycle, so that's one approximation. The other thing is that this is sometimes referred to as being constant volume combustion, and I'll show you in a moment with the PV diagram where we can see where the constant volume combustion term is coming from. So let's take a look at the PV diagram and then some of the aspects of the auto cycle itself. So what we'll do is we're going to begin at state one, and this will also correspond to where our piston is at bottom dead center. We then go into an isentropic compression process, and through the compression process, we move up to top dead center. At that point, we go into combustion, where we have internal combustion going on within the engine. This is going to be represented as a heat addition process for our analysis purposes. And then, after being at state three, we go through another expansion to state four, and that will be another isentropic expansion process, and that is the power stroke, and then we have here our exhaust, and we will model that as a heat rejection process. So looking at the steps of the auto cycle, one to two isentropic compression, two to three was the heat addition process, and that is constant volume combustion approximately. Three to four was our power stroke, and that is isentropic expansion, and then four back to one, that is where our exhaust cycle, where we will model that as being a heat rejection, and we'll call that constant volume heat rejection, and that really is expansion. So those are the four components that we have within our auto cycle. Now what we want to do, we want to be able to come up with an equation for the thermal efficiency, and the place where we'll begin with doing our analysis is with the first law. So let's take a look at the first law. We're dealing with a fixed mass or closed system. We're assuming it to be fixed mass and closed, although it really is an open system, but we're modeling it as a closed cycle. So looking at the first law for steps two to three, for two to three, that's where we have our combustion being modeled. It's going on right in here. So we have heat addition, so let's write out the first law. Heat is coming in, which we said with our convention, a first law would be positive, minus work equals delta U. Now I'm neglecting kinetic energy and potential energy because there will be none. It's not moving in the larger scheme of things, although our piston is moving up and down, the whole engine is not moving macroscopically as the entire unit. And consequently, we can neglect either kinetic or potential energy, and with that, with our definition of internal energy, we can write this as an approximation here for a low specific heat or low temperature change, so essentially constant specific heats. We can write that as being our change in internal energy. Now a question here, are we doing any work from two to three? Well, we're a constant volume, we have a fixed mass system, the volume's not changing, consequently there's no boundary work, so our work term actually disappears. So we can cancel that out. Now looking at steps four to one, four to one, if we look at our PV diagram is happening in here, and that was our heat rejection process. Again, what we'll do, we'll write out the first law. We have Q out, and I write it as being a negative because heat is leaving the control volume, and by our definitions, that would be a negative. And again, it is going to be equal to delta U. And again, we can make the same argument for work, we're at a constant volume condition when we're at bottom dead center, and consequently there is no boundary work at that point, and it goes to zero. And so with that, we have equations for both Q in and Q out that we can then use with our definition of thermal efficiency. So let's take a look at that, and see what we get for the thermal efficiency of the auto engine, or the auto cycle. And so if you recall, we're dealing with a heat engine, and our nomenclature for a heat engine, or the way that we model it, we can draw it schematically like this. We have heat from a high temperature source going through the heat engine, being rejected to a lower temperature sink. In the process, we have network coming out, and our thermal efficiency is the network divided by the heat input. And we also said the network was equal to heat in minus heat out. And we divide that by Q in. So rearranging what we have here is one minus Q out over Q in. And that is an expression for thermal efficiency. Well, looking at what we talked about with the first law, oops, we had expressions for both this and this. And consequently, we can substitute those in. So going back to the previous screen, here were the expressions for both Q in and Q out. So we will pull those into our thermal efficiency equation now. So we obtain that expression. The CVs are common, and so they cancel out here. However, we did make an approximation that they were constant, enabling us to do that. So be a little careful with that. And finally, when we go through and combine terms here, what we obtain is the thermal efficiency for the auto cycle. We obtain this expression. Now, I did have to invoke the cold air standard assumption here. So be aware of that. But nonetheless, what this provides us with is an expression by which we can compute the thermal efficiency from the auto cycle. Now, having the temperatures is not the greatest. So what we want to do is we want to find a way to be able to rearrange or re-express this, where we have thermal efficiency of auto without the temperatures in there, but with something else about the cycle. Usually, we know things such as pressures and volumes about our engines. We know compression ratio and things like that. So what we're going to do now, we're going to try to rearrange this. And the way that we're going to do that, we're going to look at our cycle. And if you recall, processes one to two and three to four were isentropic. And if you recall the definition of isentropic, that means reversible and adiabatic. So what we can do, we can use relationships that we have for isentropic processes. And we saw these in an earlier lecture. And we will use this with v2 equals v3. That's for the heat addition process. And that would be a top dead center. And then for the exhaust process, v4 equals v1. And that was a bottom dead center. And so with that, our isentropic relations would state the following. So there, what we obtain is we obtain relationships with our temperature ratio and volumes. So we can take these, these relationships here, and let's work them into this relationship up here and see what we obtain. And I'm not going to go through all of the detailed substitutions. But if you do that, what you can obtain is that the thermal efficiency for the auto cycle can be expressed in a very compact manner like that. Now r is the compression ratio, which is a value we will know for a particular engine based on the way that it's operating or design that is. Because the compression ratio is vmax over vmin. And given that the mass is the same here, we can write v1 over v2 for specific volume. And k is equal to our ratio of specific heats. So that there gives us an expression for the thermal efficiency of the auto cycle. We had to make a number of approximations there, such as specific heats were constant. However, it does give us kind of a useful tool because we will often know the compression ratio for an engine. We can estimate the ratio of specific heats, and with that we can then determine the thermal efficiency of the engine from a theoretical point of view.