 What are the reasons for appearing the shock waves in the divergent portion? Yeah. So, the question is on what is the reason why a shock wave appears in the divergent portion of conversion CD nozzle. See, what ends up happening is that if you are using a CD nozzle and let us say that you have a fixed upstream stagnation pressure and you keep on reducing the back pressure. As you keep on reducing the back pressure, more and more flow will start flowing through the nozzle and at some point the throat will choke. This is what we have discussed earlier. When the throat chokes, what happens is that the Mach number there is going to become exactly equal to 1 and the maximum mass flow rate is going to start flowing through the nozzle. Now, if you decrease the back pressure further than this, however, if the back pressure is not low enough so that a completely isentropic supersonic flow can develop in the divergent portion of the nozzle, because the nozzle has to now pass that maximum flow rate, the only possibility that this can be done is by the formation of a shock. So, when we talk about the formation of a shock, we actually talk about a sort of a final steady state situation, but the actual reason why a shock occurs during the process of lowering the back pressure is again exactly the same process that I talked about yesterday that you can always imagine a unsteady process that is going on while you are reducing the back pressure and during this unsteady state, there are acoustic waves generated which will get merged together at some point in the divergent section to form the shock. So, to be honest with you, it is not that obvious based on the principles that we know that we can formally say that this is the reason why the shock occur, but at least you can come up with a couple of points saying that once the nozzle chokes that there is that certain maximum mass flow rate that has to be passed, furthermore if you keep on reducing the back pressure, the flow cannot remain subsonic anywhere in the divergent section, but if the back pressure is not sufficiently low to have a completely isentropic supersonic flow in the divergent section, the only way the nozzle is going to function is by creation of a shock at some location within the divergent section. That is really what I can answer at this point. Thank you. Sir, what is the difference between isotropic stagnation state and actual stagnation state? So if I understood you correctly, you are asking what is the difference between an isentropic stagnation state and an actual stagnation state. Is that what you ask? There are some irreversible things, what are the irreversible things? See my understanding is that the stagnation state is defined only for an isentropic process. So that you imagine a state of the flow to be brought to rest in an isentropic flow and you will reach the values which we are going to call as the stagnation values. So in that sense I will say that when you are talking about stagnation values, the assumption of an isentropic process is built in. So I am not too sure if I can answer what is meant by an actual stagnation state. When we talk about a stagnation state, my understanding is that the assumption of isentropic process is already built in and is always built in. Thank you. The steam is flowing in a 3D nozzle, a metastable state is existing, can you tell the reason? If I again understood you correctly, you are saying that there is a steam flow in the a 3D nozzle and a metastable state is existing, is that correct what you said? Metastable state is existing in the diverging portion. I am not exactly sure if I have heard this terminology that a metastable state is existing. So I do not think I will be able to answer this question unfortunately. Let me see if Professor Bandarkar has any light to throw on. Unless it is, what is this metastable state? Is it possible for you to explain what you mean by the metastable state? Because I have not heard this terminology, I am sorry. I am not aware of the term that you use, the metastable state. So I was wondering if you can explain what you mean by that. Actually I saw the fundamentals of classical thermodynamics by Gordon von Weylen. That textbook I saw the metastable is existing while the steam is flowing in the 3D nozzle, a metastable state is existing in the diverging portion. So I think based on what you are saying, I think roughly I may try to guess what he is trying to say. See what happens normally is that let us say steam is flowing and you have reached the saturation temperature at a particular pressure. You should automatically go to the liquid state but what happens normally is that sometimes you continue beyond it and you are still in the gaseous form though you should have been in the liquid form and that is what is called as a metastable state. And a slight disturbance will just throw it into the liquid region and you will go where you ought to be. So I think this is what you are getting in the textbook that sometimes these situations do occur that instead of being in the correct state you are in a state which is still possible but very less likely to happen. Thank you. Sir, can I ask the questions in entropy? Yeah, please go ahead. Sir, if you convert heat into work at low temperatures, the more entropy will be generated. Suppose if you convert the same heat at high temperature, less entropy will be generated. What are the reasons why the entropy increases at low temperatures? So I am not very sure by what you mean by when heat is at low temperature, low entropy is generated. So how did you come up with this statement? If you can explain that then we can go ahead. Actually I saw in a textbook at low temperatures, if you convert heat into work, more entropy will be generated. Suppose if you convert same heat at high temperatures, less entropy will be generated. So this is what you have read somewhere that at low temperature if you convert Q, you will generate high entropy. Is that what you have heard? So I mean I am not totally sure why this would be true. So it does not seem to follow any logic right now but let me see if you can tell me which book it is, let me read up the context and I can tell you what this may be about. So otherwise I would not be in a position to answer this question. Thank you. One more question. What are the causes to increase the entropy? I mean one of the most common causes is purely friction. So for example, I mean that is the main cause of irreversibility anywhere. So for example, you know regularly in steam turbine or a gas turbine. So for example, let me draw a gas turbine on a TS diagram. So you know that you start at some lower pressure P1 and sorry these lines are not well drawn here. So you have now two pressure lines P1 and P2 and I know that you know as per the second law I need to go only either straight up or I go like this. Now this is assuming no irreversibility and an adiabatic situation and while flowing through the compressor blade if there is friction or if there is you know detachment of the flow over the blade you will start increasing the entropy and you will have to start putting more and more work in and that is what is going to cause basically an increase in entropy. So friction in general is always the by far the main cause for an increase in entropy. One more question sir. What is entropy transfer? How it is associated with heat transfer? So I am not very clear that I have ever heard this term of entropy transfer. For every system you can define an entropy and for a control volume you can figure out what is the entropy coming in and going out and figure out what is happening but I am not very sure that I would ever use this term called entropy transfer. Thank you. One more question. What are the negative thermodynamic temperature? So this again depends on the context that you are using as far as we are concerned we have set some absolute zero somewhere which we cannot reach and as far as we are concerned such temperature I mean it does not matter to us that such temperature can exist because we think that the lowest possible is zero that is about it. Thank you. Can you show the entropy varies logarithmically with the disorder number? No sorry actually I am I do not think I have knowledge about this disorder number at all so I would not be able to comment but probably what you are trying to say is what is called as the number of states that exist and there is some expression that you all have seen earlier which says f is equal to k l n omega and this is again just a definition and it is a definition which ensures that the number of states available microstates available that corresponds to the entropy idea. So it was a correspondence that was brought about just by defining it in this fashion so unless I go into statistical mechanics and tell you what w means and why I should define s like this I mean that will be a too long process so I would rather leave it at for some other point. So I did not know that someone used to call this as a disorder number I would rather not call it in that fashion usually it is just the number of states possible. Thank you. Thank you sir. Over and done. Yeah one zero seven zero Amrita Bangalore. Actually I have a question regarding compressible flow we have often come across the statement that in a nozzle enthalpy gets transmitted into velocity but how exactly that happens what is the I mean what exactly happens for the enthalpy being transmitted into velocity is it kinetic theory of gases. Yes so that is a very fundamental question that you are asking so let me repeat the question that we talk about enthalpy getting converted into kinetic energy in case of a nozzle and the question is how does this exactly happen and within the framework of our macroscopic say thermodynamics or fluid mechanics it is essentially impossible to describe how this is happening. If you want to really know the details of how this conversion is happening as you correctly pointed out you have to resort to a molecular level analysis where typically perhaps what we feel is that through the molecular collision process you can try to come up with a good feel of how the enthalpy is getting converted into energy kinetic energy that is. So the random thermal energy that perhaps we can represent using the enthalpy will get converted into directed kinetic energy in a convergent divergent type situation through the mechanism of molecular collision is what I can probably say at this point but more details are for me it is impossible to right now give unless and until you really study in detail the kinetic theory that is correct so I would say that is where I will stop. Would you suggest any book for this if you have to go through? As far as my understanding goes usually in nozzle type situation where the kind of that we were discussing is hardly ever covered in any of the books that I have gone through at the level of molecular dynamics let us say. So to be honest with you I have not come across any book so far which talks about such processes from a molecular point of view. My next question basically another question connected to this I am not very sure for example if you take a gas or an air if it starts entering the nozzle with Mach number is equal to 1 then what exactly happens? Suppose it is flowing through a convergent nozzle or a divergent nozzle if it starts from Mach number is equal to 1 starting with the equation dV by V is equal to dA by A divided by M squared minus 1 if the entry starts with Mach number 1 in one case it enters through a convergent nozzle in other case it enters through a divergent nozzle so what would be the difference? So if it enters a divergent nozzle at Mach number of 1 it is just like what you have in a standard convergent divergent nozzle at the throat and the flow will go to supersonic situation assuming that there is a driving potential that supports it. What we feel is that in case a Mach number 1 flow enters a convergent nozzle it must decelerate according to the area velocity relation here what I am assuming that the flow is entering perhaps fraction below Mach number of 1 if I can use that assumption but that is what I would answer this question. Yeah obviously if the Mach number is slightly above than 1 or slightly less than 1 then it can easily describe but usually that is the question which I come across when I teach this course if it is Mach number equal to 1 is there any explanation for example in the throat from the throat in the diverging part the flow becomes supersonic but starting from Mach number 1 if you use the equation velocity area equation we cannot actually show that starting from Mach number 1 and divergent nozzle will actually create a supersonic flow. That is correct I understand that you know utilizing only that equation it is essentially impossible to come up with that conclusion. So in practice the only way I can think about is that really if there is only the Mach number is little bit above or below then we can really talk about it using that equation and that is where I will really answer my answer your question. Yes I have one more question in fluid mechanics of thermodynamics I am not very clear about the concept of pressure is it same as force per unit area then exactly what happens in a boiler or a condenser when we say it is a constant pressure process or through for example if you take incompressible flow what exactly happens in a nozzle can we define using the basic definition of pressure rather than taking the Bernoulli's equation or something like that. You can definitely define pressure in the same way it is just that if I put let us say an area where I can measure or you know just flat plate there that is the pressure it will experience so you can keep on moving the flat plate anywhere and if you see the same pressure then the whole system is at the same pressure. So this is what is happening in a boiler that is in the tube. This is the force per unit area that the boiler tube will see and this is the force per unit area that is there in the boiler drum also. So it is exactly force per unit area that is the standard pressure that it will see. Actually in a boiler there is heat addition, there is a phase change, there is a change in a specific volume but why exactly the pressure remains constant in a boiler or a condenser. Okay so this is up to us whether we want to maintain the pressure constant. So for example if you use a simple pressure cooker you have put the whistle on top of it. So the pressure keeps increasing up to the point that the whistle will lift and it will exhaust steam so that the pressure inside is maintained. So this is what is done in a boiler too. The drum is such that slowly you start building up the pressure because you convert water into steam and the steam will want to now occupy that space but you are converting more water into steam and slowly the pressure will build up. So it is a huge pressure cooker and you have to now decide at what point this is safe. So there is a big vent out there and let us say you have decided that the pressure it take is 160 bar then you will safely design it for let us say 300 bar but if it goes above 160 bar there is a vent which will just let off the steam and ensure that the pressure is within a range. So it hardly ever 160 bar but it will be plus minus a few 5 bars because you will be continuously venting if there is such a process but this is how you maintain it. You know you see some gauge you ensure this is the pressure if it is more then you know it is dangerous you start venting off. So that is what you do there, thank you. But suppose if you take the condenser of a refrigerator we do not have any controlling things like this but still we consider it as a constant pressure process. No, no in a condenser to you are actually doing it because you are constantly ejecting things out. So if you start the condenser you actually need to suck out everything using pumps and then only you start the condenser and once you start condensing the gas immediately the sorry the steam the steam condenses to water and there is vacuum and you are ensuring by continuously sucking out that the pressure is maintained. And it is always pretty dangerous the condenser is by far the place where you will get the worst contaminants because any small leakage and the outside atmospheric air will try to come in and if it mixes with the steam you have a pretty deep problem. So you always have to ensure that the condenser is very, very least size and ensure that the pressure is maintained constant in the condenser. Otherwise the pressure in the condenser if you do not design a good condenser unfortunately your condenser pressure will reach one bar and you will have an inefficient steam turbine. Thank you. I have a last one question. In thermodynamics why a specific heat at constant pressure is always greater than specific heat at constant volume? No, this is just how we have defined it you know we have defined something called Cp and we are calling it as dh by dt and h is just u plus pv. So if you look at our present definition this is how we have defined it so that Cp is always defined in terms of Cv plus r. Whereas if you wanted to look at the previous definitions which were not in terms of these enthalpy and internal energy then you could have easily explained that to maintain constant pressure you wanted to do some work and the heat is going not only in whatever heat you are adding is not only going to increase your internal energy but it is also doing the work that is necessary to maintain the constant pressure whereas if you are not let us say the system expand then all it would have done was increase the internal energy and hence you would require lesser amount of energy. That is how the previous people would look at it and that is it is okay to look at it in the same fashion there is no problem thank you. So Cp minus Cv is always going to be r for that fluid and Cp minus Cv in molar terms is always going to be universal gas constant so it is the same you are pretty correct on that if you consider on a molar basis it will always be r universal gas constant so that is always going to be true thank you. One to one nine prestige institute go ahead. Sir good evening I am Srikanth Tare from prestige institute in science along with my colleagues Dr. Parip Banjavani my director my question is on combustion if I got number of gases mixture like LPG which is the combination of hello which is the combination of propane and butane in that case without actual combustion I want to know the calorific value of this mixture how can I calculate it. Yes sure see as long as you know the proportion of the gases so let us say you have 60 percent butane and remaining 40 percent as propane so what you do is you say that I have 100 moles of the gas and I say 60 moles of butane C4H8 will react with oxygen and nitrogen I find out how much energy is released for 40 moles of C3 this is butane C4H10 propane is C3H8 I will react with regular air find out how much energy is released so I will see that for 100 moles of LPG this is the amount of energy released then I can either calculate it on a per mole basis of LPG or per kilogram basis of LPG so there is as long as you know the composition you can do it is all pretty much direct proportion so I do not see any harm in you know going ahead with the calculation thank you. So by simply mathematical calculation you said that it will be possible for me to know about the calorific value. Correct yes exactly you just do a direct proportion and you should get your calorific value. Thank you over and out. 1047 SDM college. Sir my question is why the air fuel ratio is always considered on mass basis and not on volumetric analysis? I think it is just a matter of convenience. Both in gravimetric analysis and volumetric analysis it is taken on mass basis not on volume basis why? Yeah see if you express it is just a matter of convenience you realize that if you for most hydrocarbons if you express it in terms of volumetric basis then you realize that as you go for higher and higher alkanes you will require more and more oxygen and nitrogen and more moles so on a molar basis you would have far more moles as the alkanes size increases whereas what you really want to do is find out what is the actual amount needed for combustion and you need convenient number for that and it is far more convenient just to go for the mass basis because you know roughly where the range should be and that is where you are going to be comfortable thank you. Sir can we define the equivalent ratio on volume basis is it positive? Are you saying you want to define the equivalent ratio on a volumetric basis it is up to you I mean it depends on whether other people want to use the same thing but you can please go ahead with it no problem. Sir one more question there the composition of air is always taken as the mixture of oxygen and nitrogen why sir though to some extent some other gases are also present but for all calculation present we take only oxygen and nitrogen why is it I mean I would say the simple answer is that we are engineers we do not want to get down to real minute details we do not affect us so 79 and 21 we take we can get all our calculations we are not really you know trying to figure out how much argon or carbon dioxide there is how is it is going to affect because those are in some less than 1 percent population out there and we think if this is not going to affect so much why bother or put so much effort into it and you will realize the most engineering calculation reasonably good approximations we can design something pretty well and I this is my only simple answer for it thank you. Sir one more question the polynomial fit that you explained on CV that is specific it at constant volume that is developed is it this can we develop the same expression for CP for CP being the yes for sure because actually the expression is always calculated for CV rather than CP okay so see actually it is the expression for CV which is calculated and that is because CV is directly defined as du by dt and du by dt is very readily obtained and we find that for monatomic gases CV will start with 2.5 R and as you go to diatomic gases you can go to higher CV sorry you go start with 1.5 R and if you go higher you can get higher and higher CV and then finally CP is just defined as CV plus R because that is a straightforward expression so actually the calculation is always of CV first and CP is just defined as CV plus R thank you. Sir one more last question sir what is the exact relation between the enthalpy and internal energy of combustion. So I am not sure what you want to say what you want to mean by internal energy of combustion I mean enthalpy is very straightforward defined h is equal to u plus pV so that is the strict definition for enthalpy so if I want delta h is delta u plus pDv plus vDp and most of the combustion is always done at constant pressure so we do not look at this so there is always pDv work built in so rather than looking at internal energy we just go for enthalpy because it is of convenience again it is purely a matter of convenience that we list enthalpy rather than internal energy that is because we know all our reactions are such that we are trying to maintain a constant system there and it is more convenient to work in enthalpy terms rather than internal energy thank you. 1195 Mansell college please go ahead. Hello good evening sir sir my question regarding convergent divergent nozzle in convergent divergent nozzle when we are going in the direction of flow we have seen that pressure velocity pressure temperature decreases while Mach number and velocity increases why it is so happen? So the question is on pD nozzle and what is being pointed is that as you go along the flow direction the pressure and temperature is going to decrease whereas the velocity and the Mach number are going to increase and the question is why does this happen so the answer is if you want to look at it from a purely microscopic point of view you can simply go and use our mass balance and the momentum equations which we put together and we came up with that area velocity relation using which you can figure out that for example if you are dealing with a subsonic flow situation as the area will decrease you will see that the velocity is going to increase and if you are dealing with a supersonic situation you will see that as the area decreases the velocity also decreases and using this going back to the momentum equation you can figure out that as the velocity is behaving accordingly the pressure is going to behave and you will see that as the velocity decreases for example you will see that the pressure will sorry as the velocity increases the pressure will decrease because we are dealing with an isentropic flow situation the pressure and temperature are related through the isentropic law and hence the temperature will also decrease so just using the conservation equations of mass balance and the momentum equation which we combine to form that area velocity relation you can actually work out the entire mechanism as to what is happening here so this is from the framework of a continuum fluid mechanics and thermodynamics if you want to know more fundamentally one has to resort to a molecular level description which is totally out of the scope of this present discussion and I will not be able to discuss that here thank you we have seen the convergent portion in divergent portion divergent portion always longer than longer to convergent portion does it any effect on the you know supersonic region when we are since divergent portion is longer than convergent portion what happened in if you are reducing the divergent portion in the similar fashion as it is convergent see actually the reason typically the divergent portion is kept longer is so that the angle of divergence is kept to a fairly small value in order to avoid phenomena such as separation of the flow so if you want to maintain the flow attached to the wall and you do not want to get into undesirable phenomena like the separation of flow you would like to maintain the divergent angle small and if you want to maintain the divergent angle small the only way to achieve a certain area ratio is by increasing the divergent length section so that is the reason I think you will see that typically the divergent length is long sir is there any role of density is there any role of density with this sonic subsonic and supersonic flow yeah so the change in the density is built into the into the entire process through the conservation of mass statement if you go back to our slide you will see that for this cos 1 dimensional flow you will see that the mass conservation statement was for finally written as 0 over rho plus dv over v plus da over a equal to 0 so indeed the density is changing from location to location and in fact all variables the density, the pressure, the temperature and the velocity and obviously the Mach number also are all changing and they are all coupled to each other in this compressible flow situation so I will say yes that the density is playing a role and the biggest effect of that you can see through the mass conservation statement thank you thank you thank you 1079 Gita Institute please go ahead if you have a question if the alpha is the nozzle angle for a multi-stage stream turbine then what is the maximum efficiency for that actually at this point I would be unable to answer it and it could not depend only on alpha it would depend on many other factors you need steam to enter smoothly and then even you know stick to the passage and go out so it cannot depend only on alpha but there are many kinds of empirical relations which will tell you how the efficiency can be related to alpha plus the camber angle and so on but at this point I don't really have the adequate knowledge to answer that question thank you entropy of the universe is always going to increase can you tell me a point in the atmosphere at which the entropy is constant according to the layer of the atmosphere so I am not sure whether you know anything can be said like this because everywhere the state of the system is constantly changing so if you can actually go ahead and find out some place where the state of the system is not changing for a long time then you can see all its properties are the same including the entropy whereas you know steady state for a long time is really a pretty much difficult thing but let's say you can observe certain points where the state is not changing maybe you can see such thing but I am not very sure that you can actually pin somewhere and say the entropy here never changes or the energy here never changes thank you it may be possible I think yeah I mean anything is possible but I don't think you know you can actually go ahead and pinpoint such situation I mean maybe you can go in the universe at some point where no one exists and say the entropy and energy here will never change but suddenly some particle may come there and change everything so nothing is steady state for a long time so I don't think there is going to be any such point ever for a very very long time but for a reasonable period of time maybe you can always you know search for such point that is all I can say we have read in our books that entropy is always going to increase yes so what do you want to say means there is a point in that universe means we have to search that point I think so I am not very clear what do you want to say what are you talking of are you want some point where the entropy never change yes sir I want that point in the atmosphere so I am not sure whether you know you can go ahead and search for such a point so that's what I said that you know I am not sure whether you can do it thank you okay I think there are no more questions so we are closing the session again and tomorrow professor Gaithonday will come back thank you