 When we talk about thermodynamic cycles, we are often talking about the interaction between heat and work. As a fun fact, that's where the name thermodynamics comes from. But before we talk about heat and work, let's talk about water and work. If you imagine water at a high elevation, it is naturally driven, by gravity, to lower elevations. We are able to take work and push it back up the hill, but going down the hill happens for free from our perspective. If we were to build a water wheel, we could tap into some of that naturally occurring transfer of energy to produce power. In that analogy, heat is water, and elevation is temperature. When we talk about taking a high temperature and allowing it to transfer heat to a lower temperature, which happens for free from our perspective, we can generate work from it. That thermodynamic cycle is called a heat engine. It taps into naturally occurring heat transfer, by which I mean heat transfer that happens for free from our perspective, and produces work. And just like how we can take work and pump water back up the hill, in this analogy, we can take work and push heat back to a high temperature. That thermodynamic cycle is called a refrigeration cycle. We are consuming work to push heat in a direction that it does not naturally want to go. When we talk about evaluating performance of a heat engine and a refrigeration cycle, we are describing an efficiency. And like all efficiency, it is the proportion of what you get out of it to what you put into it. Desired effect over required input, if you will. In a heat engine, our desired effect is network out. And what we put in to make that happen is heat transfer in. Therefore, our evaluation of a heat engine's performance is going to be the proportion of network out over heat transfer in. We refer to that parameter as the thermal efficiency. Thermal efficiency is abbreviated with an eta subscript th. And it represents how much of the heat transfer you put in is converted into power, into network out. If you put in 100 kilowatts of heat transfer and got 60 kilowatts of network out, that means you have a thermal efficiency of 60%. Furthermore, because the heat engine itself has to obey the first law of thermodynamics, that means that you would have 40 kilowatts left over that leaves as heat transfer out. In a refrigeration cycle, what we're putting in to operate our cycle is network. And the quote, desired effect, unquote, can either be heat transfer in or heat transfer out. If you're trying to refrigerate a space, you orient your refrigeration cycle so it's pulling heat out of the space you're trying to cool. But you can also use a refrigeration cycle as a heat pump. You can push heat into a desired space and heat it. Because of that, we have two different ways of keeping track of the performance of our refrigeration cycle. When it's operating in cooling mode, subscript r, what we're talking about as the desired effect is heat transfer in. Therefore, our relevant performance parameter is the heat transfer in divided by the network in. What we had to put in to make that happen. When it's operating in heating mode, subscript HP for heat pump, our desired effect is heat transfer out. Therefore, our performance parameter is heat transfer out divided by net power input, the network in. Furthermore, even though this is really an expression of efficiency, we call it COP or coefficient of performance. The only reason we call it something other than efficiency is because it's possible to have a coefficient of performance that's higher than 100%. And that tends to be a little bit confusing when you're talking about efficiencies of 300% or 400%. So COP is chosen and referred to as a number as opposed to a percentage, just to make it easier to talk about it. For example, the refrigerator running in my kitchen might be moving more heat than I'm paying to operate it. If it moved three kilowatts of heat for every one kilowatt I put in, that would be a coefficient of performance of three. And that's not violating any laws of thermodynamics, it just means that I'm getting a lot more payoff for my investment of work, which is completely fine. So the two most common thermodynamic cycles that we consider are the heat engine which taps into naturally occurring heat transfer, or rather heat transfer that is occurring for free from our perspective to produce net power. And a refrigeration cycle where we are paying work to move heat in a direction it would not naturally want to go. Another parameter that's useful when we're talking about the performance of thermodynamic cycles is the theoretical maximum thermal efficiency or coefficient of performance. When we are considering the best case scenario, we are treating everything as being as ideal as it possibly can be. What the operation of the device would be if conditions were perfect. And in that case, we are treating it as a Carnot cycle. And specifically when we treat it as a Carnot cycle, we substitute the relative proportion of heat transfer for the relative proportion of temperatures driving the heat transfer. Note that this is written as the heat transfer on the high temperature side over the heat transfer on the low temperature side as opposed to Q in over Q out, because the direction of the heat transfer exchanged with either temperature changes. In a heat engine, Q H is Q in. In a refrigeration cycle, Q H is Q out. The metaphor I like to use to think about making this Carnot substitution is that waterfall again. If you imagine a situation where you built a water wheel halfway up the waterfall, then you can only tap into half the available power available in the waterfall. If you were building the water wheel a third of the way down, then you can only tap into a third of the available heat transfer. Therefore, you get a third of the available performance. In order to calculate the theoretical maximum, we're going to be plugging in the heat transfer occurring on the high temperature and low temperature sides so that we can substitute in the temperatures on the high temperature and low temperature sides. In order to be able to rewrite this equation in terms of just heat transfer, I'm going to have to consider an energy balance on the cycle itself. If I draw a control volume around my heat engine and I assume that it's operating steadily, then I can treat it as a closed system. If I have steady state operation of a closed system, then I'm going to be writing out the rate form of my energy balance, which I got by dividing everything by dt and then recognizing that the energy in my control volume isn't changing with respect to time. That means e.in is equal to e.out. If it's a closed system, then e.in could be q.in plus work.in and e.out could be q.out plus work.out. Since I'm describing a network out, I recognize that that's a work.out minus work.in, which, because of my energy balance, must be equal to q.in minus q.out. Similarly, on the refrigeration cycle, if I draw the same control volume, I can end up writing the network in, which is work.in minus work.out as q.out minus q.in. We can see that visually on our diagram here by recognizing that if we have, say, four kilowatts of e.transfer.in and we are paying one kilowatt of network in, that energy has to go somewhere. It has to leave as q.out, because there are no other opportunities for that energy to leave my system. By the way, while we're here, what is the coefficient of performance of this system? The correct answer is, it depends if we're using it as a cooling device or a heating device. If we're using it for cooling, then we're taking q.in over the network in, which means we have a coefficient of performance of four. If we're using it for heating, then we're taking q.out over network in, which means we have a coefficient of performance of five. Here's another flawed analogy for you. I like to think of the refrigeration cycle as a shop vac. A shop vac allows you to connect the hose to the input side or the output side. If you attach the hose to the input side, you're using it as a vacuum. If you attach the hose to the output side, you're using it as a blower, but the mechanism in the middle is the same. Whether you attach the hose to the inlet or outlet is affecting what you're using the device for. We could stick our refrigeration cycle into a room, such that it was pulling heat out of the room, in which case we would be operating in cooling mode, but we could pull that device out, rotate it around, stick it back into the room so that it was pushing heat into the room, and all of a sudden that same device is operating as a heating mechanism. But armed with this substitution, I can plug that into our thermal efficiency and write this as Q dot in minus Q dot down, divided by Q dot in. And for convenience, I can split the denominator and write this as 1 minus Q dot out over Q dot in. This substitution so far is not considering best case analysis. This is only rewriting for the case that we have a closed system operating steadily. To continue, we'll recognize that in our heat engine, the heat transfer out is the heat exchange with the low temperature side, and the heat transfer in is the heat exchange with the high temperature side, which means I can write this as 1 minus QL over QH, which, when I make my Carnot substitution, is going to become TL over TH. So regular thermal efficiency is the rate of network out over the rate of heat transfer in, which is also equal to the magnitude of network out, divided by the magnitude of heat transfer in, which is also equal to the specific network out over the specific heat transfer in. And the theoretical maximum thermal efficiency that could occur if everything were perfect is 1 minus the low temperature divided by the high temperature. And that's the temperatures of the thermal reservoirs from which the heat is exchanged. When we consider the refrigeration cycle, we're going to start with the same substitution. We're going to substitute network in, which is work in minus work out, with Q dot out over Q dot in. And then, in order to be able to write this as a proportion of heat transfers, I'm going to bring my numerator down to the denominator. When I distribute that inside of the parentheses, this becomes 1 over Q dot out over Q dot in minus 1. I can do the same thing with heating mode, at which point this becomes 1 over 1 minus Q dot in over Q dot out. When we're considering the refrigeration cycle, the heat transfer out is the heat transfer exchanged with the high temperature side and the heat transfer in is the heat exchanged with the low temperature side, which means this is going to be equivalent to 1 over Q H over Q L minus 1. And this one is going to be equivalent to 1 over 1 minus Q L over Q H. Then when I make my Carnot substitution, substituting the proportion of heat transfers with the high and low temperature, with the temperatures of the high and low temperature, I have 1 over TH over TL minus 1, 1 over 1 minus TL over TH. So we have ways of representing our thermal efficiency in terms of works and heat transfers, and a way of calculating the theoretical maximum. And we have ways of representing the coefficients of performance in terms of heat transfers and works. And we have ways of relating the temperatures, driving the heat transfer or receiving the heat transfer with the theoretical maximum coefficients of performance. The last thing I'll leave you with is, depending on your textbook, you might see COPR called beta and COPHP gamma. But COP is the ASHRAE standard of referring to those two parameters.