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Current time:0:00Total duration:14:01

Carnot efficiency 2: Reversing the cycle

Video transcript

in the last video I showed you that the definition of efficiency ADA is the work that we do given the amount of heat we are given to work with and we show that for an engine that could also be rewritten as 1 minus q2 over q1 essentially the heat that we output from our heat engine divided by or 1 minus the heat we output from our engine divided by the amount of heat we input from our engine now we apply this formula to a Carnot cycle and we said hey for a Carnot engine we could get an efficiency we get an efficiency of this so I let me write this here so the efficiency for a Carnot so ADA for Carnot is 1 minus t2 over t1 to get this result I we had to use the fact that we were dealing with the Carnot cycle to use you know we were moving along these isotherms and I was able to take the natural log of home and and do all of that and I was able to get this for the efficiency of a Carnot engine and let me be very clear this is the efficiency that can only be attained by a Carnot engine the other definitions of efficiency so when I just defined efficiency is equal to the work performed divided by the heat and we call it the heat input or when I defined it as when I define it as the change the net the net heat let me put it that the net heat in so q1 minus q2 over q1 this applies to all heat engines this is true for all heat engines including including the Carnot engines and a heat engine is an engine that operates on heat I probably should have said that a while ago and you know these this engine that I made this Carnot engine is definitely a engine that is operating on heat because it's taking heat here and later it releases the heat down here the cycle just shows what's happening to that engine I just want to make that distinction to the engine is the actual physical thing the cycle just describes what's happening to it so with that said I said that this is only true for a Carnot for a Carnot heat engine now what I'm about to embark on and I don't know if I'm going to finish it in this video it might take into the next video to do it properly is to show you that if we're operating a heat engine between two tempered sources so I have my hot temperature source I'll call that th 40 hot and I have some it's an H not an N it's transferring some heat q1 and some other heat is coming out at q2 at q2 and I'm performing some work and then q2 and then my my other cold temperature reservoir I'll call that T cold is down here and that's where I'm releasing the heat to I'm going to show over the next few videos that the most efficient the most efficient engine is this theoretical Carnot engine that no engine can get more efficient than this so this is a Carnot engine this is the most efficient engine the most efficient or this is the ideal where nothing is lost well I'll go into that in a more detail no engine can get more efficient than this then this Carnot heat engine so to get there to prove it to you I'm gonna I'm just gonna play with the Carnot engine a little bit just to show you some of the tools that it has at its disposal so one of the things let me just draw a PV diagram it's P that's V let's see whoops wrong tool P and V P and V and the Carnot cycle we've done so far we kind of always moved in one direction we had our isothermal expansion it went something like it went something like that that was isothermal then we had our adiabatic expansion and the whole time we were going in that direction then we had our adiabatic expansion and it went like that then we had our isothermal contraction it went something like this it went something like that and then we had our adiabatic contraction to get to where we were to begin with so then we went back like that and the whole time we went in this kind of clockwise direction we went in the clockwise direction and we took in heat up here because we were doing work we took in heat to keep our temperature constant and then we released heat here to keep our temperature from going up from q2 and so if I were to draw another draw this another way I could like well I just did one like that but let me draw it like this I could also depict it like this where that's my engine this is my hot reservoir let me put this is t1 t1 is up here it transferred q1 to my Carnot engine my Carnot engine did some work did some work and then left over or transferred into my cold reservoir t2 it transferred q2 this is another way of depicting what went on in this Carnot cycle and here I've actually drawn the engine now one of the tools I want to show you is that and is that this is a reversible reaction or or that we can take this and go the other way around and it's dependent upon an assumption that I threw out a long time ago so when I first drew these I kind of introduced you to the idea of a quasi-static process quasi quasi static and quasi static just means look you do it really slowly so that you can always say that you're pretty enough close enough to equilibrium that you're your macro state variables are always defined and that was the whole justification for dealing with pebbles like this but instead of just doing it wholesale instead of just moving all the pebbles and just getting to this state from A to B kind of jumping I wanted to do it gradually so that I would be defined at every point in between that's what quasi static did for us and when I actually made the video on quasi-static processes I said you know quasi-static processes for the most part are reversible and sometimes I used I use the words interchangeably now by definition our theoretical Carnot cycle is said to be not only is it quasi static but it is also reversible which means at any point in time let's say we've moved we've moved a couple of pebbles and we gotten we've gotten right here if we want to if we're in the mood we can add some pebbles back and just follow this right back to where we were that's what reversible means now I mean it means you can reverse something now what what what do we what has to be ideal about the system in order for that to be true well it means that there was no that with that the actual movement of our of our piston of this movable ceiling that it shouldn't have any friction because if some of the heat is lost to friction then when we go back we would have lost some of our heat some heat would have been destroyed just going from one state forward and back so the assumption that we have to make for the Carnot cycle is that or the assumption we have to make an order for the Carnot cycle to be reversible is that it's frictionless frictionless frictionless so the Carnot heat engine this theoretical engine is a frictionless engine which is theoretically impossible to be completely frictionless to be but well we'll talk more about that in the future so I want it so if you have a completely frictionless engine and it's quasi-static it's also reversible so if we want to do it reversible what does that mean it means I could start in this state my state a that I've labeled before but instead of going around that way I could go around the other way so what I could do first is I could I could adiabatically expand first so let me do let me do it maybe let me let me redraw it so I do it the other way so I could reverse this reaction and it would happen the exact same way and that's an artifact of that I'm always in equilibrium and that my system is frictionless that I don't lose energy just going back and forth so I could start at state a here and then I could adiabatically contract adiabatic contraction would look something like this and I'll get to that state then then I can isothermally expand so I'm going like this then I can isotherm eclis expand and as isotherm eclis expand let me so I'm going like this ice I'm all along an isotherm I'm doing some isothermic expansion so in this case I'm doing work but I'm doing worse work isotherm eclis right at some at some cold isotherm I'll call it a let's call it t - right just like this was T - so in this case if I'm if I'm expanding and I'm staying at t2 and I'm sitting on top of my t2 reservoir heat is coming he this area under the curve the work I'm doing is the heat added this is q2 and that is given to me by by my t2 reservoir so everything is going in Reverse that's the whole idea that I adiabatic Li contract I can adiabatically contract like that and then I isothermic Li contract then I isothermic Li contract like that to get back where I start when isothermic Li contract what's happening work is being done to me so now all of this area over here will be negative and in order to keep my temperature constant I have to release heat so I'm releasing heat but I'm doing it at a high temperature so I'm releasing it into my t1 reservoir so it's the exact same thing is it happened before but since when I go in a reverse direction I'm actually some work is being applied so now if when you look at it this way when you figure out all the areas the area in here will actually be negative and the reason why I'm saying that is because the positive work values are going to be this this sir this is going to be the positive what I'm doing in blue right here and the negative work values are going to be what I'm will be all of this stuff so if you wanted to figure out the total work done it's gonna be negative so what's happening if I run the Carnot cycle in Reverse so let's call it I'll call it the Carnot refrigerator or no and that's not what I wanted to do I'll call it Carnot reverse but it's handy that are also stands for a refrigerator Carnot this is the Carnot engine it does work by using heat by by by taking advantage of the heat the difference between this hot you could have you this as a tea hot and the tea cold now a Carnot a reverse Carnot engine or maybe we call it a Carnot refrigerator does the opposite that's exactly what I just drew over here what it does is it starts with the cold body it starts with the cold body I'll call that tea cold or t2 it takes some smaller amount of heat it takes some smaller amount of heat from the cold body from the cold body some work has to be input into this system in order to do this and then it puts more heat kind of you could have a kind of use a combination of this work and this and this heat taken from the cold body and it gives it to the warm body sorry this is q2 and it gives it q1 so everything just happens completely in Reverse and that's just a byproduct of this is reversible so I can just go and I can do this this is the way we went before when we're an engine if we want to be a a refrigerator we go the other direction and everything just gets reversed and I want you to really understand that this is doable that this is there's no there's nothing wrong with this you might say hey doesn't this defy the second law of thermodynamics we're going from we're taking heat from a cold body to a warm body and my answer will be the same thing I said on my entropy videos I said well no we're applying some work this is a refrigerator so some work has to be done in order to do this and whatever object that is doing the work maybe some in the case of your refrigerator it's a compressor that is adding more entropy to the system than the entropy that's being destroyed or more entropy to the universe than the entropy that's being destroyed by our refrigerator so this does not defy the second law of thermodynamics now I want to make another I want to make another point about the Carnot engine let me take the reverse Carnot engine I'm going to call it a real Carnot refrigerator so if we take that and this is really just more math than anything else if we take that if we're taking if we're taking in q2 adding in some work and producing q1 we can scale this up arbitrarily we can scale this up arbitrarily if we take two times if we take let's say we take x times if we take x times q2 in and we do and we put in x times W in then we're going to put x times q1 into our top reservoir and that makes sense because these are just arbitrary numbers so if we do if we have for example if we have two Carnot engines in parallel you can just kind of view those view that is that whole thing is two Carnot engines doing it together so all of these would be twos if we had three car engines doing it together all of those would be three and you could just view them as one collective engine now with that said I think I've laid the framework for for at least the the ideas that led us to a let us show that the Carnot engine is the most efficient engine that's abled that can be produced and given that the Carnot engines efficiency is this and we're going to prove that it's the most efficient engine this becomes the upper bound on efficiency for any engine that anybody can or will ever make and I'll do I'll kind of do the crowning touch in the next video