Welcome back. So we wanted to understand how would our analysis change if we used a small compressor. Which is another way of saying, how would our analysis change if the mass flow rate wasn't constant? Because again, maybe my compressor couldn't keep up with the supply that was required. or, if we drained a tank. So let's say the the pressure within a tank was decreasing as a function of time. So the state conditions and the mass flow rate could be changing as a function of time. Well as we saw before, this transient analysis is pretty complex when we have steady conditions from our fill chamber. If they're unsteady, the analysis is even more complicated. So let's take a look at the specifics. So if we have our, conservation of mass equation recall before, we had the change in the mass in the control volume. From the initial to the final state was given by this expression for the mass flow rate into the system here. So what we need to know, again, is how does a mass flow rate vary as a function of time? And for this, you'd say, well, let's take the draining take example. There, you would expect the mass flow rate to decrease as a function of time. Probably exponentially or asymptotically, so you might imagine where the mass flow rate is something like this, as a function of time. And we would have to decide before we did our analysis where am I on the slope, what's the magnitude of the initial, mass flow rate. So this would be at, let's say at the start of the fill. And this would be the mass flow rate at the end of the fill. And the same types of approximations we were using for the specific heat. When we wanted to invoke assumptions of a constant specific heat we would consider here. For example oh, well let's say for only filling for a very short period of time maybe I can assume a constant value for the mass flow rate. And that's just an average between let's say this initial T1 and this initial T2. On the other hand, if it really is something that, if it really is conditions that span from this really high mass flow rate at the initial point. To really low mass flow rate at the final point, I might need to do a polynomial fit to that process. I might want to do a piece wise linear. But you can imagine you now have to take that information, and you need to integrate it in this expression. Now the next step would be then to consider the conservation of energy. Which is also effected by non steady state condition within the fill chamber. So remember before we said during the fill process, that the change in the internal energy in the control volume. Was balanced by the enthalpy from the fill chamber, and then the mass flow rate in. Well, in these conditions again, doing a little bit of simplification here, from the initial state to the final state. So this would now be, sorry, integrated over a function of time, T1 to T2. And you could, potentially, have two things changing, two variables changing simultaneously as the function of time. So the state within the fill chamber could be changing. And typically, we would say, okay, we'd look at how the enthalpy changes as a function of temperature. So, how does the temperature within the tank change as a function of time. And we'd have to have that convolved with. The mass flow rate changing as a function of time. So you can imagine these get very sophisticated, these transient analysis with not just time varying within the control volume. But these fill problems or draining tank problems, also become very complicated. when we consider the system to be changing, during the process that we're concerned about. Okay, so again, we try and design the systems or interpret or make assumptions that make them as simplified as possible. Make them tractable. So, when you consider transient analysis you know, try and invoke appropriate assumptions. So that you can make the system analyzable, so it's tractable. You can do the analysis without overstepping or overreaching the assumptions. You could also consider limiting cases. Choose a constant mass flow rate, but the highest mass flow rate. Choose a constant, or assume a constant mass flow rate, but the lowest mass flow rate. And those can bound your solutions. So those are all sorts of methods that we can use to make the system tractable for our consideration. Alright, so we're pretty empowered right now. We have thermodynamic state relations at our disposal, we have simplified models. We have like, ideal gas model, incompressible substance model, we have the two most powerful conservation equations. The conservation of mass and the conservation of energy. So let's take a step back. There's another, really critical application for thermodynamics which is power generation. So that's typically either stationary power or power in the mobile sector, so transportation market. and we want to see how those systems are analyzed and being considered from a dynamic standpoint. So the first thing we're going to consider are power cycles and cycles have some pretty unique features and let's start with those. So these are generic features of cycles. A cycle, by definition, has the initial state equal to the final state. So, we'll write that down, by definition, a cycle Ends where it begins. That sounds very philosophical. What we're saying is that the initial state for a cycle from a dynamic state is by definition equal to the final state of the cycle. Okay, if that's the case, lets consider some power generation, a power plant or it doesn't matter if its stationary or a mobile power plant. Like a diesel generator or diesel engine, we both consider them from this perspective in the same way. Lets look at that system, and consider the application of the conservation of mass. So, excuse me, conservation of energy of that system. So the conservation of energy, we're going to assume its a closed system. A closed power cycle and remember in it's most general form, the conservation of energy says. The change in energy is equal to the heat transfer, the net heat transfer in the network transfer out of the power cycle. So this is generic for any closed system. Well now we can just add subscripts to say hey, this is specifically for a cycle, and we look at the definition for a cycle. Well heck, the the initial energy, and the final energy of the cycle are, by definition, equal. So, that means this is true, and we look at the right-hand side of the expression for the conservation of energy. And we say for cycles by definition, the net heat transfer has to be equal to the network transfer. This is a statement of the first law of thermodynamics. It is always true, so as long as you can define a system as being a closed cycle This is a statement on the conservation of energy. And it has to be true, it doesn't matter what the fluid is, it doesn't matter what type of power plant it is. It's always true, it's a statement of the conservation of energy. So, again, always true. [SOUND]. All we have to do is make sure that our system can be described as a cycle, and that it does indeed have the initial state equal to the final state. Okay, so that's the generic description of a cycle. there are essentially three categories of cycles, and it's more like two categories, because two of them look a lot alike. So, in power cycles, the goal is to actually generate power, or work work ra, a rate of our transfer out of the cycle. The other flavors of cycles are refrigeration and heat pump cycles. So the goals of those cycles are actually to move heat around to move for heat transfer. So to provide a particular type of heat transfer rate. So under the Power Cycle category, our goal here is to provide net, work, transfer to the surroundings. Okay, and the generic power cycle of, which is also referred to as a heat engine, and we'll explain why, a little bit later. Why they're called heat engines. But for power cycles, we just set them up for our initial conversation as being some generic, some arbitrary system. And that system is supplied with heat transfer from a high temperature reservoir. So we have heat transfer that goes from this high temperature reservoir into our power system into our power cycle. And we generate work that's the goal of this cycle. And of course we're considering this analysis from the viewpoint of where the power system, the power cycle is the system. And that power cycle has to reject heat to some cold temperature reservoir. So, this is a high temperature reservoir, and this is a low temperature reservoir. Okay, now, I can tell you that every power plant that generates stationary power electricity fits within this category. There are some very small fraction that don't fit within this general description. But the majority of them, over 90% of them look like this, kind of generic description. And the only difference between the power plants, whether it's a solar powered power plant, a nuclear powered, coal fired, natural gas fired. Is in the details. But they all have the goal of generating heat, that's this high temperature reservoir. So in the coal fired power plant, we're going to burn coal. And we're going to use that to heat a working fluid. So our working fluid is within this box here. And that working fluid is going to then generate power. So lets be even more specific, lets take our natural gas. Natural gas is a relatively clean burning fossil fuel. So we burn the natural gas that provides us with heat. We use that heat to heat water. And then that water is, goes from a solid, excuse me from a liquid phase to a vapor phase. And that vapor phase has a lot of energy internal energy associated with it. We saw that when we looked at our turbomachinery analysis. and specifically steam turbines. So I take my liquid water and I heat it with the heat from combustion of fossil fuels, that takes it to a high temperature steam or a high temperature vapor. That high energy vapor is then expanded through a turbine. The turbine spins a shaft and out of that shaft we connect it to generators, then we connect it to the power lines and that gets us power out into society. Okay, all right. We know that as we move that steam through the turbine we go from a high energy state to a low energy state. And then we want to close this loop so we take that low energy vapor and we condense that into a liquid. And in order for us to condense the water, we need to remove heat from that working fluid. And that's this portion of the power cycle. So we reject heat out of my power plant and into some cold temperature reservoir. Okay, so that's the generic description. I gave you the example of a natural gas, fired, steam, cycle. And we will go through the details of these cycles. That's what we're going to spend some, some time on in the next few segments. What I want you to think about before we come back, is where would a nuclear reactor fit in this little description of a, of a power plant? So what does the nuclear reactor do? The actual nuclear reaction itself, and where would we kind of plot, plot down that nuclear reactor in this little schematic diagram. And that's what we're going to start with next time.
