 so yesterday what did you do about recycle reactor smaller and larger i think still you understood that know smaller and larger why it comes ya but the other areas anyone prove you got it excellent very good one person you have done a good i think two people atleast good okay so we will now go to auto catalytic reactions ya i tell you there are lot of equations that you have to derive for recycle reactor for first order, second order, second order with epsilon and all that and reversible also reversible also a going to r reversible reaction all these things it is only just mathematical exercise so that is why those mathematical exercises you have to do on your own if you are not still believing then you know i cannot conduct 100 tests you know only maximum 3 or 4 only we thought okay first only 3 we will say first afterwards we will see depending on if you are writing very well why 4th all of you for one person i cannot okay there will be always one person who has not done well for that one person alone i cannot conduct test one, test two, test three, test four and all that so on the average if all of you have not done well then we will conduct another test okay that is not a problem so that is why these mathematics are very important those derivations so now let us take auto catalytic reaction recycle reactor ya okay recycle reactor auto catalytic reaction that means you know why recycle reactor is better for auto catalytic reaction but till now we have not discussed about auto catalytic reaction let us discuss also about auto catalytic reaction by name we know that auto catalytic reaction means the product would have catalyzed the reaction okay so the general scheme that is normally written for catalytic reaction ya r is the product that is for the balance okay so if that is not there then it is not auto catalytic reaction because r is again participating and you know the reaction takes place because of that and if it is irreversible i can write this one as C A C R okay and there are also many equation many reactions that are available in particularly organic chemistry where some names also some of them i will give you later right so this reaction is possible so now we would like to find out whether which react is the best for this so before that we should also have an idea what kind of rate versus conversion graph we get so from this rate what we can first discuss is that initially this is C A not to start with and there may be very small amount of product that would have formed very small amount okay so then that small amount is represented by C R to give a value this C R may be point not not 1 and C A not may be 10 for example so when you multiply that then K is constant for a given value then R is smaller then as the reaction is proceeding R is falling i mean R is increasing more and more and C A is decreasing also along with the reaction right so that is why this product we will first go start with lower value then go on increasing reaches a maximum where C A and C R values are sufficiently large and the product is larger then afterwards C R will be increasing but C A will be decreasing and finally it may go to almost point not not not 1 so that it will again rate will be decreasing so that is why if i plot minus R A versus C A other thing i will also tell you ask you this is rate versus time these two okay so first tell me this one how does this look like yeah initially it may be because we are this side or this side i have to start this side why because this is C A not okay that side C A not then it may increase like this and then finally it may go to 0 also because when C A equal to 0 C A equal to 0 so that product will be 0 so the direction we can show like this with the time how does that look like same parabola yeah so at time t equal to 0 i may have some reaction because i have some C R left otherwise it will never react okay some C R is there so it may start and then go to like this it may go to okay so this is the one okay yeah so it seems in nature it is also found that you may start with A there is no product right you are not adding any product there and on its own there is a route where a little bit of R is formed and then that R once it is formed then it starts again auto catalytic okay so anyway different schemes are there but i think what we think is that anyway whether we have R in the beginning or later our reaction starts only when you have sufficient amount of R which is now started catalyzing the reaction good okay and if i want to plot this as 1 by minus r A versus X A 1 by minus r A versus X A now tell me how do i S shape yeah puja says like this thereby and says like this S shape yeah any other shape there are infinite shapes yeah which is right why did you say yes you saw in the book which book is that leon spiela do not blame leon spiela i said poor fellow huh yeah puja why did you say this way you have to explain no otherwise simply you cannot say but this is C A this is X A no that is why this is X A that is C A huh starting will be left butter what is the shape yeah okay so it also goes like this okay but it need not be symmetrical that depends on stoichiometry and all that it need not be symmetrical okay but under some conditions you will also get very symmetrical good so this is the one now when i have this kind of 1 by minus r A versus X A which react is the best which react is the best because you know this i tell you i also told you earlier that you do not have to learn anything new great concept now onwards it is only delta X here and there right so i think whatever you are supposed to learn in C R U already learnt basic concepts but those things only either you are combining or subtracting and you know the concepts and then finally you have to solve the problem that is all what you have anupriya which react is the best anupriya i was asking which react is the best huh X A X A okay agree with you there where depends on X A C C i said which reactor is the best first one one reactor afterwards we will see P F R okay so P F R what is the area i have to take where so if i have let us say this is X A F yeah that is X A F so total area under the curve i have to take right yeah that is not bad okay that is not bad someone was telling that depends on conversion or something to anupriya yeah so what do you mean by that say if we be the minimum of that curve then if we are taking M F R then that area will be lesser than yeah definitely what you said is right it depends clearly on the conversion at which you are trying to do the thing okay so if i take that the conversion is corresponding to the maximum rate somewhere here this is the maximum rate so that is why 1 by R is minimum okay 1 by R is minimum so here if i have somewhere here then C S T R is the best so we think that always C S T R is very bad no normally it is giving large volume and all that this is one case where C S T R is the really best okay so that is why normally what we can say here is the reactor selection depends on whether you have small conversion or large conversion so when i have small conversion so that means i have similar versus X A so then you have something like this and till here if i take anywhere between you know this point this is the conversion which is corresponding to maximum rate X A 1 i say X A 1 corresponding to where X A 1 corresponding to maximum rate so till that point always mixed flow is the best this is mixed flow M F R and if i take P F R here this is the total area this entire area okay P F R okay good so when i have intermediate conversions intermediate conversion means you know somewhere here till here so when i plot that so X A versus 1 by minus R A like this so my conversion corresponding to maximum rate is somewhere here X A 1 but i am interested somewhere here okay so that means anyway again for P F R i will have all this and for M F R from here to this is M F R so sometimes both may be both areas may be same then which reactor you choose if conversion is intermediate where this this area okay i have here again M F R and P F R both areas are same equal to okay sometimes you may get that you know in between yeah which reactor you choose Swami yeah you tell me whichever is more economical which is more economical how do you say that it is giving same volume M F R is why what do you mean even even for P F R also one cylindrical portion only required that is what he is telling Abhishek telling you know go for whatever is the cheapest yeah but other than cheapest do you have any other criteria for what temperature control okay temperature does not matter i think you know i can control then what is the thing i am removing that restriction also okay temperature control is not very that much crucial for me somehow i will manage is there any other point yeah which is easy to operate M F R why P F R is difficult to operate because i have to i have to explain all the yeah there is no ideal P F R anywhere it is only in our mind correct no it is only in our mind ideal P F R where is ideal P F R right because as long as you have walls you have some liquid which is sticking to that the velocity profile is different there near the wall so that is why theoretically there is no particularly for you know liquid and gas flowing through pipes right as long as but whereas if you have coal and then you are burning on a what is that conveyor belt and all that so then you will have definitely no problem i think excellent plug flow you can have that is why whenever you have this kind of pipe flow for homogeneous reactions theoretically there is no i mean plug flow the reason is that the velocity flat velocity profile you get only when you have velocity equal to infinity theoretically infinity okay so that is why reason why whenever we have a choice between P F R and M F R people jump to M F R okay so that is why so the next one is next alternative is high conversions so that one we have 1 by minus R A versus X A similarly i have again somewhere here we have X A F now which reactor is best P F R because only this area that is next that is next yeah so this is P F R M F R will be this one very good okay now we will go to combination of reactors these are the individual reactors what we have either P F R and M F R so now individual now i think you have become master so i think i do not have to take much time to explain this is 1 by minus R A versus X A so the best thing is till maximum rate yeah maximum rate M F R and then till this X A F P F R yeah first C S T R then we have M F R okay so here i have X A 1 X A 1 that is the best setup 2 right there is another best setup also if you bring mass transfer and reaction together what is that best setup yeah up to maximum rate we put 1 by minus R A versus X A this is the one so i will put a C S T R okay so then i have here that is X A F right this can go back again that is okay right so yeah able to follow sorry yeah what is happening here i want my 90 percent conversion that is true my conversion may be 90 percent but what i do is i conduct the reaction only till X A 1 and use either absorption or distillation procedure to that is what you are doing in distillation no you are only just increasing the product or one of the components purity right so that way you can now go to X A F and this is X A 1 so if you are able to combine these 2 then we have the best thing because reactor volume is decreasing it does not mean that this is economical as you said finally finally economics only either this or this or individual reactor or the next one is recycle reactor okay good so physically discussing why at this point for example okay this one why it is giving that is the best reactor what you have if i operate this one as at X A 1 okay X A 1 why what is really happening there why now suddenly this inefficient fellow has become very efficient or in this insert now we are saying that mixing is better for this reaction correct no earlier we say mixing is not good at all now we say mixing is good you know that much physics we should we should be able to imagine a little bit why earlier we were telling that you know mixing is very bad for reaction and all that but now we suddenly say that mixing is very good you go for only mixed flow but you can tell any idea not able to imagine can you tell what is the reason what is the reason why mixed flow is very good now for autocatalytic reaction what do you mean by probability and we are deliberately mixing both because there you know when you are continuously feeding what is happening the fresh reactant is mixing with the product where we have that product conversion is yeah X A 1 not very high X A 1 very high means again it is becoming very inefficient okay at only one point it is excellent okay yeah because you are mixing product with very simple thing is this equation combination of this C A C R is the best in in uh mixture flow reactor and how many how many rates you have in that how many rates are there in this system only one rate which is equal to the maximum rate right so if you take any other rate it will dilute correct no you will have different conversions so that is also the reason why plug flow is bad here you know all this area plug flow is bad because there are so many rates those rates are not good here right so that is why uh this extra area will come because along the length of the p f r you have so many rates whereas here i have only one rate which is the maximum rate and i am able to operate the system only at that maximum rate so it is not you know no contamination no dilution anything nothing so that is the reason why this uh you know every dog will have its own days people say you know so this dog has its own day mixture flow reactor best so at least it will be proud i mean every one of us in the nature you know should be proud so that is why this is the one of the best reactors okay if you understand this then going to recycle reactor is easy another possibility for autohydrolytic reaction is one reactor okay but you see here here i have two if i want to go for slightly higher conversions right and here also i have two maybe mass transfer and reactor both this is mass transfer equipment maybe distillation column and this is the one but if i go to actual recycle reactor then we will have only one reactor but only thing is recycle you have to you have to put a pump and then recycle it to the inlet that is all what we have there so why it is better than as one reactor sometimes it is better sometimes only it is better you have to again calculate the overall economics why it is better is that for autohydrolytic reaction you need some product for reaction now if you are able to choose that recycle ratio where you will get the optimum recycle ratio so that inside the reactor you are going to have it is a plug flow reactor the average of all those reactor all those rates are very high the average rate you know i do not know still you are able to get me or not when you are doing integral d x a by minus r a that means you are nothing but you are averaging all the rates you are trying to average and then you will get that so in normally n greater than zero that average is more for plug flow reactor that is why you get more conversion for a given volume or vice versa and in mixed flow because you have only one rate and that rate corresponds to your outlet rate which is x a f that means rate is smaller no x a f is 90 percent rate is small so you are able to operate the mixed flow at only one rate which is the lowest rate possible for that reactor because you have fixed x a f equal to 90 percent so now what is the volume 1 by r a correct no x a by minus r a equal to v by f a not so that is the reason why you should have automatically very large volume for the reactor okay i mean again i am repeating the same concepts only okay that is the reason why and because you should not forget you know these concepts very simply beautifully can be explained all these right so that is the reason why we are not able to get very high conversion for a given volume or for a given conversion you will get more volume this is for n greater than 0 n equal to 0 which reactor is the best both are both are best now you have to give me why physical reasoning why both are best that is what is always required i say that physics that physical reasoning i have been telling you know very typical i ask her everyone answers right it or wrong except her e by f a not will be same why i am asking you are talking mathematics i am talking about physics we should have chemistry together no better chemistry they say you know if two people work in team means you should have better chemistry that is what everyone uses i do not know whether they understand or not but anyway so everyone says that yeah that team has excellent chemistry that is why it won okay what about physics we do not know what about mathematics we do not know so i think see me you are telling about v by f a not equal to something mathematics i am asking physics unless our chemistry is good you cannot answer that is also there yeah okay i think puja puja try to say something that is very simple see why what is happening is you know you would have got that point because at this point of time you should have not got that to your mind but we know that we feel that why because for zero order reaction the reaction is independent of concentration and how p f r affecting concentration how m f r affecting concentration m f r affecting concentration because it gives the lowest concentration okay corresponding to that outlet concentration whereas in plug flow you have a concentration history from starting to c a not to c a f there is again concentration history the overall rate is again definitely better than this m f r that is why we are getting better for zero order it does not matter concentration is not coming into picture so whether you mix or do not mix it is same that is why you have zero order for both same because these things physically you have to explain not mathematically mathematics also required but i think first physics then mathematics right so that is the reason so that is why if you take the recycle reactor now okay recycle reactor now what is the objective now you see we have two things now we understood what is recycle reactor we have an equation v by f a not equal to that entire thing right and now we have also understood what is autocatalytic reaction and we now say that autocatalytic reaction definitely needs some product to be formed now the question we have to ask is the conditions that are going on inside plug flow element of a recycle reactor depends on how much you are recycling yeah how much you are recycling that r now you have to choose that r such that your volume is minimum for a given conversion what is that r what is that r which gives me minimum volume for a given conversion we can do the other way also for a given volume what will be the maximum conversion okay i think either this problem but i think let us see only the volume thing puja you are able to define the mathematics now i am talking about mathematics okay yeah so now what is the mathematical condition for this first of all you understood the condition yeah why why why why recycle ratio i have to choose i am asking sometimes stupid questions yeah autocatalytic reaction needs some amount of product now this product is coming through recycle what is that recycle ratio which will minimize my total volume that means you know maximizes their r throughout the reactor throughout the because it is average okay throughout the reactor that is the mathematical problem now so what is the mathematical condition double derivative okay double derivative will come first what is the first derivative yes so that means the condition here is i have d v by f a not by d r equal to with respect to what i have to differentiate with respect to r because that is what what is the optimal recycle ratio optimal recycle ratio which will reduce my volume to i mean lowest minimum volume so this is equal to zero okay and one can write this also as tau by c a not that is okay right so that is the mathematical problem now so how do you solve this problem so whenever you of course i have where v by f a not equal to zero anyone remembers to smitha r plus one into excellent r plus one into well r by r plus one x a f to x a f d x a by minus yeah so now you have to differentiate okay which is not a simple task that is why i think really like levenspiel he gives again some mathematical clue for us there is a formula called leibniz formula for this kind of integration so the reason is here this d r also present in the limit in the limit okay so the condition levenspiel gives for this kind of if i have a function f r this is function f r right yeah so then with limits where the variable with which we are differentiating okay is present here right because the boundaries also are changing this is boundary this is boundary limits okay when you have that kind of thing so yeah this one will be x r d x so now the differentiation of this function is d f r by d r equal to integral you have to sincerely follow this and then substitute and tell me the answer a r b r dou of i mean differential partial differentiation of x r by dou r into d x that is one term plus this function this function this function b r that is upper limit into d b by d r b is the upper limit d by d r minus i also have a r into d a by d r so what you have to do is now you have to get this term for this function okay and i have the lower limit lower limit upper limit even lower limit is very i mean upper limit is very happy because it is not a function of r so that will become zero we like zeros except in the question paper answer paper okay answer paper we want zeros but you know one zero zero okay so now you have to have this term is not present for us now you differentiate have this term and then this term tell me quickly because i am asking you to do because you have to participate in the class you take this function as it is you know that function is now one by minus r a that is what is function that function is one by minus r a you do not have to substitute r a as first order second order third order that i will give you in the examination but this is different this is only this is only straightforward now you identify that function first what is that function one by minus r a okay this you have to take inside r plus one by one by minus r a i mean that r plus one outside so i forgot sorry that has to go inside that is the function okay so now this f of r f of r is now r plus one by minus r a right that is the one now it is must be easy okay tell me what is the first term that is the first term right because i think only r plus one you have to because it partial differentiation with respect to yeah with respect to r so that means dough actually you have to only r plus one dough r d of r plus one by dough r dough of r plus one by dough r that is one so then the remaining function is only minus r a okay this one one by minus r a that is one yeah next one zero that is very happy next one r plus one by minus r a this is at x a one that is at the entry which is equal to of course r plus one and all that also let me write this one is x a one lower limit because i am leaving sometimes with lower limit yeah this multiplied by there is r by r plus one this term this term is only r plus one r plus one divided by minus r a because your function you know what is this function this function is this as it is that is all copy we are experts in copying right i think maybe not from the board from others answer paper maybe okay so this comes as it is no i mean there is nothing wrong anything great thing that is involved but only thing is at x a equal to x one that is all yeah so that is the one this is the equation what we have but now you have to differentiate this fellow what is this d x a one by d r equal to this is d by what is the equation for this r by you differentiate this very good so this is equal to x a f divided by r plus one whole square this one differentiation very simple differentiation no i think they are not difficult at all okay so now but anyway i have to substitute this is equal to zero finally right to be optimum then second derivative we can check whether you are really getting minimum or maximum okay depending on the sign you will get but this is the one so now there is a nice exercise of saying and arranging this equation so i will substitute here first of all x a f by r plus one square so one this one this will get cancelled so then what do i have here this a is this one minus a no minus minus that is right it is minus only is right yeah so this is minus so then i have equal to zero if i put i am just simplifying this one x a one to x a f d x a by minus minus r a equal to this one is r plus one yeah this is at x a one this is important yeah okay okay i am just writing again yeah so how do i eliminate this fellow r plus one i do not like that fellow intensities to calculate r but i you know for me to get one condition i do not want that r how do i eliminate that then that will give you a very beautiful condition you have to eliminate that condition i mean eliminate that r from the derivation v by f not is coming there okay i will give you the equation can i say r plus one equal to x a f by x a f minus x a one how how do you say yes because i have written there yeah yeah yeah please do that you know i am a very dangerous fellow this alone will be separated the small portion five minutes okay so that you have to arrange that and then only you have to find that okay good so now i want to know happily remove this that means what is that i have to remove x a by r plus one now x a f by r plus one equal to yeah it will not get cancelled okay cancelled means i am just taking this x a f by r plus one equal to x a f minus x a one so now the condition is if i do that wonderful condition one by minus r a at x a one okay x a one equal to x a one to x a f d x a by minus r a i am just writing this equation minus r a divided by x a f minus x a one what is the meaning of this condition this you got no no problem i think after substituting here and rearranging who can tell the physics of that this mathematical equation yeah this is again you know simple calculus only how do you find out the average of like for example temperature varying with length if the temperature is varying with length how do you calculate the average average temperature temperature is varying along the length what weight average how do you find out temperature into temperature into temperature into length that is sigma of yeah okay that sigma if you are able to extend to integral integral t d l again another integral that is zero to l okay that is just you know we are only talking in air but i have not written there but are you are you really familiar with that because this is where is missing you know from you not only from you from all these students that relationship between mathematics and physics okay and this is the one average you should have done many times like exactly what you said you know yeah like if i have some temperature changing with length okay so the average temperature for example t bar how do i calculate integral yeah so some function this this t is a is a function of x because t is changing along the length okay so then i have dx where this is zero to l divided by l here or simply one by l divided by l or the same thing no yeah so that is all i mean that is what is exactly told here so that means what Ramakrishna told is correct okay so this condition is a wonderful condition i think please take down the condition okay in words the above equation i think you are writing below this you know that is the above equation okay yeah in words the above equation indicates that the optimum recycle ratio introduces to the reactor a feed whose one by minus r a value equals the average one by minus r a value in the reactor as a whole i know normally you would have written without thinking what i am trying to say okay now someone has to explain this in words the above equation indicates that the optimum recycle ratio introduces to the reactor a feed whose one by minus r a value where is that whose one by minus r a value which minus r value are talking right hand yeah what is left or right left hand side exactly this is the rate okay yeah yeah at x a one at x a one right that is what whose introduces the value whose average rate value average rate equals equals the average one by minus r a of the entire reactor this is what this is x a n x a n x a one entering the reactor x a two or x a of leaving the reactor this is the average that is for the entire reactor so the optimal recycle ratio introduces a rate because when you mix the recycle ratio as well as the f a not coming from the fresh stream those two rates must be combination of that must be this rate okay so because i am mixing two streams know so i have a rate also there average rate so that rate must be equal to the average rate throughout the reactor when when when that happens you will get the minimum volume for a given conversion it cannot go wrong because it is only mathematics mathematical equation because this is the first one and if you want to prove that whether this is really true take this same thing second derivative you will get negative slope second derivative must be positive or negative positive for minimum yeah right right negative for maximum okay here minimum volume okay so that is the one