[BLANK_AUDIO]. Welcome back. So last time we talked about work transfer as a form of energy transfer across the system boundary. Now there are a few key points we need to remember. Work transfer is not a system property. Where transfer depends on the process path and there are many different types and forms of work transfer so last time we introduced expansion and compression work and that's the work to raise or lower a piston. Now that type of work is very important because that's pretty much the power generation that's used in all of the transportation sector, is the expansion and compression work. And so we looked at that as a specific example and then we left with a question of what would a constant pressure compression process look like on a pressure volume diagram? So we're starting to put some of our tools together. The PV, pressure volume diagram are state diagrams, we're going to discuss those quite a bit in the coming lectures. But we also talked about work as being, in it's most general form, an integral expression. And then, we said, okay, there's a specific form of that expression for expansion and compression work. So, let's answer that question now. Okay, so we talked about the PV diagram. And again we said that work in its most general form could be expressed as an intergral moving from one state condition to a second state condition so that's S1 to S2 and the most general form of that expression is one where we use just some force. Times the distance, that it, that force has executed in that process. So from state one to state two forced integrated over that distance, over that process distance. And we said for expansion and compression work in particular, we would move from state one to state two and that the form of the expression was pressure times the differential D volume so remember the V with the cross hash is a volume expression okay so that's our expression if we have a constant pressure process. We know that that pressure is constant, and we know from our math that we can take the pressure outside of that integral. The state conditions here are now volume at state one, and the volume at state two. So, integrating that expression is pretty simple now. We have the work is simply equal to the constant pressure times the volume at state two minus the volume at state one. Okay. And, I'd like to keep track of our state conditions, our process path, by using the notation one-two. So, we're going to say we start at state one and we end at state two. So, now we come back to that question: what is constant pressure compression look like? Well, it's constant pressure so we know that's going to be pretty straight forward - it's a straight line. And we know that we have an initial state, and a final state. We have to decide which is which on this diagram. So we know its compression. So we should start at a higher volume and end at a lower volume. And we know to emphasize that we'll give an arrow to show the direction. So that's supposed to be a strait line there. But that's what constant pressure compression process would look like on a pd diagram. Cool. So hopefully that was pretty straightforward for you. Now like I said there are lots of other forms of work. And which form of work is relevant to the problem we're considering depends on the problem itself. So we're only going to define in detail one more. And that's shaft work. Shaft work is the work that we use to generate power by rotating machinery. So, all of your jet engines, your gas turbines, any type of turbine, generates work by spinning a shaft. There's also extension work. That's when you might extrude a metal or pull on a piece of plastic. Electrical work we sort of use. electrical work is, of course the power. Generally we consider it power instead of energy transfer, so it's energy, the rate of energy transfer. But that'd be like if we plug in a television set for example. The energy that that takes, that energy transfer process it takes to power the television, is electrical work. So we don't tend to look at the details of the electrical work in this. In thermodynamics we just take that as an input condition and we recognize that it's electrical work. There's also film work and many other types of work. There'd be like, if we were looking at something like a plasma, like we would find in a fluorescent light bulb. We would have to consider these different forms of work. But again, for the big, primary sources of power on the transportation stationary power sector, in your home, all those types of work are really going to be covered by shaft work, extension, excuse me. Shaft work, compression and expansion work and electrical work. So let's take a look at shaft work real quickly. Again, because we want to understand that one because it's the primary way we generate power in the stationary sector. So in a shaft system, we have something that's rotating. And again, if we were looking at a turbine, these would be the turbine blades. Or compressor or a pump or anything that has rotating machinery like that so forgive my artistic rendering here but what we want to show is that we have a shaft that carries the transfers that energy and its rotating. And so we go back to our general description of work is equal to the force times a product of the distance that is executed. And in this case the force that's executed is a tangential force, because we have a rotating piece of machinery. And so we define that [NOISE] as a tangential, force. And more specifically, we know that the velocity at which we rotate is given by the product of, this is a, I'll try and make this a script W so we can understand that that's actually the angular velocity. Of that shaft, and r of course is the radius. And we're going to take that velocity, so this is a velocity but it's a radial velocity here, at the point Point of application of that tangential force. Okay, so if we put that together, and now I'm introduce another term here, I'm going to write at least a form of an expression. We're going to take that W and we're going to put an overdot on that. And that is always going to mean that the variable is on a per unit time basis, or rate basis. So this means rate basis [SOUND] or per unit time. [SOUND] Okay. And after we go through that exercise I'll use a script T here to denote the torque we find that the power generated for a shaft is simply the product of the torque times that angular velocity. Okay, so again that's the only other work term that were going to care about in, in detail in terms of the expression. Because we want to know typically how to correlate the angular velocity with the power out of the system. So, let's review. We talked about sign conventions in the previous lecture to. So what does that mean? Work out of a system is positive. So, that's greater than zero. Our work term here would be greater than zero, and that's consider the phrase we would use is that work is done by the system. So my turbine if it generating power, which we kind of hope all turbines are generating power. That that would be work out of the system, it would be a positive term. If the work is into the system, like when we compress a gas like we would use in a, like we do in a engine, then that term is negative. That would give us a numerical value less than 0 and the work is done on the system. Again, this is a traditional sign convention for thermodynamics and it's not uniform. So you go to other disciplines and you'll find different sign conventions. It's one of the first things you need to understand when you're reading somebody's analysis or when you're doing your own analysis, is what's the sign convention you are going to use. And it's your discretion, you can use a positive sign convention for positive work out, or you can change that around, but you have to be consistent. And in this class, this is the sign convention that we are going to use. So you have to consistently use this sign convention. So, I already gave you my speech about units. The units should always help you in your analysis. The sign convention should always help you in your analysis. So, just remember your sign convention, be consistent and it will be a tool to help you understand how your system is behaving. Okay. So this should be easy now. If a turbine generates power by shaft work. What is the sign for the work done by the system?