 And now today we go to our today's scheme, today it is Venice day 22nd, 8th day of our course. And now we are going to talk about topic 12 combined first and second laws. If you see what we have done so far, we have first studied first law, solved problems based on that, then studied second law, solved problems based on that. Of course, you would have noticed that while solving the second law problems, we cannot neglect the first law. But whenever we solved problems requiring second law, whether for closed systems or for open system, we sort of separately applied first law and separately applied second law and went ahead in solving the problems. When it came to second law, we either directly or indirectly determined the entropy produced for closed system or rate of entropy production for an open system and checked that this is this as the required sign entropy production. I think it was mute from this side, so I will say we are now going to look at combined first and second laws. And if you see what we have done so far, if you see what we have done so far, we have studied first law, we have studied second law, applied them to both closed and open systems. But when we applied second law, we had to apply first law, but the first and second law applications were separate, they were sequential. You apply first law, determine all interactions, then you apply second law. In the second law, we computed entropy production or rate of entropy production and checked whether it is non-negative. If it turned out to be negative, we said that the process as proposed is impossible. If it is positive, we will say that it is a real life process and if it is 0, we will warn ourselves that look it is a possible process, but it is a reversible process and we know that reversible process is very very difficult, almost impossible for us to execute. We did not give much value or much importance to the magnitude of the entropy production or entropy production rate. Of course, we have looked at one question. For example, when it came to second law, exercise SL 10, where we had to determine the wastage of thermal energy, there we combined the equations algebraically, equations for the first law and the second law and derived the expression. There we derived an expression for the minimum amount of heat transfer or heat absorption from the steam, q min. Then we said q by q min was the wastage of steam. Today, we are going to combine these two laws formally and undertake something which I specifically did not include in the title. What we are going to do today? Going to do today is essentially availability analysis or exergy analysis. The reason I did not use these words in the title of this section is the following. In my personal opinion, these two words are simply buzz words. They are defined for convenience. Unfortunately, unlike other properties or other parameters in thermodynamics, basic parameters in thermodynamics, temperature, energy, entropy, etcetera are very properly and very precisely defined. For convenience, we have very properly and very neatly defined properties like enthalpy, specific heats, Gibbs function and Helmholtz function. These are properties because they are derived properties. Enthalpy is u plus p v. Wherever you put u plus p v, replace that by enthalpy and vice versa. Absolutely no difference occurs. Unlike that, availability and exergy are not first class properties of the system. They are pseudo properties because they depend on the state of the system and also on state of the so called surrounding. Surroundings for me may be different or surroundings for you. For example, today I am in Mumbai. It is raining. The temperature may be just about 27 degree C. Ambient pressure will be low because clouds are coming towards Mumbai. So, maybe ambient pressure is 0.99. Atmosphere temperature is 25 degree C. I will say that is my ambient condition here, surrounding condition here. Somebody who is in Jaipur where rains perhaps have not reached, I do not know what is the temperature, but I would not be surprised if it is 45 degree C outside and pressure may be reasonably high 1.01 atmosphere. That means the state of surroundings differ from place to place. If you go on an expedition to the North Pole or the South Pole, you will say my ambient temperature is minus 40 or minus 50 degree C and my pressure will be may be nearly 1 atmosphere. When you travel by air, the pilot usually announces halfway through the flight that the outside temperature is minus 40 degree C. It does not say what the outside pressure is, but the outside pressure is very low. If you want to study the behavior of the jet engines, then naturally the state of the system, the state of the jet engine may be very same as it was on the ground, but the state of the surroundings is different. Now availability and exergy will all be different. That is the second reason. The third reason is that we have never so far defined availability and exergy formally. If you look up three different books, you are going to get three different definitions of availability and exergy. It goes to such an extent that what is availability is exergy on the other plane part of the Atlantic and what is exergy is availability on the other part of the Atlantic. So, you cross an ocean and you flip the definitions. That is the reason why at least in the title of section 12, I did not use the word availability and exergy. So, what is availability and exergy? As the title says that for a given system and for a given process in the presence of surroundings or the so called environment, we are going to apply the first law. We are going to apply the second law and when you do this, we can always determine the entropy production or the entropy production rate for an open system. But we are not going to stop here. We know that the moment any one of these is greater than 0. We know it is a real life irreversible process and we are always interested in something ideal which is fantastic. For a simple process, the ideal is a quasi-static process. For a thermodynamic process, the ideal would be a reversible process in which this would be 0. So, what would be the ideal and because we are not executing the ideal process, the difference between the ideal process and the real process. This difference means that we have lost something and what is it that we have lost? How much is it that we have lost? That is the basic purpose of this combined analysis of first and second law or the availability or exergy analysis and although in the exercise which I referred to that is S L 10, we looked at the wastage of thermal energy of steam or loss of thermal energy. Typically, in availability or exergy analysis it is the lost work that one talks about. So, actually we are going to derive no new relation, no new basic relation. We are going to use first law, we are going to use second law and we are going to see what is the difference and what is it that we could have done but did not do. Now, the basic idea is like this. Let us first look at a closed system and let us say it executes a process, some process 1 to 2. We have delta E is Q minus W, no doubt about it. We have delta S is, let us say that this Q is made up of two parts. Usually when we consider this, we say that the system which goes from the initial state 1 to a final state 2, let us say that the Q is made up of two parts, one, two or more parts. One is Q from some other system, some sources or to some sinks with which we know everything about but we always say that there is an environment which is at P naught, T naught and we will say let there be an interaction Q naught as required with the environment. For an adiabatic process, adiabatic with respect to environment this will be 0 but we consciously make note of this Q naught and hence when it comes to first law, we write this as Q plus Q naught minus W. Not only that, this W also we split into two components. We say W could be W expansion plus W other. This split we have already done for many of our discussions and exercises but we also notice that for a fluid system, W expansion will involve a delta V, a change in volume of the system and a change in volume of the system means that if the system volume increases by delta V, it has to push back the atmosphere which remains at the pressure P naught. So, if you sketch a cylinder piston arrangement and let us say this is our system, there may be some stirrer work but if it expands, there will be some expansion work but we say that if the system is at some pressure P, then we will say for a small amount of volume change P dV, P dV will be the total work but then we say that look on the other side is going to be the ambient. So, when it moves, a work equal to P naught dV will be done against the ambient and the remaining part of the work will be some force which will be displaced by some displacement dS. So far, we did not split this force into f dS and the pressure of the ambient acting on it but now we do because that is how we do exergy and energy analysis and this is the total work plus there will be a dW other. So, this is the total work done by the system P naught dV is the work done in pushing the atmosphere and we define this as the useful work dWU. So, with this we come back to our system 1 to 2 Q naught Q, this is P naught T naught and our first law naught now becomes delta E is Q plus Q naught minus W useful plus P naught delta. This was first law. The second law delta E is replaced by delta S on the right hand side. This I will simply write Q by T and you can put either a summation here or an integral here with dQ as required depending on the process details of the process plus you will have Q naught by T naught because this interaction is at the temperature T naught and there is no corresponding term but here you have an SP with SP greater than or equal to 0. Now, notice that WU is related to SP in a partly direct partly indirect way. If you change SP that means you are changing at least one of the parameters here either you are changing delta S that is changing the end state of the process or you are changing Q the heat absorbed from some sources or you are changing Q naught the heat transfer to the environment. If you change either Q or Q naught you are directly changing those terms in the first law and hence you are changing WU. If instead of that you are changing delta S then naturally a change of state would mean you are changing delta E and that means even if you have not changed Q and Q naught you have changed WU. So, that means a change in SP definitely means a change in WU and you can show which we will do in by analysis that as you reduce SP from positive values to 0 WU goes from whatever its value is to algebraically higher values. That means as you reduce is reduced from positive to 0 that is the limit you cannot go beyond 0 WU increases and naturally from whatever is initial when this is 0 you get positive WU max. The purpose of combination combining first and second law is to determine WU max and we determine that by analyzing the situation by first law analyzing the situation by second law combining these two equation obtaining a relation between SP and WU of course through the process and through T naught and then finding out what would happen if we set SP to 0 that would give us WU max. Now remember as I have already said to reduce SP to 0 we have to change at least one of these three terms 1, 2, 3 either change in the interaction with the environment or change in interaction thermal interaction with the environment or change in thermal interaction with whatever other sources you may have or change in the end state itself. The choice is with us we may do one of these three or we may do a combination not all choices lead to the standard availability or standard exergy analysis. But since many text books talk about the standard exergy analysis and tend to define something called exergy of a system or exergy of a state and availability of a system we will do the standard analysis. Now instead of a standard analysis I would now say that this is a common analysis we will first do it for a closed system and then we will do it for an open system. Again I warn you that the definition of availability and the definition of exergy are not unique you pick up the number of books you pick up the number of different definitions. Sometimes you will find that the definition of exergy in one book matches the definition of availability in the other book and may be vice versa. But in the common analysis what we do is the following we have first let us take it up for the closed system open system is similar and we will do that later. This goes from state 1 to 2 it may have q from various sources but it definitely has q naught from t naught also there is a p naught. The common analysis of availability ask the question what is w u max if only q naught is q naught. And of course because and if at all because of this there could be some minimal consequential other requirement and then it defines lost work w u max minus w u this is for the specified process this is what is computed. Now first the basic analysis let us consider the closed system and let us say that we have q naught from t naught there is a q available from some t we can sum it up if there are more than one q's. The change in the state of the system is delta e delta v delta s it does work w u plus it does work which is p naught delta v. So, this is the total work w u may contain expansion work it may contain stirrer work shaft work electrical work what you have we apply first law this is w plus w work done against atmosphere. So, w u plus p naught delta v this is first law equation one. Let us apply second law delta s I will simply write q by t but remember that this q by t is not just one q over one t if need be it will be summed over different q interactions or integrated over the process or processes involved plus q naught by t naught this is a proper ratio plus s p with the proviso s p greater than or equal to 0 second all that we do is multiply equation 2 by t naught t naught delta s is equal to 0. t naught by t q plus q naught plus t naught s p. Let us say this is equation 3 which is nothing but 2 multiplied by t naught subtract 3 from 1 because there is a q naught here there is a q naught here we want to eliminate that and when you do that you will get delta e minus t naught delta s is q into 1 minus t naught by t this gets eliminated you have minus w u plus p naught delta v minus t naught I do not think I have missed any term. Now, we will rewrite this as w u equal to I am taking this w u term. On the left hand side it has a negative sign on the right hand side so it becomes positive sign on the right hand side equal to q into 1 minus t naught by t this term then I have a delta e minus t naught delta s here and if I transpose this p naught delta v with a negative sign to the left hand side I will get delta e plus t naught delta s plus p naught delta v but I am going to write it on the right hand side. So, minus delta e plus p naught delta v minus t naught delta s. So, I have taken care of all terms except this which remains t naught s p. Now, this equation I call equation 4 and fortunately I cannot do any cut and paste. So, I will rewrite this on the next page and I will rewrite this in different lines first we had q into 1 minus t naught by t then I will write plus minus delta e plus p naught delta v minus delta s. Now, notice that our s p has to be greater than or equal to 0 that means t naught s p is a positive number this minus t naught s p will always be a negative number. Now, what does it mean if I make my s p equal to 0 then this term will be equal to 0 if s p is 0 and s p is 0 is the limiting case the ideal case the reversible case and in that case when s p equals 0 these two terms will give you the maximum work maximum work done. And if this is the maximum work obtained which is when s p equals 0 what is this term without the negative sign or t naught s p we will now say is the lost work defining lost work by the relation w u is w u max minus w lost and remember w this is the definition of w lost and we get w lost equal to t naught s p this provides some sort of a physical significance to s p using our second law we have defined s p we have computed s p for various problems and we said that s p must be positive for a real life process which is irreversible at that time we did not or we could not assign any field to the value of s p entropy production for entropy produced using this analysis we can say that s p represent lost work the magnitude of lost work is environment temperature on the Kelvin scale multiplied by s p and with the condition that if we were to re execute the process in a reversible way but now when you make a real life process reversible you are changing some parameters of the process but we say that we execute the same process from the same initial state to the same final state so the final state is not being disturbed our destination remains the same however during the process we have readjusted the heat transfer to the environment q naught. Remember we do not see it because that is the term which we eliminated between the two equations representing first and second law so if we keep the end states the same so that our origin and destination states are the same we re execute the process by readjusting the heat interaction with the environment in such a way that entropy production is 0 then the additional work which we could have obtained is this lost work represented by t 0 s p that is the first part the significance of s p so what we have done is we have defined lost work we have shown that it is equal to t 0 s p so we have provided some level of significance to the numerical value of s p as t naught s p is lost work but remember the weakness here t naught is a property of the environment environment for me environment for you is likely to be different environment for me today an environment for me after a few months is also likely to be different so w lost is not the change in property of our system now coming to the w u max w u max you will notice is made up of two components one part let me say is component a the other part is component b I will rewrite this here w u I will now write max because I do not want to write the third term is q into 1 minus t naught by t the a component and second minus delta e plus p naught delta v minus t naught delta s the a component and the b component what is a component a component represents max work obtain or obtain able when we absorb q at t the environment for me the environment is at t naught and no change in state of our system no change in state of our system that means we are executing cyclic processes q naught is only allowed at t naught other than q so this is equivalent to running a 2 t reversible heat engine between t and t naught the efficiency of that heat engine will be this 1 minus t naught by t and naturally the work which you can obtain is q into 1 minus t naught by t that is the understanding of the term a and you would notice that the 2 t reversible heat engine efficiency term or karma efficiency term 1 minus t naught by t it sitting there that should not be a surprise the term b is max work obtain able due to process 1 to 2 which is represented by change in energy delta e change in volume delta v and change in entropy delta s when only q naught at t naught is allowed no other heat interaction is allowed that is the part b and notice that so far apart from entropy production and lost work we have not used the word availability or exergy. Now up to this point the mathematics and algebra is the same now we come to the definitions of availability and exergy again I will use a definition where for closed systems we will be talking of availability for open systems we will be talking exergy again this is on one side of the Atlantic and one side of the Pacific on the other side of the Atlantic for closed systems everybody will be using exergy or those who talk will be using exergy for open systems it will be availability but we will restrict availability to closed systems and exergy to open systems. Part b notice is minus I can write a delta of e plus p naught v minus t naught s and this term in bracket is a common definition of availability so our term which I will write first fully delta e the second term the maximum work is which we can obtain just purely because of change of state and when heat transfer is allowed only with the environment is delta e plus p naught delta v minus t naught delta s which can be written down as minus delta of e plus p naught v minus t naught s this is the written down as the availability phi phi is given the name availability so this term is simply written down as delta phi and then phi or availability it is considered remember our derivation of the say the Helmholtz function we said Helmholtz function is like a potential the decrease in which is the maximum work we can obtain in an isothermal process and similar relation for the Gibbs function so similarly the availability is something like a potential the decrease in which notice this negative sign the decrease in which represents the maximum work that can be obtained over a process which is allowed to have heat transfer or maximum useful work which can be obtained in a process which is allowed to have heat transfer only with the environment at t naught and because we are talking of useful work this p naught is hiding inside the availability function and because the environment is at t naught and we are allowed to have an adjustable heat interaction with the environment that t naught is also written there now the second definition and this is the definition of something called a dead state which is often used although availability function or availability can be defined like this as shown here quite often something called a dead state is used the idea of a dead state is something like this suppose I have a system at some pressure some temperature fluid system and I leave it to freely interact with the environment when I leave it to freely interact with the environment that means I allow work interaction including expansion compression work and I allow it to have a heat interaction what will happen the system will come to a stage where the pressure will be p naught its pressure will be p naught and its temperature will be t naught. So dead state is the state of the system at p naught t naught where p naught t naught are pressure and temperature of the environment and let at this state let the energy of the system be e naught let the volume of the system be v naught and let the entropy of the system is s naught then we can say that if we have a system and we allow it to interact with the environment in such a way that eventually it comes to the dead state and we will during this interaction we may allow we will try to extract extract the maximum amount of work but we will allow heat transfer only with the environment in which case what is the maximum useful work that you can obtain I will write it as u max 0 saying from the current state we are allowed to come only to the dead state this will become minus delta e will now be e minus e naught delta v will now be v minus v naught and delta s will now be s minus s naught this thing is also known as availability quite often. So on this page you see two definitions of availability this is availability definition one where you do not talk of a dead state and when you talk of a dead state this is availability definition two in either case the relation w u max for a process which is allowed to have heat interaction with the environment remains minus delta phi that remains that is because this is a change in availability or decrease in availability whether you define availability by definition one using this definition or definition two using this definition it is just a here you have a particular reference state here we do not talk of a reference state we talk only of a the change of state. So that is the basic idea of the availability analysis. So finally the whole thing reduces to availability analysis reduces to the w u is w u max minus w lost w u max is made up of two components the component when you obtain when you have a heat interaction at temperature t sum it up or integrate over appropriate heat interaction and the second component which is minus decrease in availability. So this depends on heat interaction at t and this depends on change of state and also on p naught t naught because p naught t naught is sitting there then w lost w u max minus is p naught delta sorry t naught s p phi is defined as p plus p naught v minus t naught s this is definition one or phi is defined as p minus p naught plus p naught plus p naught v minus v naught minus p naught s minus s naught this is definition two where p naught v naught s naught is so called dead state p naught t naught that is when the system pressure is p naught system temperature is t naught. So before I go to a similar derivation for open systems I am open to questions n i t 3 chi good morning over to you. Sir I have this question this w u is equal to q into 1 minus t naught by t plus del E plus p naught del v minus of t naught del s it almost looks like our first law is correct sir over to you sir. Yes it will look like first law but remember there is a delta s so there is a second law also hidden there and that is not surprising because this is a combination of the first law and second law as I said in the title of this lecture all that we have done is written first law written second law and remember the unit of every term or dimension of every term in the first law is energy units will be joule the dimension of every term in the second law the way we write it in terms of entropy is energy per energy divided by temperature because it is heat interaction divided by temperature. So it will be something like joule per Kelvin we multiply the second equation by temperature so its unit also becomes that of energy and then we just algebraically add or subtract one from the other. So if some terms look like first law some terms look like second law some terms look like the Carnot efficiency term there is absolutely nothing surprising that is in fact what one should expect over to you. Actually what I want to ask is say even if you consider only the first law the moment you consider about the du the change of state or the interchange arise that indirectly mean that there is a change of entropy will be added to this. So in that case then we should bother more towards this equation is concentrated on first law only or it tells that same law is already hidden in it that is what I want to ask is what is there. See nothing special it is a combination of first and second law if you were to write only first law entropy will have no place there if you were to write only the second law energy will not have a direct place there. So there is nothing special about it this is just I may even say that this is algebraic jugglery using first law and second law. In fact that is my personal opinion but if I write a book or write a paper I will not use such terms like algebraic jugglery but between you and us just keep it I am not writing it down at jugglery but you can hear jugglery and I can say jugglery during the course of this course over to you. Next question sir is it possible to reuse the unavailable load sir over to you sir. I did not understand your question is it possible to reduce the unavailable what over to you. Unavailable work. Unavailable okay okay okay I understand I understand what you mean is see the lost work is something which you are calling unavailable work the question is is it possible to reduce the unavailable work. Remember that the actual situation as analyzed will have that lost work or unavailable work the actual situation will also have some entropy produced in it or entropy production associated with it. If you want to reduce the unavailable work by some magnitude or ideally to 0 you will have to reduce the entropy production by some magnitude ideally to 0 that means get rid of entropy production. There are methods by which you can do this there are schemes or there are ideas the ideas would be may be you reduce pressure drops when something is flowing through pipes. If there is friction between the piston and cylinder reduce that friction if there is a leak past the piston reduce that leak. If the process is non quasi static which definitely makes it irreversible execute it in such a way that it is quasi static. But if you do this then the cost of design and operation of over system will shoot up that is why we try to do it to a certain extent but we do not go very far enough. I have had problems from industry where they try to do this and if I plot a graph something like this say cost and this is let me say entropy production. That is S p for a closed system or S dot p for an open system and say this is entropy production 0. So, this is the reversible case actually he would call it thermotopia that means thermal utopia thermal paradise and what happens is you have the current state of the process is like this as you try to and remember entropy production also means this is lost work or lost power or wasted heat or whatever. If this is the current state and if you try to reduce it you can reduce it but only at an additional cost and as you try to go near this thermal utopia you find that the cost increases it shoots up. So, finally it is economics which says that it is not worth going beyond a certain cost. So, you stop your efforts at some stage like this I think this provides enough ideas to you over to you. K. K. Wagnashik I can see you so over to you for any question. Sir my question is on availability definition 1 and definition 2. In definition 1 it is again delta e and in definition 2 also e minus e 0. So, by in first definition you have only written as a phi is equal to e plus p 0 e while in second definition it is it seems as a delta e. So, is there any difference as e minus e 0 and only e in definition 1 and definition 2. Over to you sir. See I gave two definitions because these are the two common definitions in text books the first definition phi was defined as this is e plus p 0 v minus t 0 s this is definition 1. For this your values of e and s will depend on whatever is your definition of the reference states of e and s and the absolute value of phi does not mean anything when you use this definition. Whereas, for the second definition I will rewrite the second definition in a slightly different way I will write it as e plus p 0 v minus t 0 s minus e plus p 0 v minus p 0 plus p 0 v 0 minus t 0 s 0 this is definition 2. I have just you know rewritten in a different way. Now, in either case this thing remains w u max because of change of state is the minus that means equals the reduction in availability whether you use the first definition here or whether you use the second definition. Finally, this remains correct only thing is here for the first definition we cannot assign a direct meaning to phi when you use the second definition you can say that if phi is defined by the second definition the availability is the maximum useful work one can obtain when you allow the system to come from a given state to its dead state that is full equilibrium with the environment by having interaction of heat only with the environment. In that case the maximum work you can obtain is the value of availability by definition 2 and that is the reason why many people define and use definition 2 because you can say that availability of a state means something whereas, for definition 1 it does not mean anything it is only the difference which means something and that is this. But this equation is valid in either case I think this explanation should satisfy you over. Sir there is one more term that is called the energy can you tell something about that see there are various terms one can define I am restricted I am restricting myself to lost work sometimes you will find lost work or something related to that is will be called energy or energy saying it is unavailable energy I do not want to get into those definitions because you can define many terms just for your convenience or just for your understanding. Now let me say over and out let us have our cup of tea.