 So, degree of segregation or micro mixing, this is what we have been discussing yesterday and we also for, yeah, then we took batch reactor and also ideal plug flow reactor and then we found that whether you have micro fluid or macro fluid, there is no, yeah same conversion, there is no difference. So whenever you have batch reactor and plug flow reactor, if you have micro fluid or macro fluid does not matter, right and then we went to mixed flow reactor and then we again started thinking about micro fluid and macro fluid. Micro fluid is the one which you have already done and we know the equation for first order, second order or even third order if I have mixed flow reactor, micro fluid, micro fluid is the one with individual molecules and all that, okay. So that is the one and then we went to macro fluid. Macro fluid it behaves differently because the molecules are not communicating with each other, each pocket only stays sometime and the conversion in that pocket depends on how much time it has spent inside the reactor and RTD gives me that information, how many pockets, how much time, right. How many pockets means it is the fraction of material between time t and t plus delta t is ET dt. In that ET dt I have let us say 10 percent of the fraction. So in that 10 percent of the fraction I will have may be millions of pockets because pocket size are very, very small and I am saying this big but it is never, right. So that is why. Then we wrote the segregated fluid design expression, okay. What was that? This is CA bar by CA0, this is averaging only 0 to infinity CA by CA0 batch ET dt, right. So this is CA by CA0 at batch is the one which I can imagine my pockets are batch reactors and those batch reacts are spending sometime and that time will be given by this ET dt. So correspondingly now I can find out this, you know, the conversion in a segregated fluid, okay. So micro fluid, ideal PFR, I mean ideal MFR, ideal PFR, all that we know how to write the balance. That is also no problem, good. So then for second order reaction we have developed an equation for N equal to 2, I just gave that integral expression before leaving, this is 1 by T bar, correct, T bar M, T bar or T bar M I am writing, okay T bar, okay. So then 0 to infinity and for second order reaction we know, yeah. But for ET it is an ideal MFR, okay, this is for MFR. PFR and all that we said there is no change between micro fluid and micro fluid. That is why only for this we are deriving equation N equal to 2. So this is E power minus T by T bar divided by 1 plus CA0 T into dt, right. So this cannot be integrated, so only thing is it has to be transformed into the form of exponential integral, okay. So exponential integral is defined as E of I X, we have already done once is X to infinity, I may have minus T by T dt. If I am able to convert this format into this format, right, so then I have an exponential integral and depending on the value of X there because E A X, right, if this X equal to 2 that means the value depends on only on lower limit. If X equal to 2 I will go to my mathematical tables and then see what is the value of X equal to 2, right. And the latest addition of Levenspiel he also given you know expansion X equal to I mean you can substitute for any X and then calculate, he has given an equation also in that, right, okay. So otherwise tables are available then you can just go and read that. So now our idea is to convert this into this format. So the first thing what we do here is let alpha equal to 1 by K CA0 T bar, in fact this is inverse of demkohler number, right, so this is the one. Now let us say that we have K CA0 T, this also can be written as 1 plus K CA0 T bar by theta where theta equal to T by T bar, okay. So I just multiplied and divided by T by T bar divided by T bar and multiplied by T bar so that became theta, right. So now this equation is also I think 1 plus theta by alpha, correct, no, alpha is this definition and that is also equal to alpha plus theta by alpha, okay. So did I give any equation numbers earlier for this, okay. Then I think you put this one 1, this one 2 and this one 3, okay, good. So now let us also take because I have to convert this, you know the entire thing is now to convert this equation into exponential integral, that is why, right. So now let us also take alpha plus theta as alpha plus T by T bar and now let us differentiate this, okay, also equal to X and let me differentiate this, so if I differentiate this, d of alpha theta equal to how much? Yeah, this also, d of alpha, alpha is a constant please remember, correct, no? It is a constant value. When I do that what I get here is d theta by, okay I will write here, d theta, this is equation number 4, yeah, d theta equal to, yeah, d theta equal to dT by T bar, right, good. So now substitute all these values here alpha, okay, so substituting equation 3, 4 and 5, this if I take as 5, what do you get? Can someone quickly tell? C A bar by C L, substituting equation 3, 4, 5 in 2, 3, 4, 5 in 2, no? Got it, you did not do your notes sir, okay. But I said also we have to only till here, yeah, someone quickly, okay, I will write may be slightly difficult for you. Let me write that you know I have alpha, E alpha, E power minus alpha theta divided by alpha theta into d of alpha theta. I added here E power alpha and minus alpha because here I have only T by T bar, right, so I multiplied by, d by T bar is nothing but theta, that is no problem, E power minus theta. Then I added minus alpha and also multiplied by plus alpha, so that will become 1, right. So that manipulation, why all that manipulation is only to get this, now I also told you so alpha theta equal to X and alpha E power alpha is it constant? So they will come out, so I will have here alpha E power alpha, limits I will just put there, then we have this is minus X by X d X, okay. Now taking here the lower limit when T equal to 0, theta also 0, then that will be lower limit will be alpha, lower limit will be alpha, upper limit anyway theta infinity also, okay. So now you do not have to take this one, when theta equal to 0, X equal to alpha because X, everything is written in terms of X, no, these are X, right. So when theta equal to do not, you need not worry about that one now, theta equal to 0, X equal to alpha, that is lower limit. So theta equal to infinity, what is X value? Also infinity, yes, also infinity, yes. So now this is an exponential integral for us. So what is the solution for this? This entire thing is definition of exponential, so we have A of alpha. So that is what is the conversion in a segregated fluid, second order M F R, right. So if I now tell what is the value of alpha, alpha is nothing but 1 by Damkohler number, right. If I say that Damkohler number equal to 1, 1 by alpha also equal to 1, Damkohler number, yes. So then you will have 1, E power 1 and E I 1, E I 1 value I have to get from, right, okay. So if you see Levenspiel, I think I will write also that one for the sake of completeness, yes. So E I X is given as, yes, of course, integral is same thing, X infinity E power minus T by T dt, because these are limits or any variable I can put there, because this is definite integral, so that is why. Then we have the values minus 0.57721 minus ln X plus X minus X square by 2 into 2 factorial plus, I think I have given this one earlier also, but I am sure you will not remember that minus, etc. So now if X equal to 1, you can substitute X there and then calculate what is X. Otherwise tables are given, directly you can go to the tables and then get it, okay, good. So this is what, now if I say that I have alpha equal to 2, that means damkohler number equal to 0.5, damkohler number equal to 0.5, I have the values here, what is the conversion? CA by CA naught, I will also give you that values, good. If I have alpha equal to 2, that means alpha equal to 1 by CA naught T bar, okay, that is equal to 2. So that means damkohler number equal to half, right. If I have that value, how much is that? I think I can also give you the value of E of I2 is 0.0489, 0.0489. The other values you can check. By the way what is that you have to calculate? Yeah, CA bar by CA naught in equation 6, that is what you have to calculate. Yeah, 0.7 Xl, so C bar A by CA naught equal to 0.7 226 and XA equal to 0.2774. So we are now talking about macro fluid that is there in ideal mixed flow reactor. So now we have to compare this with micro fluid. What is the equation for micro fluid? What is the equation for micro fluid? Micro fluid is the fluid which you have already done in your B-tech, mixed flow, we are only talking about mixed flow reactor, you cannot say easily because it is the second order reaction, it is quadratic, right. So for micro, for micro fluid, okay, otherwise you will never forget, you will never remember, you know. So that means you know B-tech, you know that means you have already studied that in B-tech, right. Yeah, micro fluid B-tech, it has got degree also now. So N equal to 2. So the equation is quadratic equation CA by CA naught equal to minus 1 plus square root of 1 plus 4 K CA naught T bar divided by 2 K CA naught T bar. So can you tell me this value? What is K CA naught T bar? 0.5, yes. So K CA naught T bar equal to 0.5 which is also equal to 1 by R. Okay, quickly. Yeah, CA by CA naught is 0.732, 7321 correspondingly conversion is 0.2679. What is the conclusion? This is XA micro, correct no? So this is XA macro, macro fluid, that is what the meaning of that. So what should be the conclusion? Which should be more? Micro is more. Micro is more, why? Mixing is less. Yeah, Rajasri, you are trying to say something? Yeah, because for second order or N greater than 1, you have to maintain the concentrations as high as possible. So here in the pockets, the concentration is maintained as high as possible but still it has residence time distribution. That means some packets coming early, some packets coming late. But if you are able to put all these packets again in plug flow, you will get still more conversion because all of them would have converted into same conversion whereas now depending on its residence time distribution 0 to infinity exponential decay, now you are getting this much conversion. If you do now for half order reaction which we are not doing, so again you will get the reverse one. That means XA micro here for second order for N equal to 2, XA macro is greater than XA micro, right. So I think you know at least these things will again reiterate and also reconfirm what we have discussed in earlier B Tech and also of course beginning of this course, right. I do not know in B Tech anyone told you like this, you know all that concepts like why we should have for second order, more conversions, okay. That is all, yeah, when you have second orders you have to maintain the concentrations high and all that, right. But I think most of our system is only examination oriented. So but here at least I think you can appreciate now why, what is really happened at that time, now at least we try to understand that, okay, good. So this is the one. Now what is the conclusion for degree of segregation or micro mixing? Micro mixing is one of the components of the overall mixing when you are talking about direct use of RTD to estimate conversion, okay. What was the real problem? Real problem was that other than first order reactions we have to use either late mixing or early mixing, late and early or simple humanities words, correct, no, humanities, English department words. But how late if someone asks, technically you should be able to tell. Now to define that how late we have now defined a fluid called macro fluid which will never mix, that means I think any number of years it may stay inside the reactor but it will not mix with other molecules, that is what the latest one. So our one, our definition is that as long as it is there inside the reactor, if it is not at all reacting with any other molecules or communicating with any other molecules that is the latest mixing, right, that is the one. How early also is the one, the question because in our, this is a wonderful model, Netherlands I think it is, I am not able to recall it must be Crammers, you know who told that first, right, Cramer, K R A M E R Crammer not fine Crammer I think only Crammer, so this, this example, this is a wonderful example and yeah. So here we say early mixing and here we have late mixing and second question early is also answered by saying that we have a micro fluid which can mix any, the moment, the instant it enters the reactor, provided the reactor is allowing mixing, that is why that macro mixing is defined by reactors or reactors define, sorry macro mixing is defined, okay reactors, RTD is defined by macro mixing, reactors, how late, how early because here no, these two reactors only tell me how late or how early because this tells me that the reaction is late, okay mixing not reaction, mixing is late and this one early, right. So that is why the macro mixing which is connected with RTD that will clearly define saying that okay if I have a particular reactor it may allow mixing, it may not allow mixing because we have two reactors which allows perfect mixing, the other allows zero mixing, no mixing, it will not allow any mixing at all. So that is why to suit them we define this micro fluid and micro fluid, right I think the connection must be clear now, right. So only to answer this question early and late and early we have defined two extremes, one is a fluid which can immediately instantaneously mix, provided it is allowed to mix because that mixing is not taking place the moment I put the same micro fluid in plug flow reactor because by definition plug flow does not allow any mixing, mixing means overlapping of ages that is also another way of defining mixing, okay overlapping of ages. So that means if first year M Tech people and second year M Tech people if you do not meet them at all so that is one packet and second year M Tech is another packet, okay this is segregated flow but if you both are allowed to mix so that means your age inside the IIT you just join now so that may be six months and your seniors will be one and a half months, one and a half years, one and a half years and now their age is different your age is different but still you are able to mix. So that means overlapping of ages is also indirect way of saying that we have mixing. What is happening in an ideal mixed flow reactor the moment you have under steady state conditions something is entering something is coming and inside you have zero to infinity residence time distribution the moment you send any fresh batch so now the molecules can mix with any other molecule irrespective of its age, correct no? So one molecule may go just entered may go and attach with a molecule which is now you know fifteen minutes back it entered one here one minute or one second I said yeah, okay one second and fifteen minutes back some another molecule entered both may mix together and then just come out, okay. So that is what is the life expectancy so after they come together then life expectancy is yeah same for them because both of them are living together, right that we cannot have a good example in M text, right yeah because even though you this is plug flow that is why you do not have that example, right because you have to spend twenty four months and your seniors also have to spend twenty four months to get the degree, okay. So that is what that is what is another mix another definition of mixing I think that also I will just quickly discuss very quickly ten o clock yeah so that so this is fine no this is very clear and lesson from here is do not worry when you have batch reactor or plug flow whether you have macro fluid or micro fluid you do not have to worry so micro mixing macro mixing will not come into picture and all the all things like degree of segregation will not come into picture maximum mixing as will not come into picture all these things will disappear problem is only with M F R, right. So M F R also micro fluid you have already the equations B tech, right I mean R M tech beginning, right, okay and only segregated fluid we have the problem just to imagine that how do you what is happening for the segregated fluid inside the mixed flow reactor that is what what we have defined here so depending on how many fractions are how many packets are spending within this time the time that is may be one minute two minutes three minutes like that you have the conversion in that fraction like that all fractions are added together this is nothing but sigma, right yeah so then you will get the average conversion or average concentration that is left inside the packet, okay good. So the other one what I thought I will just quickly tell is about okay that I will tell later then the next one we will take you know, okay that alpha gamma I will tell you know that is age and life expectancy I can tell a little bit later also so, okay. So what is the next one we had two parameters micro mixing and macro mixing so micro mixing we have discussion now we have taken three reactors, okay and then we have discussed the first two reactors batch and plug flow no problem at all only MFR segregated fluid is the problem now we have proved that even here macro fluid gives more conversion than micro fluid because micro fluid allows mixing whereas micro fluid does not allow mixing so that is why in fact the Damkohler number what we have taken is very small point five correct no yeah if you take larger Damkohler numbers then the conversion difference will be slightly much better here we have only how much you said oh this is point this is 27 point 7 percent the other one is 26 point 7 oh yeah that is all that means only you know 0.01 percent, right you know 0.01 difference, okay 1 percent difference but if you go to larger and larger Damkohler numbers you will see clear difference, right good nice so now what is the next one that is left now maximum mixing as, okay maximum mixing as or what is the other word for this in the bracket macro mixing macro mixing is the word that is given there good yeah let them settle and let me also draw this this is the degree of segregation this is segregated maximum mixing as the other one is this, okay this is also of course micro mixing and then here I have P f below no P f top P f below M f top so this is R T D equal to no I have not written earlier 0 and this is R T D equal to 0 to infinity, okay that is the two extremes and macro mixing is increasing in this direction macro mixing, okay and I also have all intermediate R T D's so all these things are intermediate R T D's these are the extremes, okay good so these are the things only what we are talking now macro mixing macro mixing should define R T D, right macro mixing should define R T D that means on this scale I have a 0 R T D and on this scale here at this end I have 0 to infinite R T D macro mixing is a name that is given to describe R T D's, okay good so now I will give you the extremes first like because now we are talking about R T D only we are not talking about reactors I will say that I have a system with direct delta function R T D I have a system with direct delta function R T D so what is the meaning of that I idealized idealized pulse or in the bracket or delta function I can write delta function R T D, okay this is case 1 so one extreme we are talking what are the how many kinds of reactors can give me this kind of idealized pulse R T D we are asking the reverse question if I have plug flow definitely I will get idealized pulse, okay right if I have plug flow I am not telling anything about the reactor now I am only telling about I have a pulse what are the systems you can bring under this pulse that can describe R T D what are the other reactors or whatever possible reactors, okay that will give me idealized pulse what is the meaning of idealized pulse what is the meaning of idealized pulse not mathematical meaning physical meaning yeah zero time what what do you mean by zero time idealized pulse this we are talking about outlet only we are talking about heat is that is the one each and every particle must spend exactly same time I think in this semester I could have told the same thing I think thousands of times if someone would have counted, okay yeah so here also in a different way only we are trying to tell again so if someone tells me that I have an idealized pulse R T D the meaning is that each and every particle spending exactly same time so what is the reactor that makes this kind of zero residence time possible now we are asking reverse question if I have plug flow I will definitely get this because that is the definition of plug flow, okay Savita left no, okay good so if I have idealized plug flow then definitely I know that I will get their delta function why by definition each and every particle should spend exactly same time now I am asking the reverse question I do not know what system I have but someone came and told me that this is the pulse I got from my experiment that is exit pulse only what we are talking that means ET only, right so otherwise I can also write this is ET equal to delta T minus T bar because I cannot write T bar P it can be anything, okay so at T bar all the things are coming out so before that or beyond that I think delta equal to zero good yeah so what is what is that possible which system will give me direct delta function as ET yeah only system that is possible here is only plug flow no other system can give because direct delta function thickness equal to zero what is the meaning that means width equal to zero distribution equal to zero so when can you get this unless each and every particle spends exactly same time you will never get that so when each and every particle spending exactly same time what must be the reactor plug flow and if you consider the other one you know you cannot say it is it is not a flow system in batch also residence time distributions are zero correct no in batch also now when I have this kind of idealized pulse ET equal to delta T minus T bar and do I have to worry about micro fluid or macro fluid you do not have to worry so again even in macro mixing scale in this scale this end is very good end for me trouble less end because even in macro mixing scale if I have RTD equal to zero that is ET delta T minus T bar again I do not have to worry whether I have micro fluid or macro fluid why we have to worry about micro fluid or macro fluid micro fluid allows mixing whereas macro fluid will not allow mixing right so because it is a plug flow reactor whether I have micro fluid macro fluid both should spend exactly same time conversion will be same in that so when I take average micro fluid or macro fluid or batch reactor should give me exactly same at least you know number of times when I am telling it should be permanently somewhere adsorbed in your brain till you die even if you do not join you know chemical engineering job because I think so many times you heard so I think you cannot forget unless you take a brush and then really rub it out okay I do not think you can reach that with a brush the moment you reach it out again so that is the reason why so many things you know again it is only repetition I told you know we are not learning anything new concepts these are the same concepts but by number different number of ways when you are learning your learning will be perfect otherwise only once if you tell only that aika what is that called aika santha grahi aika santha grahi and I think in Sanskrit they call them because only once they hear and then they can remember it is like computer computer is aika santha grahi because one click enough it stores correct no it will never unless you go and delete it is there permanently till computer dies unfortunately our brains are not like that so that is the reason why I have to repeat number of times right okay good so again when you have this delta only P f plug flow reactor will give idealized idealized pulse okay hence macro and macro fluid and micro fluid give same conversion same conversion in plug flow right it is a continuous there is no comma there and there is no pulse stop there is no comma hence micro fluid macro fluid will give the same conversion excellent okay good now you can store this one in a separate file okay in your brain so now next one next question is let us take now exponential decay curve that is other extreme in between I think you know we do not have to do anything okay so now for I am not saying we have MFR I have simply exponential decay curve second one is exponential decay curve right that means I have here ET versus T then exponential decay okay so now what is the system which gives me this a plug flow yeah so ideal mixed flow is one this is one okay ideal MFR any other system gives are only this is possible difficult to answer I will tell you there are many many many types of systems which will give the same exponential decay right let us see there let us see how many are there few of them we will take and you will appreciate this okay now let me take to create this kind of exponential decay I will take plug flow small one slightly bigger one slightly bigger one slightly bigger one slightly bigger one to match all this so that means I may have a system where still down like this like this because theoretically this is possible like this any number I can draw so now and again so this is volumetric flow rate this is volumetric flow rate this also gives me same exponential decay correct no because this particular one which is coming very very quickly may be between 0 to 1 minute is coming here so next one 1 to 2 minutes is coming here right so like that the entire thing I can simulate using any number of parallel PFRs these are parallel PFRs yes see what is your idea idea is only to get that much fraction right so between fraction spending a time between T and T plus delta T so to get that fraction you have to naturally change the flow rates right the idea is whether you are able to get that kind of ET versus T graph or not right this is one possibility now you see another very beautiful possibility another possibility is I have ideal plug flow okay this is ideal plug flow so then what I do here is this also gives me same exponential decay correct because I have a parallel pipe and then I am introducing here which is just coming out where is this point Swami here because that is coming very quickly right yeah and also that is coming very quickly at this point the life expectancy of a molecule which is coming here and also its age two ages are different right this is just entering and it is continuous flow so the other one probably which has entered here would have come and then just join both will just go out right so that is what is exactly what we call as mixing mixing is whenever you have overlapping of ages I told you know example if you are mixing with your seniors then you have mixing and if you are if you are not talking to them and then if you don't know also who they are right your mtech class is a one packet and your senior mtech class is another packet that is segregated for it right so when you mix then it is yeah mixed it is I think you cannot say it is maximum mixingness maximum mixingness is from 0 to infinity okay residence time distributions and 0 to infinity life expectancy is when they overlap each other that is maximum mixingness that is this corner that will happen only in case of which reactor only mf or for micro fluid that is why it is called maximum mixingness and micro fluid will give that kind of thing if I have that kind of overlap between life expectancy and the what is that other one age right and also that is why again beautifully we say that inside contents of mixed flow reactor is exactly same as outside outlet that is why the distribution of residence time inside and residence time distribution outside again exact otherwise that condition is not fulfilled without knowing all this in the beginning itself what we say in the btech yeah assume that temperature is same concentration is same conversion is same inside the reactor and outside the reactor I also did not tell you at that time because if I tell you all that at that time by this time you could have totally confused okay unless otherwise you know I teach it totally in different way tell the concepts first and then you can go that is another way of telling so normally teaching is two styles one is that you teach generalities and go to specifics like transport phenomena you take navier stokes equation right that is the generality so any system can be described under navier stokes now depending on which system you are talking throughout some terms that will be specific applicable to some specific system otherwise you can now start with specific system assume simplest one with so many assumptions simpler system and then remove one by one assumptions later slowly complicate take another system with less assumptions another system still less assumptions another system still less assumptions finally we will end up with whole navier stokes equation so here also we can do the same thing in reaction engineering also first giving the theory of all this micro mixing micro mixing and all that then apply to first order reaction second order reaction separately mixed flow separately plug flow all that we can say luckily we have only two systems here plug flow and mixed flow okay good so this is the one and not only this there is another one you can see or if you have some more things also you can tell me so this is also an ideal PFR PFR is always ideal PFR now you can see how nicely this can also be managed I will now send the flow like this and withdraw here this system also gives me exponential decay so what is that we have learnt there nothing that may be the truth but truth hurts given this exponential decay if I block all these you can never save what system we have that is what exactly this also tells me the residence term distribution curve for this is simply like this correct no ET versus T but now if I give you this one and ask you to calculate conversion you can take some system with early mixing you can take some system with late mixing okay good can you tell me in that four systems which one will be early mixing which one will be late mixing ideal MFR is early mixing but you are perfecting another condition yeah that also then you have to say now you see you are becoming more and more knowledgeable no so definitely yes okay because the question comes when she says that ideal MFR will give me perfect mixing it won't give if you have micro fluid so now we have to say that if we have micro fluid this system will give me okay let us take first micro fluid and then discuss four systems okay so which one will give me early mixing if I have to take micro fluid this gives me micro fluid this gives this one let us mix no mixing okay so here micro fluid behaves as if micro fluid so that is the one so this one it is yeah micro mixing early mixing or late mixing here yeah why because these streams are mixing with the other flow where they have different residence time distributions right so this one is early mixing okay for micro fluid and this one late mixing because inside you are not disturbing at all you are only simply taking out the streams and this is an ideal plug flow right this is an ideal plug flow simply you are withdrawing without disturbing these are all theoretical this is what I told you that jittering paper that fellow really thought I say I really appreciate those minds why I think at least one millionth of that kind of mind is not there for me also really because they simply sit down and then try to think and you know most of those those times most of the papers were single other because of that late mixing okay most of okay so most of the papers were single other papers because he is the only person sitting and thinking that is all and he gets the idea and then may be checking with friends and all that whether there is logical error okay and if there is no error then finally write a paper and then send wonderful so that is why but we have to know still we do not have to worry for this kind of ideal thing because when I have micro fluid right then I have to identify whether I have late mixing or early mixing and if it is late mixing or here segregated flow this is segregated flow this is segregated flow then I have to use this right and in this case and in this case I have early mixing right so here I have an equation we take if it is first order or second order third order you know that normal V by F not equal to XA by minus RA that is for micro fluid please remember that that is for only micro fluid so I can find out that there is no problem at all good so this is the one and yeah okay so now we have seen exponential decay for any other RTD for any other RTD it is not that easy to find out but there is wonderful information which is not useful but information is good useful in the sense that you know you cannot use them for calculating conversions because they are very very complicated so that is why people are it is not popular those papers there are few papers Levenspiel also gave in his chapter and any kind of intermediate RTD is that means some late mixing and some early mixing okay some late mixing and some early mixing so these two can be managed using only these two systems so this is early mixing this is late mixing right so that means let me say that I have few packets and few individual molecules this individual molecules will give me early mixing packets will give me latest mixing that is possible so in a real system how do I know how many are there how many packets are there I do not know but if there are really some packets and then some molecules now you imagine this model you see individual molecules will give me early mixing that is micro fluid and packets will give me the late mixing so you take this is 30 percent this is 70 percent and then calculate overall conversion right like either parallely you can put or series you can put these two systems either series in parallel and then you try to calculate by a fitted parameter called the percentage of the same number of packets and the number of individual molecules in terms of some percentage okay 10 percent of the fluid is in packets 90 percent is in individual molecules that means where are you on the scale where are you 10 percent I said 10 percent packets 90 percent individual molecules only 10 percent packets and 90 percent so similarly now you take this one as 90 percent this is as 10 percent and you calculate conversion do the experiment find out whether that conversion is same as this if that is same by hook or crook you got the values so then it must be same that is what what we confirm so that is why other than the extremes we are not doing anything but you should know how the extremes also should be manipulated using only these two systems so this is ideal plug flow and ideal plug flow but this gives me early mixing that gives me late mixing but both the systems I can manage such that my external my actual conversion is matched with some adjustment between these two things you know flow also is a parameter how much is going in that how much is going that that is one parameter and fraction also is one parameter so both you can just try and then try to find out it is very difficult that is why none of us are trying to do that but knowing information there is nothing wrong okay good so that is what so this is what is completed RTD but only thing I just wanted to tell you was the other one what is that I left it because I think some late comers may also use this I think I will again so this is clear no so we have defined two components of mixing one is micro mixing one is micro mixing okay micro mixing gives me degree of segregation I have to vary only when I have mixed flow reactor and that gives me macro fluid and micro fluid some difference whether it is first order second order sorry whether it is second order or half order for first order it is absolutely no problem okay that is why in 6th order of lounge field we say that if you have first order any order of you know either plug flow and mixing flow for n equal to 1 so reason is this good and then we came to maximum mixing as the scale P f and M f and this RTD equal to zero distribution we discussed that is possible only for P f r right so P f r means again I do not have to worry about whether I have micro fluid or macro fluid or segregated flow or I know early mixing or late mixing because both will give me exactly the same conversion and in exponential decay when I have I can have any number of possibilities but you have to identify them which is late mixing which is early mixing and if it is the latest mixing that is possible we have this equation right and if if this is M f r alone we already did it it is B take in between any RTD if I take in between any RTD I take jweitering has given that equation that equation I will just write we know we are not deriving that that equation I will give you yeah any RTD okay this I can remove so now we have seen two extremes but any RTD in between any RTD between P f and M f equation given by jweitering I sent that jweitering equation to all of you this is dCA by d lambda equal to minus rA directly he has calculated conversions because our idea is direct use of RTD estimation to estimate conversion okay plus CA minus CA naught e lambda by 1 minus f lambda so this is the equation and the boundary condition is lambda equal to infinity CA equal to CA naught lambda equal to life expectancy that is why the boundary condition lambda equal to infinity means do I know what is the life expectancy in a general reactor when the molecules are entering do I know that value but definitely it is not zero correct no definitely it is not zero because it has to enter the reactor and then come out unless you have 100 percent bypass 100 percent bypass means you don't have any reactor because it is not it is supposed to go to reactor it is not going into reactor so that is why we take that value but lambda is lambda equal to infinity is the large one of the largest value you have to take and then slowly solve that that is why I told you it is not very popular so which value you have to take and then solve this particular equation and for first order yeah this is e lambda is again you know the exitage distribution function and f lambda but in terms of life expectancy in fact I told you in the beginning for e t it is also a probability a probability function correct no you measured there at the outlet saying that the fraction of material coming between t and t plus delta t is e t d t the same thing I can also tell in the beginning itself correct no so this is a probability the probability is that this e t delta t the fraction of material which is coming which will come between time t t and t plus delta t will be e t d t but where you are telling normally we take only at the exit because we would like to calculate conversions but same thing can be imagined in the beginning also as if that much fraction is going to come only between second minute and third minute or fourth minute and fifth minute sixth minute or seventh minute so that is why so this is not a difficult problem to understand so that is why but this is 1 minus yeah what is 1 minus f lambda high lambda of course t bar high lambda will come there okay so anyway that we do not write normally so this is the one for first order also you can solve and you will get immediately for first order m f r first order p f r one can easily solve this okay so that means minus r a into k into c a and what is e lambda and f lambda for ideal c s t r you know the equations that equations we can correspondingly put and then one can solve okay good so that lambda to understand a little bit then little bit jewellering defined a function called residence time okay I think I will write okay or let me write residence time residence time t okay can be imagined as alpha plus lambda is lambda will you know gamma okay alpha plus lambda where t is the residence time that means any molecule which has entered definitely will spend some time to come out let us say it has come out in five minutes so somewhere two minutes before if you if I would have looked into the reactor then I would have seen that it has three minutes age alpha and remaining two minutes is life expectancy again easiest way of understanding is our own age the moment we come to this world our age starts so and life expectancy we do not know assuming that a person is living only for 45 years so when you look at him at 20 years life expectancy we can now say that 25 years left but I only an ideal plug flow every particle will have and every age will be correctly associated with the lambda that means definitely I know if a molecule is spending one minute totally is 10 minutes so nine minutes definitely that will spend and I also know that if a molecule spent already age three minutes so we'll have again another seven minutes to go but all other real systems you can never expect this lambda right so that is why what he says is if you imagine that I have the reactor at this point right just before entering alpha equal to zero correct no just about to enter age it has not yet entered just zero plus the age equal to zero and when it is just about to leave lambda equal to correct no so at this point you have alpha equal to t t is the residence time and at this point alpha zero you have life expectancy we do not know how much but still you have the life expectancy what he says mixing is it is a kind of transformation from that means all the molecules alpha equal to zero here all the molecules have alpha equal to zero so I can call this there is a uniform environment of molecules with alpha equal to zero right uniform environment all molecules will experience the same thing that is what uniform okay good at this point now I have different ages but life expectancy equal to zero so that means again in this corner we have another uniform environment called age correct no because age it is already come out alpha alpha of course alpha equal to t because residence time right so he says that some kind of transformation is taking place from this uniform environment where alpha equal to zero to life expectancy equal to zero so if it is as quickly as possible if the transformation is as quickly as possible then we have what is called it is taking as quickly as possible we have what is called maximum mixing that is m f r okay no no my micro mixing is a overall scale we are talking about only this point maximum mixingness okay you know once more so I have an uniform environment here where alpha equal to yeah where alpha equal to zero and I have another uniform environment where lambda equal to zero this is zero life expectancy and this is zero age environment so now if this zero is environment as quickly going to lambda zero environment then it must be because of maximum mixingness otherwise it cannot go no if it immediately taking place otherwise if it is plug flow what will happen that at least you understand if it is plug flow each and every particle must spend exactly time t okay we are talking about individual molecule time t right it is not t bar it is time t but t bar equal to time t for plug flow reactor so that is what so that means in any other reactor if the transformation is very fast if it is the quickest means that should happen only maximum mixingness and that is possible only for micro fluid and m f r okay that is why he has used this lambda in his equation he discussed this first and then derived this equation and the derivation is given in Foggler which I am not asking but it is only just for your information okay for examination you do not have to worry so this lambda life expectancy is no from here that means if I have alpha zero t is lambda right here alpha zero t is lambda at this point and at this point when I have lambda zero t equal to alpha so from this lambda transformation to alpha transformation is if it is very quick then we have what is called maximum mixingness if it is delayed as much as possible excellent that is the other extreme segregation okay as much as possible means I mean within the reactor that is not happening then you have segregation so based on that he has now derived this equation and that equation is only for your information good I think this is over and there is a very nice problem given in Levenspiel that is a beautiful problem in third edition second edition also it is there that is problem number 16.1 16.1 that problem I think all of you can go through that but I think there is it is very easy to understand because we already discussed in most of it there the problem is something like this okay that is a very beautiful problem in Levenspiel that problem is really wonderful problem right so the problem is given as this is 16.1 I told problem it is not problem actually it is solved example example the problem is given like this ET curve ET versus t and we have here the exponential decay yeah exponential right yeah so of course here A equal to 1 and all that somewhere here I have t and this is t bar by 2 okay what are the systems you may you know this is only exponential decay what I have I know this is only ET versus t curve only what I have now what are the possibilities I can imagine here the first one is ideal PFR 1 1 A is sorry not ideal mixed flow by PFR that is 1 B is just reverse okay just reverse and the same thing here I have okay you see RTD again cannot differentiate whether you have micro fluid or macro fluid so this can be only for micro fluid okay same two schemes I can also have for macro fluid C is right yeah but I have to show here pockets so that is the one and is there any other possibility like exponential decay just now I have drawn earlier E I have yeah I have PFR okay now I also have parallel PFRs so like it goes that is also possible right yeah these are the possible schemes now out of all these which one is the latest mixing which one is the earliest mixing D is the latest mixing D okay next one that is all only one C also is the latest mixing no no no no no no no no no no no no no no no someone is saying SS I will ask them would you say SS see again you are forgetting definition by definition of micro macro fluid it can never mix okay so here I have macro fluid here I have macro fluid so both are late mixing or early mixing late mixing and what about here it is definitely late mixing and we do not have to care whether that is why we have not written separately for micro fluid macro fluid either macro fluid or micro fluid both will give me exactly same thing so that is why all these three we have to use segregated model what is segregated model this equation this equation so if you know ET and you have second order reaction you can you can solve it beautifully and the first two A and B here early mixing because it is micro fluid but this is not equivalent to maximum mixingness please remember that maximum mixingness will be only for this one but you also have some other non-mixing component correct no you have a plug flow in between so that is why this is early mixing and this is fairly late mixing why we say fairly late mixing compared to what yeah compared to these three this is the latest mixing possible this is late but not as late as this right because here I have plug flow then I have mixed flow where mixing is taking place there at molecular level so this is the wonderful problem please go through that and very nicely solved simple and these two solutions already we have in the last class we have given X macro X micro and all that okay so this we have and yeah so now we summarize I requested Kannan okay whether you can tolerate if I send these people 15 20 minutes late he said no problem okay Kannan class you you are whose class is yours is your you would like to go not now or then you have to go anyway okay then anyway the so if you want you can leave now or not now after your test after five minutes also you can go no problem okay now let me tell the summary of what we have done micro mixing and macro mixing okay PFR maximum mixiness maximum mixiness or segregated flow yeah so both will give me okay micro fluid plus macro fluid macro fluid behaves behaves same okay then MFR again we have maximum mixiness okay that is mainly micro fluid yeah this is normal behavior normal behavior normal behavior instant like your B Tech behavior okay yeah design design equations derived in B Tech yeah okay then we have this one is segregated fluid segregated flow okay macro fluid this is one extreme correct no that is why you are assuming that you know segregated fluid is one extreme okay one extreme equations derived in M Tech okay now yeah yeah this is MFR and in general segregated fluid if I have segregated flow you have a design expression 0 to infinity CA by CA bar batch you know ET DT so segregated flow any RTD any RTD any RTD you can use that equation absolutely no problem that is 0 to infinity CA by CA bar by CA not equal to 0 to infinity CA by CA not batch ET DT okay so here this is a design equation derived in M Tech M Tech at IATM that is important because you may forget where you have done your M Tech later possible no what is the sir Ramakrishna you are very angry sir Ramakrishna angry yeah oh that is why no okay good or of course M Tech IATD or IATM address for him okay so now maximum mixiness maximum mixiness any RTD yeah where did you learn something anything about this maximum mixiness any RTD jwetering jwetering equation mentioned in M Tech IATM why M Tech MS M Tech MS VHD so that is right M Tech CHE chemical engineer oh I think Arya also is there where is Arya oh she is there okay okay jwetering okay M Tech we have written so M Tech okay jwetering equation that is all good yeah see when compared to jwetering equation there is another technique which you have already done you may not know that you have done that already okay when you have micro fluid there is another way of doing things what is that what is that before this direct use of RTD what is that we have done what did we do there what is the fluid we assume it was micro so if it is macro fluid does not matter you have a beautiful equation 0 to infinity that one okay if it is micro fluid you have both the extremes PFR you know MFR you know in between RTD is difficult but oh jwetering or jwetering equation I thought still it is there jwetering equation is valid but if you want to avoid jwetering equation go to axial dispersion model or go to tanks in series model or go to CSTR with dead sparse dead space and bypass and also number of tanks you know you can have three tanks with dead space and bypass one after the other so that will cover but that is a two step process but this is a direct use of RTD jwetering equation is direct use of RTD that is why we called zero parameter model there is no parameters there in jwetering equation whereas if you go to dead space and bypass you have two parameters and if you take three tanks in series like that then you will have another another parameter so n is a parameter and the other two also parameters okay so that is what so and the equations which we can write now I think I should listen to Rahul because he was telling sir we will take Xerox and give but I thought you know let me write here okay equations for for Kannan students of people I think you know students do not worry I think he said he accepted last night I called him equations for single fluid yeah okay one for micro fluid please take this for micro fluid P F R M F any kinetics otherwise I think I will distribute this one to a later with this only summary again I think I will distribute tomorrow also I can distribute okay I think we will take today tomorrow design class also I can distribute it is only just summary because I have another other things to do so that is why okay okay you can give this one this one this one I think some here some here some cartoons yeah and also now I think we can give sweets you know what this I have to explain yeah this I have to explain and then afterwards this is two pages what happened clapping afterwards okay you got this now or what okay everyone got this yeah now you see we will explain the first one stage one okay stage one is input kinetics ideal contacting because you know only M F R and C S T R sorry P F R and C S T R okay and you see this guy is very happy very bright eyes and then saying that I know reacted design fully but I did not it did not work X I is different and now started weeping Y Y Y and you see lot of tears and bottom also there are lot of tears there okay so that is one then you can see when he thought that you know how I know reacted design fully the collar up and all that is there you notice that okay the collar collar also up there yeah so that is the one and next one you see afterwards again he thought that non-ideal parameters you know a bulb came and then thought that non-ideal parameters is one you know these parameters may affect the conversion and you introduced them he introduced them recirculation dead zones axial mixing channeling bypass and all that because he was confident second time also again collar up there because you know thought that he it will work so then again second time also he did not get same conversion then you know did not work X A again is different Y Y started weeping lot of water and all that then again bulb came then he thought that he has to now take into account macro fluid and macro fluid that he has taken so then I think he is okay in the third stage because collar gone by the time because all pride has gone because thought that there is no use of being no use of being very proud and then started okay introduce the micro fluid and macro fluid and that also did not work because he does not know there is early mixing and late mixing okay so finally in the last one he has introduced kinetics RTD micro fluid macro fluid late mixing early mixing everything then his eyes also got stand now it is straight looking you know no happy and all that that also now you see from the beginning to the end by the time he came to fourth stage no head only solid head that's all so this this cotton will definitely tell you you know so beautiful things now okay and I thought just keep it with you and this is very nice so that you know you will have some you won't forget if you look at a cotton so what are the so important things that one