 So, we are on our fifth and final day and two topics remain, we have decided to de-emphasize cycle. So, we will spend major amount of time on combined first and second law. Traditionally the name of this topic is availability analysis, exergy analysis. The differences historical, sometimes availability analysis was pertaining to the close thermodynamics system, exergy analysis pertain to open thermodynamic system. So, also possible that exergy was a word coined by East Europeans and Russian engineers, availability was used by Western Europe, Western workers. But the fact remains that there is absolutely no new basic principle of thermodynamics involved. We use first law and second law together. So, that is why it is a combined analysis and that is the reason why I have kept the topic as combined first and second law. Somebody quickly goes through the topics, they will say where is availability, where is exergy? Exergy is nothing but combined first and second law. So, what is this analysis? This analysis helps us do the following. We have a vague idea that sp or s dot p which is always positive or at most equal to 0 is not a good thing, that is the basic idea. We should reduce sp or s dot p, why? Maybe it is better that way, I will put a question mark, why is it better we do not know but that is our mental set and that is based on the again a mental set that a reversible process would be an ideal process, that means it would be best possible and the reason for that is simple. One of the aspects of life which tells us that nothing is reversible is the progress of time. And it is the second law which tells us that in any thermodynamic systems behavior there will always be a state before and the state after whatever happens. If you have a situation in which everything is reversible, remember that when a system executes a reversible process or a pair of systems interacting with each other execute a reversible process, in principle it is possible for us to reverse everything and bring the systems back to the original state without any other thing changing. So, if a system has executed a reversible process and then reversed it back and if we were not observing that a process was taking place, we would never know ever that a process has taken place. If in a particular zone say in this room everything were to take place reversibly and that means our clocks would have to stop because there is no before there is no after. There is no before there is no after so time would lose its meaning and everything would become you know amar in our as we say undying and perhaps that is the idea right from our philosophers. So, amruta was always an attractive thing the elixir of life. That gave us an idea that something which extends life, something which takes meaning away from time would be important. In thermodynamics that attracts us to the ideal of a reversible set of processes. We have used it for defining entropy and doing many other good things. That has given us a mental behavior, mental set that a reversible process is ideal since an irreversible process produces entropy, entropy production is bad and that has given us led us to the quest which has led to exergy analysis. For every process let us calculate sp or s dot p. We know how to do it from our second law analysis. But by combining the first and second law we are able to provide some physical meaning to the value of sp. And then we can make efforts to see how it can be reduced and when you reduce it what happens or what do you have to do to reduce it. So our thing is presence of sp or s dot p would lead to loss of work. That means you could have obtained a work say maximum amount of work you obtained only the actual amount of work W. So this would be the lost work. What is the maximum amount of work that we have yet to determine or it could lead to you can look at the other way. Waste of heat, we know heat has to be supplied to do some work, to do some job, to provide a change of state, to accomplish a task. So the thermodynamics will give us the minimum amount of heat which is required to do the task. But if we consume something more than that this is the amount of waste of it. You could have done with this but you are unnecessarily using something more. So the combined first and second laws together of the exergy analysis would give us an idea of either the loss of work or lost work or the waste of heat or wasted heat. Usually people look at it essentially from the lost work point of view but we can always turn it around and look at it from a wastage of heat point of view. How do we do this? We do the following. We use first and second laws for the actual process. Then we think of an ideal process and naturally for an ideal process our idea is an ideal process must be reversible and that means for such a process S p should be 0 or S dot p should be 0 whatever way you are looking at it. Then we define using the difference between the two typically lost work and link this up to S p or S dot. And while doing this we often come across the components of the process which contribute to lost work. I will take a diversion, we will go to our exercises and we go back to the last exercise in second law 4.10. I skipped it at that time because although it uses first law and second law I think it is appropriate to do it at this time because it is related to what we are going to do now. But just to emphasize that nothing more than first or second law is to be used we will do it now. So exercise 4.10. So we have a closed system it undergoes a process between fixed initial and final states say one and two such that the overall changes in energy entropy and volume are delta E, delta S and delta V respectively. That means you have a closed system which executes a process such that change in energy is delta E, change in volume is delta V, change in entropy is delta S. The only input of energy to the system is from condensate steam at a temperature T s. So there is a temperature T s. Let us say the heat absorbed is given amount of heat absorbed from steam by the system is Q from the steam by the system so our direction is all right. The only outputs of energy are the heat transfer to the environment it is given that T naught is the temperature of the environment. So let us say the interaction with the environment which is at T naught is Q naught and work done in displacing the environment because of this delta V the environment gets displaced and that is P naught that means the work done is the expansion work and since it is only against the environment it would be P naught delta V. Apply the first law to the system, apply the second law to the system show that the irreversibility in the process gives rise to a wastage of thermal energy that means wastage of heat of steam by an amount as given. So let us do how do you apply first law to the system? First law to the system simply means Q equals delta E plus W but there are two components of Q here. So the first law would simply say Q plus Q naught is delta E plus W, W is P naught delta V. If you really want to be particular call this Q s write the first law as Q equals delta E plus W then write Q s, Q s plus Q naught and W as P naught delta. I am taking a bit of a short here. Second law, what does the second law says? Delta s must be greater than or equal to sigma or integral Q by T plus SP. Let us put it the turn it around we will put delta s on one side but on this side Q by delta Q by T has two components Q by T s plus Q naught by T naught. This has to be change in entropy of the system delta s plus any entropy produced. Actually this is not second law this is a relation linking SP to all this second law actually say SP must be greater than or equal to Q. This is definition of SP this part is definition of SP this part is this is the real second law. So let me call this equation one let me call this equation two it is minus SP. Let me correct it and say this has to be plus SP here equal to delta s. Multiply two by T naught what do you get? You will get Q into T naught by T s plus Q naught plus T naught SP equals what is on the other side T naught delta s this is two A or you can call it three. We had the first equation Q plus Q naught is delta E plus T naught delta Q plus Q naught is delta E plus T naught delta. You can eliminate Q naught between the two and you will get for example you subtract two A from one. So you will get Q into one minus T naught by T s minus T naught SP equals delta E plus T naught delta V minus T naught delta s solve this for Q equal I will put everything on one side I will have delta E plus T naught delta V minus T naught delta s plus T naught SP this is T s minus T naught by T s. So this will be T s by am I right? Now notice that SP will have values which are zero or positive thermodynamically the ideal value is zero and notice that as I reduce SP I will come to a minimum value of Q. I am assuming here that T s is greater than T naught and that is a inherent assumption here. So that gives us Q min which is obtained by putting SP equals zero this would lead to Q min which is delta E plus T naught delta V minus T naught delta s into T s and Q by Q min is Q wasted that is what the answer we have Q minus Q min the value of Q min is as given all that we have done is we have used first law we have used second law and notice here that the interaction with the environment Q naught that we eliminated and we said that using the link between Q and SP we determine Q min which is at SP equal to zero. So anything which is Q minus Q min would be Q wasted. Now the question that arises this is what thermodynamics and even the exergy analysis does not directly tell us what shall I do to reach this state of Q min. Notice that the first law should also be valid. So if you want Q to be reduced to Q min on the right hand side delta E and P naught delta V it is something you cannot change change in the state of the system pressure of the environment invariant you want a particular change in the state of the system. So you must execute in such a process in such a way that the if you want to minimize Q you must manipulate Q naught change Q such that Q goes to Q minimum and Q naught goes to the appropriate value which still keeps the balance. How to do that thermodynamics does not tell that is for other engineering ideas to be implemented. Now although this was in the second law section this exactly is what we do in exergy analysis and although textbooks tend to indicate that exergy analysis is some standard analysis exergy analysis is not a standard analysis. However to define properties for pseudo properties called availability and exergy the general idea which is used is essentially this but then we have it straight jacketed. So suppose we have a system and we have the interactions heat absorbed work done and let us say the system during the process goes from state 1 to state 2. What does the first law tell us? First law tells us simply delta E is Q minus 30. This is first law. The second law tells us that delta E is either integral summation whatever of Q by T plus S P with S P greater than or equal to Q. Now aim reduce S P to 0. Now notice that if you want to reduce S P to 0 this equation has to be satisfied and as you change S P either you will have to change Q or you will have to change delta S that means either you must give up some or modify some interaction or say that the system should go to a different state because I want a reversible process. You cannot say that I must have this interaction the system must go to that space in which case you cannot change S P. So if you want to improve situation quote-unquote improve by reducing S P you must allow yourself to change either the end state or the interaction or both. If you do not allow a change in the state but allow a change in the interaction so then naturally Q will change, here Q will change and then your W will automatically change. Usually a reduction in S P would mean because delta S is the same a reduction in S P generally means an increase in Q and then increase in Q here generally means an increase in W algebraically. So generally a reduction in S P if end states are kept the same improves the work interaction that means the work extracted and hence the idea of a maximum work has come about. But remember to reduce S P to 0 we must either change interactions, change in state or both. The standard availability or exergy analysis which we will now derive allows us not to change end states but change interactions to such an extent that even an adiabatic process would be replaced by a non adiabatic process. Because if you say that your original process is adiabatic that means Q is 0 and if you say you do not want to change end states that means delta E and delta S did you do not want to change but if you want to reduce S P then naturally Q has to be changed from 0 to non 0. So even to that extent one has to consider in our standard availability or exergy analysis. So we will now look at what is known as the classical analysis availability slash exergy. Since availability and exergy are used interchangeably we will say that availability generally pertains to closed system exergy analysis usually pertains to open system. And in this classical analysis what we have is the following. We have the idea of an environment usually means the surroundings and which is usually modelled as a large system or a reservoir P0 T0 unchanged. Then you say that any amount Q0 with environment is possible and this is the one interaction which we will be changing to go from the real situation to the ideal situation. Then third we say there is the idea of useful work. Here the idea is useful work is defined as the actual work done by the system during a process minus P0 delta V. Why P0 delta V? Because if the volume of the system changes by delta V or increases by delta V then the atmosphere is pushed back by that amount. So the process during the process the system will have to do work P0 delta V against the atmosphere that work will be absorbed by the atmosphere and is not useful to us for any other purpose. So that is this idea of useful work. And we do this by assuming that we have fixed initial and final state and fixed interaction except Q0 which is the energy transfer with the environment and W. Because we want to go from W to W maximum so that we must allow and the only change we allow is Q0. And now we proceed with an analysis which is very similar to that in exercise 4.10. This is our system goes from state 1 to state 2 delta E delta V delta S. If that some work W, W useful is W minus P0 delta and let us say that the other interactions are one interaction Q0 at T0 and the other interaction it say at T1 so much parallax Q at T or Qi at Ti whatever you want apply first law what do you get Q plus Q0 equals delta E plus W and W we will write as W useful plus T0 delta okay. Second law Q by T plus Q0 by T0 plus S P equals delta. Equation 1 multiplied by 2 by T0 standard trick Q into T0 by T plus Q0 plus T0 S P equals T0 delta S and eliminate Q0, eliminate Q0 between 1 and 2A. Here we will subtract 2A from 1 and we will get Q into 1 minus T0 by T Q0 gets eliminated minus T0 S P equals delta E plus W U plus T0 delta V minus T0 delta S. Let me call this equation 3. Now transpose equation by putting W U on one side and everything else on the other side W U becomes Q 1 minus T0 by that is one term W U will have minus T0 S P on the other side but what remains is the negative of delta E plus T0 delta V minus T0 delta S minus delta E plus T0 delta V minus T0 delta S minus T0 S P. Check that the algebra is right. Is that right? Okay. Now what is this? What can be the value of S P 0 or more than 0? In general it will be positive. So in general this T0 S P or minus T0 S P will be a negative number. If I reduce T0 S P or reduce S P the value of the right hand side will improve. When S P equals 0 the value of the right hand side is maximum that is our W U max. So whatever is here this equals W U max and what is this? T0 S P is W U max minus W U maximum possible work under the condition that no change in end states. Only change possible is change of interaction with the environment heat interaction with environment. This is known as lost work not with a negative sign. Just this is lost work. So the W useful is W useful max minus lost work. All that is it does is tell us that S P has some significance. Multiply S P with the environment temperature and that represents the amount of work which you could have obtained perhaps by executing the whole thing in a reversible way but did not obtain or could not obtain that is why it is known as lost work. Now in this itself there are two components. First component is this. The second component is this. This Q into 1 minus T0 by T. What is this? We have come across this earlier. 1 minus T0 by T is the efficiency of a reversible 2T heat engine working between T as the source and T0 as the sink. If Q is the heat absorbed by such an engine this is the maximum possible work one can do. So this is W U max from Q at and this now that we know this textbooks on availability we will say this is the availability or this is the exergy of that heat which is available to us at T and then they will say that look if T is higher for a fixed T0 higher the T better is the availability of Q and lower the T poorer is the availability of Q because if T0 is 300 Kelvin T is 600 Kelvin only half of Q is available as work whereas if T is 1200 Kelvin then 3 fourths of Q will be available and it so happens that what however high be T unless you go infinitely high Q will never be completely available to us as work. And that is why the idea that Q is a low grade energy cannot directly be fully converted into or comes out of this nothing to do with availability and analysis something to do with our second law of thermodynamics but this is known as the availability or exergy of Q and then those textbooks will have cranky problems that demonstrate that the exergy of heat depends on the temperature at which it is available shown by this expression. Now the second term I will write it down as minus delta E plus P0 V minus P0 S and this E plus P0 V plus P0 S it is not a property E plus P V minus T S will be a property because only properties of the system is involved but E plus P0 V minus T0 S I can call it or people call it a pseudo property because it depends on the property of the state but also depends on the environment this is given the symbol so that this can be written down as minus delta phi or minus phi 2 minus phi 1 and what is this this is WU max due to change of. So, remember this is like a potential remember yesterday's calculations in property relations during a constant temperature process the maximum work that one can obtain is the reduction in the Helmholtz function. So, during an adiabatic process in which there is no Q maximum useful work maximum useful work which can be obtained is the reduction in this property or pseudo property phi perfectly analog or in a gravitational field if a body changes the height maximum work that can be obtained would be the reduction in the gravitational potential energy perfectly analogous to that. So, this means that phi is something like a potential or why not phi you can consider to be a potential you can write I am getting both reduction in which represents the maximum useful work that can be obtained due to change of state and during that change of state what interactions are admissible the only interaction admissible apart from doing work is heat transfer with the environment this Q is not admissible because that itself will add to some work, but that Q not which you have eliminated is admissible. So, this is a potential which represents the maximum useful work due to change of state when only Q not 2 from t not 2 or from t not Q admissible. Now, is this idea clear this is simply algebra we have used first law we have used second law and we have decided that what happens if sp is reduced to 0 which gives us w u max apart from that only definitions and nomenclatures are involved nothing more nothing less and now to confuse you some more definitions and some more nomenclature phi is known as the availability of the system or sometimes it is known as the availability function or sometimes the even exergy of the system because availability and exergy are used interchangeably, but there is a modification we notice that w u max I will say Q not only would be minus phi 2 minus phi 1. So, only differences in phi are of significance just the way differences in energy only are significant and only differences in entropy are significant, but the modification to this in the definition of phi not to this is often you consider this what is p not t not what have we defined it to be pressure and temperature of the surrounding then if we say that look I have a system and I will leave it free to do whatever it feels like with the surrounding. So, I have a balloon with warm air or cold air in it and I leave it to do whatever it feels like with the surrounding feel it to exchange energy with the surrounding finally what will happen to its temperature it will come to t not I say that look I remove all stresses make it freely expandable balloon allow it to expand to its hearts content what will be the pressure that it will reach p not ok. So, p not t not which is the pressure and temperature of the environment also equal the so called state which the system may eventually reach if left free to interact with the environment this is also known as a dead state for some reason nothing to do with death it is known as a dead state. So, you leave sufficient you leave a system sufficiently long to interact with the environment and do not put any other constraint do not put it in a pressure I cylinder let there be a piston which is free to expand till pressure equalizes what do we get finally the system comes to a dead state quite often phi is defined not as e plus p not v minus p not s but it is defined as this minus e not plus p not v not minus p not s not where e not v not and s not are the properties of the system when it reaches the state p not t not. So, this definition sort of eliminates the question what is the availability of steam at 100 bar 400 degree c. Now you will ask with respect to what one can say with respect to whatever is the reference of the energy whatever is the reference for entropy in principle these two references could be different but if we define it like this then with respect to the corresponding dead state some books and some authors tend to follow this definition this is an alternative definition in which this is an additional constant term and which sort of gives us a reference this remains the same because phi 2 minus phi not nothing no analysis changes only the reference for phi changes. Now I think before I go to the exergy analysis for open systems it is proper to do some problems and that brings us to exercise sheet combined first and second laws which is exercise sheet 7 but in the assignment to you I have called it CL. So, when you submit it instead of 7.1 7.2 let us call it CL 1 CL 2 that way the topic numbers and these numbers are. Now notice assume that the environment is at p not 1 bar p not 300 K unless specified otherwise that is the default environment 7.1 because this gives us an idea that we means vacuum means nothing soon in our philosophy but can we extract some work out of that the question is vacuum in the absolute interstellar space we may not be extract work out of because it exists naturally but in the earthly environment where the surroundings is at one atmosphere we need to spend power and effort and money to create a vacuum. So, naturally vacuum is useful and there must be some work extractable from a perfectly evacuated space of volume V vac. So, first let us look at it this way this is our system it has a volume V vac it has no temperature associated with it its pressure is 0 there is no mass in it. So, first absolute formula use by keeping our brain power away for some time what is the maximum useful work this will be minus pi 2 minus what is pi 2 E naught plus P naught V naught plus P naught S naught now forget P naught V naught there is no mass. So, there is no E there is no S but what about vacuum in the final state this is 2 final P this is going to be E 2 plus P naught V 2 minus P naught S 2 that is the first part the second part is going to be P 1 plus P naught V 1 minus P naught first let us look at this part what does it contain nothing does it have any energy no. So, even is 0 initial state. So, initial state this is state 1 we will then sketch the state 2 when we really realize is there any entropy associated with it nothing there is no mass nothing no entropy. So, this term also goes P naught atmospheric pressure what is the initial volume V 1 is V vac. So, V 1 is V vac. So, this particular term becomes P naught into V vac. What about here this is the state 2 which is equilibrium with environment what do you do when you bring it in equilibrium with environment does the vacuum remain nothing no nothing remains the system simply vanishes if it were a balloon because it is vacuum it will just go to 0 volume and anyway it does not contain any mass. So, E 2 would be 0 as earlier S 2 would be 0 as earlier even V 2 would be 0 as earlier. So, what remains it is simply P naught V 1 with a negative sign with another negative sign and that is P naught V vac this is the maximum useful work according to our formula the way we understand it. So, 2 equilibrium with environment essentially means the system contains nothing does not even have a volume right is this right can we think of an experiment we can there we can demonstrate that this can be done let us look at it let us say that I have the vacuum of V vac and I have created it in a cylinder piston arrangement and I have because the pressure here is P naught the area is A and let us say the so V vac is product of L and A I must hold the piston against this pressure otherwise piston will simply move into the vacuum principle used in vacuum breaks. So, all I do is I apply a force here by putting it on a friction less inertial less pulley in extensible unbreakable string and put an appropriate mass here. So, that my mass is such that Mg equals P naught A forces balance now all I do to bring this in equilibrium with the environment it simply tap that piston slightly tap tap what will happen I have provided a small impulse. So, slowly the piston will move there is no force difference across the piston that small impulse which I have provided allows it to move very slowly steadily into the vacuum when will it stop it will stop when it reaches the dead end at the other of course if it is an elastic collision it will bounce back, but I will see to it that I tap it again stay there what has happened piston has moved in by a distance L what has happened to the mass mass has moved up by an elevation L basic definition of thermodynamic work how much work is done P naught A into L which is P naught into V. So, W because it is useful I have used it for raising the weight is M into G into L P naught into A into L P naught into V understood this is basic thermodynamics on the previous phase what we had done is a dump application of the availability and exergy analysis and mind you something like this you can always do this is a very simple situation. So, it was very obvious what to do, but in a more complicated situation also you can do because all that you are doing is applying first law applying second law and considering the option by changing SP by changing an appropriate interaction and mind you the standard exergy analysis does not allow you to change end states it allows you to change only the interaction with the environment and that sometimes leads to a problem because sometimes we want the process to be adiabatic adiabatic means no interaction of heat with anything. So, how do you apply the availability analysis the standard availability analysis cannot be because the standard availability analysis says that to go from W u to W u max change the interaction with the environment, but that means if initially it was 0 now you have to make it non-zero and you will say that look I need an adiabatic process for some reason why do you force me to have a heat transfer you know that is why availability and exergy analysis only talks about the thermodynamic ideal it does not take into account the ground reality what we have this will become clear when we look at you can use you can calculate the lost work, but how are you going to extract it that is precisely the problem that comes up nothing in my absolutely personal opinion this is so that some people can write books or chapters in books and I can spend time about two three hours explaining to you what all this thing is about otherwise apart from first law and second law there is absolutely nothing new except certain ideas and certain nomenclature and definition your answer should be what I have done is exergy analysis without saying so if we do this we will call it exergy analysis if you do this what will you call it will call it basic thermodynamics, but that is what it is it is basic thermodynamics I will make a personal from very powerful in the process if they don't extract it I want you to say for so I will ask him to register as a student in the June workshop that is what anybody you know who comes and meets you saying oh I know everything about it oh then why do not you attend the June workshop that should be it It is dangerous to mix two disciplines unless you have a person who is equally comfortable in both the disciplines or one expert in one discipline another expert in another discipline who are equally comfortable with each other and that you know either a pair or a set of people may be a few economists and a few engineers equally comfortable with each other or a few people who are equally comfortable in both economics and engineering that that is a very very rare combination. I think Bejan here Bejan's books. Yes Bejan's books is perhaps the only place where there is a balanced thing but he does not talk much about economics he remains within the thermodynamic domain actually one of the characteristics of Bejan's books is that he does not make any non-thermodynamic comments say it is an excellent book but you can notice that he is restraining himself by not making certain comments. Is there any difference in available energy and availability? Availability is nothing but a short form for available energy similarly exergy is a short form for excluded energy that is how this nomenclature has come. Available energy means available for what? Available for converting into useful work that is it. So that is what we have said here we have said that this is availability or exergy of Q which is available at T that is what he is saying.