Okay, so last time we talked about a generic power cycle. So, we need to have some arbitrary high-temperature reservoir to provide heat transfer at a high temperature. We need some arbitrary low temperature reservoir, what we're going to reject heat. So, we have our heat transferred into our power cycle and we have our heat transferred out of our power cycle. And then, we have our kind of black box as it were. That is the power cycle itself and that's going to be how we generate the work, or the power, out of this system. So, where does a nuclear reactor fit? Well, if you aren't familiar with the nuclear physics, the typical uranium reactors that we use in the United States and most of the reactors around the world, those are used purely to generate heat. So, the nuclear reaction itself is incredibly exothermic, and an exothermic reaction is one that releases heat. So, they have huge amount of heat that's released with each nuclear reaction in those reactors. So, when we look at this generic description, the nuclear reactor is the high-temperature reservoir. And then, it's used to heat steam, so here's my little nuclear power plant, those are the cooling towers which would actually be down here. But to represent our nuclear power plant, here is where the reactions are occurring. And again, that heat is then directed into a steam power cycle. And we'll talk about those in detail in just a couple of segments. But before we get into the details of the, of the components, you know, what's the hardware in power plant? What we want to talk about is how do we, what's the criteria for evaluating a good power plant from a thermodynamics perspective? So, what we're really talking about, what metric we're defining is the efficiency, the thermodynamic efficiency of our power plant. And so for a power cycle, the generic way that we define efficiencies is what you want divided by what you paid for to get it. So, in general, we would say our performance metric is what you want and what did you paid for to get it. So, this is, you know, kind of a really vague way of describing things. What do we mean by what did you pay for to get it? Well, in the thermodynamic example, that we have right here. What you paid for to get it was this heat transfer. And if we want to be even more specific, we can say, okay, well we know there's only so much heat transfer, that let's say a ton of coal can, can provide. Or a ton of uranium, or a ton of natural gas. So, we can be very specific on the heat transfer from our particular heat source, okay. So, that's what I paid for to get it. We're going to go ahead and, and call efficiencies. We're going to use the Greek letter eta here to represent the efficiency of a power plant. And so, this is, we can put a p on it, just to show us, emphasize that this is the efficiency. And so, that efficiency, we know that what we paid for is this heat transfer. Okay, what we want out of the power plant seems pretty obvious. It's the power, right? So, that going to be the work out of the cycle. Okay. Now, these energy transfers, these two heat transfers, the high temperature and low temperature heat transfer and the work transfer. Are all the energy transfers that we need in order for us to make this heat engine, or this power cycle? So, if we go ahead and apply a conservation of energy analysis to my system here. So, this is my power plant, and we say, again, you know, it's a closed system, the conservation of energy says, okay, well I have to balance the internal energy. The change, excuse me, in the total energy with the heat transfer, the net heat transfer and the net work transfer. Well, again, it's a cycle. So, by definition, the, there's no change in the energy in the system, so all I have are these heat transfer and work transfer terms. So, I'm going to, for cycles, this can get confusing. So, I'm going to pause here and, and try and anchor people. we're going to suspend our use of the sign convention for heat transfer and works, work transfer. Or, maybe suspend isn't the right word. We're going to impose what we already know should happen. Okay. What does that mean? Well, we know that the work transfer for the cycle, by definition, has to be equal to the heat transfer. Net work transfer, net heat transfer. Well, in my little generic description up here, I have the work for the cycle, and I only have two heat transfers. If this work is out of the cycle, which we want to be positive, then by definition the heat transfer into the system has to be larger than quantitatively the heat transfer out of the system. So, I'm going to write it like this. And recognize, our sign convention would say, hey, Q out should have a negative sign. But I'm already taking that into consideration here. I've already imposed that negative sign. So again, I guess you could say we're applying our sign convention on priority and that these heat transfers and these work transfers are all going to be absolute values. OK, so hopefully that doesn't disturb people too much in terms of understanding the sign convention, we know there should be work out. We know that the heat transfer in has to be larger than the heat transfer out. Okay. so then we take this information here, and you'll notice I'm moving back and forth between heat transfer. And heat transfer rates, and we should be completely flexible. Typically, we model these systems as having one mass flow rate, so we can use either convention very easily. If there's more than one mass flow rate in this system, then we have to be cautious about moving back and forth. Interchangeably between work between power and heat transfer rates, and work and heat transfer. Okay, but we understand, hopefully we understand how to make those transitions very well. Okay. So, then we plug in this expression here. For changing everything into heat transfer. I'll go ahead and put everything on a rate basis here. And what we see is that the power, the efficiency of my power plant is given by this very simple expression. Okay, We also recognize for efficiencies 100% is ideal and of course achievable. But this is kind of a typical definition of efficiency. Right? No one can be 100% efficient, as much as we'd like to be. So, we recognize that there are bounds on the efficiency. We have a lower bound of zero. We have an upper bound of one. We also recognize, we'll never get to 100% efficiency. So, we know that the power cycle, the range of the efficiency should be somewhere between zero and 100%. And we'll talk about what sets the limit for the best we could expect out of power plants. that's also governed by very explicitly thermodynamics. But under, by using some very advanced topics. So, we're going to take the outcome of those advanced analysis, but we're not going to discuss how the, the specifics of the analysis. But it will give you the ability to take critical view of every power plant and see what's the maximum efficiency you might achieve. Okay, so these are power plants, this is the generic definition for a power plant regardless of whether it's nuclear, whether it's a diesel engine, whether it's a solar. I'll be very careful we'll do some solar examples. Some solar power plants are in fact heat engines. Solar cells can be part of a heat engine process, but aren't themselves a power plant. So, that again, gives you, hopefully that will wet your appetite for some of the applications we'll consider coming up in the next couple weeks. So, let's consider refrigeration and heat pump cycles. So, these are examples of these are also cycle systems. But the goal here, instead of being power, like the power out of a power plant is our goal, for refrigeration and heat pump cycles, the goal is to actually move heat. So, in refrigeration or air conditioners, right? Your goal is to actually cool your house or cool your building. So, it's not a, the desired output isn't work. The desired output is the heat transfer. Okay. So, for these refrigeration and heat pump cycles. We're going to actually, their described by essentially taking our power plant and operating it in reverse. So, we are going to still have a high temperature reservoir, so here's my T-High. And here's my low temperature. and what we're going to do is, we're actually going to use power to drive the refrigeration cycle. So, we actually have work transfer into the cycle. So, this is my refrigeration or heat pump cycle, and I'll tell you why we bundle them together like this in just a second. So, we have heat transfer since we've reversed the direction of the work transfer, we also need to reverse the direction of the heat transfer. So now, I have heat transfer from the low temperature reservior into my system. And I have heat transfer from the system and into the high temperature reservoir. Okay. Again, network transfer for the cycle has to be equal to the net heat from the cycle. Again, we know that we have systems where the work transfer, if we write it in this direction, we'd have a negative sign because it's worth transferring into the system. But we like to work in absolute units when we consider this power plant in heat pumps and refrigeration cycles. So again, I'm going to, I'm going to kind of suspend the sign convention and say, hey, we're smarter than that. If you want to, you could put an absolute value sign around it. But we understand that this is work transfer into the system. And so again, we just have two heat transfers in the system and that's the Q out, which is also Q, called Q high here. And this is Qn, so Q out minus Qn. And again, work transfer for the cycle. Okay. And again, I'm writing this because where the heat transfer out again would be considered a negative sign. And heat transfer here again using a sign convention would be a positive sign. But we know for this system to actually function, the work, the absolute value of the heat transfer out of the system has to be greater than the heat transfer into the system. So I'm going to just emphasize that here. So again, this really kind of wreaks havoc on our sign convention. But this expression really is just meant for us to be able to apply this cycle definition. Okay. So, for performance, we don't use efficiency as our metric for evaluating the performance of refrigeration in heat pump cycles. Instead, for refrigerators or for heat pumps we define coefficients of performance. So, the refrigerator, the coefficient of performance is denoted by a Greek letter, Beta. And again, we're still going to use the same general description of evaluating performance. Which is what you want. Divided by what you paid for to get it. Well, what do we pay for? Well, the power that we draw from the grid, or the work into the cycle. In the case of a refrigerator, what we're trying to do is actually cool a particular space. So that's actually Qin. In the case of a heat pump, we're trying to move the heat transfer, out of a particular space. So, in that case, heat pumps, the coefficient of performance, again we're going to use a Greek letter. In this case, a gamma. And here, what we're trying to achieve is to move the heat transfer out, Q out. and the cost, what we pay to for to get it is the work out of the cycle. Okay. You can go through this kind of logic analysis of, well, we know that the heat transfer out has to be greater than the heat transfer in. And you can do these order of magnitude and what you would find at the end of the day is that the Gamma, the heat pump coefficient of performance, can never be less than 1. and probably more important and more relevant, we always want the efficiency of the power plant, the coefficient of performance of the heat pump and of the refrigerators to be as large as possible. High, large big. And it's the second law of thermodynamics. Which tells us what the limit is for these devices. So again, second law, we won't cover the details, but we will use the outcome of the second law of thermodynamic, thermodynamic for us to define the limits on these systems. Okay, so I've described a power plant. Last time, I asked you to tell me where the nuclear reactor, where the actual core of the nuclear reactor fit within that system. Now, what I want you to do is take a look at this generic refrigeration system, and I want you to think about where is the actual, let's say, ice box portion of the refrigerator? Where is the cool, inside portion of the refrigerator in this generic sketch? How does that fit? And that's where we'll start next time.
