 Okay, so what is the problem? As r tends to 0, v by f equal to r plus 1, integral x a f, this is the first one you have to write it correctly, okay, then you have to show as r equal to 0, this goes to P f r, right, straight forward, right, v by f not equal to 0 to x a d x a by minus r a, okay, good, yah, so as r tends to infinity m f r, okay, so within the time given there is one of the simplest derivations what you can have here is using the physical concept, when divide by r, why should I divide by r? By the by r, divide by r means where do I divide? Which one I divide by r? Which denominator? You cannot do that, this you cannot take it to infinity, okay, that will not happen, so the simplest concept is that when r equal to infinity what is rate throughout, rate, same, okay, that means is it constant? Throughout the reactor, correct no, so when it is constant you pull out that, here v by f not equal to r by r plus 1 divided by minus r a, now integrate, what is that to be integrated? This is r by r plus 1 x a f, x a f d x a, so d x a equal to x a, right, yah, so now this will be r plus 1 by minus r a, yah, so this is x a, so these two why not, yah, why not, that is what only I must, yah, you have to integrate if it is changing with conversion, changing inside the reactor, it is not changing, it is throughout the rate because that is you know in ideal pure C S T R you have only one rate, right, so that rate is constant in a plug flow reactor where you have r equal to infinity, I mean this is what I also explained to you when we were discussing about recycle ratio equal to infinity, so r everywhere is same, that is why throughout the reactor you have one rate, which means it is m f r, now mathematical also you can prove that, so now if I write this v by f not r plus 1 by minus r a, this is substituting lower limit, upper limit x a f minus r by r plus 1 x a f, yah, how much is that, this is x a f by minus r, so what is the equation now, v by f not equal to x a f by minus r, so that is what is m f r equation, this is wonderful, again this is wonderful, okay because this is what is the physics we have discussed when we are really talking about recycle reactor when r equal to infinity means what, okay, it does not mean that you are bringing back everything back and then you know what happens, if you eat more and more and more and more and more you do not know then stomach will explode, okay, same thing there, I think you know you are not, there must be outlet, there is outlet here by drop by drop for the recycle reactor where I have infinite r, okay, only small amount, 1 ml is taken out, I also gave that example, 10 meter cube is recycle, that means throughout the reactor I have almost same rate, so that is why you can take it out and then do it, this is wonderful problem and you know the marks allotment, for the first one it is 0 and for the other one it is, ya ya see people will laugh at us if I ask you to put r equal to 0 here, r equal to 0 there and you want me, you know m tech, MS, PhD, okay, minus LKG also we will do that, that is memory, that is there already in hard disk, you are not doing anything, you are not doing anything else, it is there hard disk, okay, so that is why, ya I think marks cannot be simply equally divided I say, it cannot be equally divided, all do not have, they could not come here without hard disk, may be you know megabytes may be are storing place be less or more, okay, for that you are not made on your own, only God made it and then gave it to you, okay, good, so that is what, what is no, there is no 5 there, if at all I give it will be 1 9, that is all, it is 1 mark and 9 marks, for remembrance okay, 5 5 is too much, 2 otherwise 2, that is all, 2 or 8, 2 or 8, no no, I think you know you see, I mean people will really laugh at us, I say when you want to substitute here 0 and 0 and say that you give me marks, you give me marks, okay anyway marks does not matter, so I am going to conduct many, so now this you may write off, okay, you are not got sufficient marks here, okay, so then we will see at the end you know how you can see the marks, marks is not the criteria, but you see again once more I want to remind you the beautiful concept, now you do not forget so easily, right, at this point of time you do not forget now so easily about this proof why recycle reactor r equal to infinity is a mixed reactor, when you can turn, you are not very happy 5 marks you want, okay, marks you will see later, good, so we will come to now this plan to push the reaction, okay, so now tell me, this one the other day also we have written here, now tell me because you have to think you know Gopi, anything you see, oh this one you could not see, oh, okay, so you know the answer now, okay, good, you know only know that is all, you know surprisingly what I did earlier was yesterday I gave one exam and gave the solution and next day morning again same thing repeated, same question paper, okay, still I think only 4, 5 people out of again may be at that time 60 were not there may be 45 or something was there only 4, 5 wrote that, that is what is the use of odd disk, okay, because not able to store that also, not able to store that, so that is why I try all tricks you know to make you think and then to make you remember, remembering also is important, okay, at least main concepts not everything, main concepts, okay, so now you tell me what is the real problem here, why this reaction should be discussed at all, what is the conflict, so you think and then tell me, Avinash, what is the conflict in this equation, may be hell you have to discuss that equation as a great equation, information, R is the desired one, that round is desired one, there is parallel with respect to A, A no, so parallel with respect to A, then what is the rule, A is the rule, A is the rule, A is the rule, A is the rule, A is the rule, A is the rule, so parallel with respect to A, then what is the rule, and one is first order, one is second order, see A is going to R and A plus B or 2A is going to T, what is the rule, you have to keep high concentrations or you have to keep low concentrations, that you remember, good, that is excellent, right, so you have to keep, because R is desired one, that is lower order, you have to keep the concentration as low as possible, then which reactor is better for us, T S T R or M F R, okay, then what is the conflict in the second one, I think, is there any conflict now, so happily I can use M F R and then go to sleep, again R is going to, yes, but our idea is to maximize R, so then what is the conflict now, no, no, no, let us just assume first isothermal, so can I, what is the conflict now, you have solved that parallel conflict, is there any other conflict now, if you want to increase R we have to increase A, but if you increase A the second parallel will increase, yes, why we should increase A, if you want to increase R, increase R we have to, more A should react, no why should I increase R, A for R, what is the problem, your answer is right, but explanation is not beginning, please think A, please think please think, you are only asking marks, then what is the conflict, I mean what is that, is there any conflict or happily I can now take M F R and then go to sleep, what is the other conflict, is there any conflict as of first of all, I am not saying there is conflict, I am only just trying to think you, I mean make you think, why, why, why, why give me the reason, different positions are, for what reactions, that is the one, that point only I just want to get from you, the first one is A going to R, R going to S, R is the desired product, so actually it will tell me that use P F R, you have to keep the concentrations as high as possible, that is why use P F R, now there is a side reaction for that A and then it is higher order than the other, you know the main series reaction, it is a parallel reaction, now this fellow says that no, no, no do not go to P F R, go to plug flow, sorry, mixed flow to keep the concentrations as low as possible, right, now there is a conflict, which one is the better one, right, how do you, I mean do you use first P F R or do you use first M F R and what do you really do, that means now you can extend your thinking a little bit, you have one extreme P F R, another extreme M F R, right, but may be somewhere in between you have controlled mixing, controlled mixing means R equal to 1, you have, you have some kind of control over mixing, right, R equal to 10, you may have some kind of mixing, that is the reason why this kind of reactions wherever we have the conflict between, one says that use plug flow, the other says that use mixed flow, but in between there may be some conditions where recycle reactor is the best and also unfortunately we cannot make it as a rule, even this statement use always R recycle ratio for this kind of reaction is not a rule, because it cannot satisfy, right, so that is why only straight forward reactions like A going to R, R going to S, S, I have to use plug flow reactor, high concentration, otherwise if I have two parallel reactions without R, I mean without R going to S, I have only A and A plus big going to T, that means I have no parallel reactions, one is second order, another one is first order, I know the rule clearly, but when you combine this you have the conflict, okay, the first reaction says that you can go to P F R, second reaction says that no, no, no, M F R is better, because that means you have to also suppress this, then only you will have more formation of R and then you have to also see how do you suppress this next, right, so but in general if I do not have this, this will give me maximum, okay, if I have only series, series reaction, so now you can see the recycle reactor is also useful in such cases where there is a conflict and if you do not know which reactor to be used because this says mixed flow, the bottom one parallel, that series alone if you want P F R and somewhere in between probably you have to use that also depends on K 1 value, K 2 value, K 3 value, that also depends on K 1, K 2, K 3 value, why it depends, this I think a little bit of more thinking, you know, deeper, you have to scratch, you know like that Inception movie, first dream and dream within the dream, then dream again within the dream, third layer, so now you only scratch with the first layer, now scratch with the second layer What is that? Even though it is a second order reaction, its rate constant will be very small Yeah, that is why I think, you know, that is why it depends on K values, now you can just imagine, like I say that I have K 1 equal to 100 and whereas K 2 equal to, K 3 equal to 0.01, which reactor is best, obviously P F R, right But on the other hand if I have, K 1 is 0.1 and K 3 is 100 You see, you know, you should not think that that is going to totally revolutionize the, you know, yield No, the overall will be definitely much much less But within that, you will get more of R That is why someone has these doubts, I think after the class some people are asking me, sir by maintaining this alone, how do you get more No, within that the overall rate forms, falls Definitely if I do not use plug flow, A going to R, R going to S and then if I use mixed flow, the overall yield definitely falls But within that again, if I am able to maintain concentration of A as low as possible, that is mixed flow reactor then I will have, you know, definitely more R when compared to T T formation I have to suppress That is why the conflict in all these reactions is this particular one That means which reactor is the best one Is it only one reactor or many reactors? How do you do that? You have to write the equations You have to solve them And you have to write equations in two fashions Like one is, design equation like this Other one is time independent equations What are time independent equations? What you have done many times till now Like for example DCR by DCA or DCR by DCS I know just for comparison I am just telling you But most of the time DCR by DCA can be used because when you are talking about yield So those are time independent equations but still they are differential equations So you have to use and you have to be expert in learning or in the solutions of differential equations Now let us write the differential equation for this and then see what will happen, you know, can we do something I am not giving the final solution but I think only few tips for you so that, you know, I just want to pose because we have many things to do I cannot alone spend time on this alone but I have to expose you to those problems at least And simple problems I can always give in the examination because all the time I have to blackmail you with the exam That is the problem Otherwise immediately you delete all the files regarding multiple reactions So that is why I think our deletion activity is very very fast The moment you know that this is not coming in the examination simply switch off that part of the brain So that is why That is why now I have to expose you to some equations where I think, you know, the overall problem That is what we say in MSPHD and also MTech and all that when you are doing project If you understand what is the problem you have to solve 50 percent is already solved Most of us do not know what is the problem to be solved first That clarity is not there The moment you have the clarity, okay in this particular problem I want to only look at this point That means you have the clarity what you are exactly You know, that is not that easy to find out which point you are going to concentrate That is not that easy the way I am telling you You should have tremendous amount of background information on that particular problem definition Then only you will know what is missing, what is not missing and what is that you have to do very very clearly Okay, because all of you are going to do projects That is why I am always trying to tell whenever there is an opportunity You understood, no? I think particularly the PhD scholars, MS scholars already the problem Did you understand your problem first? The moment you understand your problem you got 50 percent of the solution It is very easy now to take it Okay, when compared to not knowing the problem at all and my experience with many faculty many students in this institute as dean academic research is that because those people who are taking more time more time, you know, like I am not telling everyone, there may be special cases like you know the problem but still you cannot do Problem itself is very very difficult to solve It is not easy to solve But in most of the time when I talk to the students the problem they tell me sir my guide did not define the problem I do not believe the student 100 percent that also is there because guide also would have given very clear problem but this fellow would have not understood because human mind, you know, they never think twice to lie to escape some situation That is our defense mechanism Okay Our defense mechanism to escape for life is either lie or run or jump whatever you want to, you have to do it So here a life and death question when dean asks Okay, why you are taking more time Then blame someone, not blame yourself If you blame yourself only you are starting thinking about your problem because who is, you know even here examinations also you write the examination if you are not doing well, who is the first one to be blamed? The teacher Question paper And God for the God who created this question paper is teacher So that is why I think teacher is blamed But on the top of, on the other hand you just think on your own Okay, what did I do yesterday Okay, how much I have prepared Right, in any subject Then blame goes to you and that will improve for some people for some people Again you know for some other people I have not prepared well but I have not prepared well because in my family there is a problem Okay, my friend is not well and yesterday my friend's girlfriend did not call him he is worried I saw a paper I think yesterday paper there was a suicide because the girlfriend did not call this boy and then he hanged himself somewhere in some other place Okay Yeah, I read yesterday yesterday or day before yesterday in the paper So that kind of stupid reasons always you can find out Okay, so that is why all these things I have to tell you also whenever there is an opportunity to change your mind I tell you teaching is not only just making you understand the subject what you are teaching changing the mind also is teaching if you are able to tweak at that point you know mind at here that is why I told you if I become Leonardo DiCroppi I would like to enter all your brains and change their inception Okay, yeah so I am not possible for me right now otherwise if I am matrix hero Neo then put CDs in everyone CRE CD inside your brain load CRE Okay so but there is not possible now to only way to learn subject is hard work sweat always sweat that is what you know the famous Edison quotation Gopi, yeah anyone else you know that quotation Thomas Hallway Edison quotation thousand ways of there is another one equivalent one no none of you know that I mean really what you can tell one percent inspiration and ninety nine percent perspiration perspiration means what that is what I am doing right now Okay, yeah that is what so work they know the inspiration and also concept is very small but to understand that you have to work work work work Okay now let us write this equation Okay so now minus r a how do I write minus r a equal to d t again minus is there catch me if I am making mistakes yeah it is forming in both the yeah it is decomposing a minus r a then yeah yeah you see temporarily you went out of your mind know out of class not mind you are just looking at that tell me now k 1 c a plus c a square because it is second one reaction okay now r r this also of course I can write here minus okay okay r r this is like series reaction okay k 1 no d c r by d t equal to k 1 c a minus k 2 c r okay of course c s also I can write but let us correct no it is normal series reaction I say that is why you know you have too much information also you get confused that is what exactly google is doing us right yeah one click 1 million and 25 in one second all those sites will come now you try to find out in 1 million sites what is there okay margaya I think you know that is why too much information is so confused right so that is why I told you know remember that centipede yeah that is what you are doing that is what you are doing when you are opening google also because so many sites you do not know what to do with them okay good okay so now I think these two are just sufficient for us now let me write d c r by minus d c a equal to yeah I think I am writing the final equation but only thing is you have to divide one by the other and adjust divide and if there is a common factor you can remove that means for example when I am dividing yeah d c r by okay let me write the first step k 1 c a minus k 2 c r whole thing divided by k 1 c a plus k 3 c a square okay so this also can be written as two terms that is k 1 c a k 3 c a square minus k 2 c r k 1 c a plus k 3 c a square so here all I am telling common thing you can cancel out right so now you will have here k 1 by k 1 plus k 3 c a now this equation this is 1 2 3 4 this equation can be written as d c r by d c a minus is there is minus equal to 1 by 1 plus alpha 1 c a by c a naught minus mm alpha 2 c r by c a naught divided by okay c a by c a naught c a by c a naught 1 plus alpha 1 c a by c a naught c a by c a naught you can try this I think you know this is only just algebraically where alpha 1 is no alpha 1 is k 3 c a naught by k 1 right and alpha 2 is this you can guess k 2 by k 1 now the greatest of at that time in 64 solve this problem this is a differential equation we made this one into a dimensionless groups this is a dimensionless group because k 3 is a second order constant right k 3 and you know the I mean units units is concentration inverse multiplied by concentration and time inverse this is also time inverse both will get cancelled first order so that is why it is a dimensional this is also k 1 k 2 both are first order both are inverse time so it is only homogeneous reactions constant density so that is why they get these are the dimensionless quantities and this is the differential equation which you have to solve and the solutions are given in this journal and there are also some more solutions by our carberry I am saying our carberry as if he is our grandfather okay so he also he and his student I think his PhD student Gillespie and carberry in fact I have to also give the initials but anyway I think Gillespie and carberry this also is CES now I think you know I do not have to write chem inside separately CES means chemical engineering science this language also you have to use CES you have to tell okay so the moment I think if I hear that kind of discussion in our department I will be really very happy first of all you know what is the meaning of CES and second also I know that you have seen that latest journal and now you are informing your friends but how many of you have seen CES journal seen very good how many of you read out of that seeing is reading seeing is different and reading you have done sir some article from nanofluid nanofluid good okay so CES and no no no I have to believe him why are they laughing they were not good no yeah so 21 is the volume 472 is page number this is how we write 1966 okay so they have produced some graphs actually phantofluid for only the two extremes like I think recycle reactor he has not attempted but he has only told that under what conditions plug flow is better under what conditions mixed flow is better that is phantofluid but later these people have told that okay now because there is a conflict between two extremes intermediate mixing may be better now you see actual use of recycle reactor is only to control mixing not to increase conversion that is the first answer which you gave earlier forget about that delete that file okay that particular file where you have stored the information that you know recycle reactor is always used for high conversions no it is not used for high conversions you will never get high conversion right but high yields you can get high conversion means we are comparing two ideal okay yeah so that is why you have to now remember that recycle is used to increase the yield in case of complicated reactions and what is the other use to control the temperature if it is highly exothermic reaction you know again you see even though many times I found common sense is not common it is really uncommon okay but we will say very simply common sense and also we also use another word recycle reactors so what I found is after 60 years of my life is that obvious is not obvious common sense is not common that is why I repeat many many simple things okay so that is the reason why I think I also repeat many times good so what these people have done was they used recycle reactor and then showed that under some conditions recycle ratio is very good okay recycle reactor is very good to plot the very complicated equations are there and you know this all chemical engineering science papers journals we have online in IIT Madras so that is why you do not have to go anywhere no physical movement sit in your room click science direct go to chemical engineering science see those numbers you will see those papers okay whether you read it or not just go through and then appreciate the kind of mathematics that are involved okay then if you are interested automatically you will try to solve okay so now what they plotted here was this is from Carberry CA not and this side I have CA by CA not most of the multiple reaction graphs will be something like that not tau in reacted design you have K tau or K1 tau or you know that kind of things right or if it is second order K tau CA not that is the term column number but most of these multiple reactions you have this kind of graphs where this is the desired product this is conversion right 1 minus 1 minus conversion okay good so I will just give 2 examples like I mean 2 cases where alpha 1 equal to 005 and alpha 2 is 0.5 there are many graphs but I have chosen one graph 2 graphs for you just to compare so this is also same thing CR by CA not CA by CA not and here alpha 1 equal to how much yeah 10 here alpha 2 equal to 1 good so now oh my god timer okay so this is varying between 0 to 1 may be 0.2 0.4 0.6 0.8 this is 0.2 0.4 0.6 0.8 here also I have this is the advantage of using dimensionless numbers you got 0 to 1 here 0 to 1 here 0.4 0.6 0.8 and 1 conversion varies from 0 to 1 anyway yeah but here the graph which I want to show here is that this is p of r and very near to that there is another line r equal to 0.1 afterwards we have may not be exactly to scale but anyway to illustrate the point it is okay MFR and I have another line here where r equal to r equal to 20 excellent so now another one I just want to draw here so I have MFR line here also I have to put 0 to actually I want to show here CR by CA0 this is 0.1 this is 0.2 okay that means maximum here atleast I could go to 0.6 and here it is only maximum 0.1 and 0.2 okay but your graph will be only in between this so here if I take alone MFR it will be something like this okay this is MFR let me write later MFR so now if I have ideal p of r you will have like this this is p of r if I have recycle ratio of 1.5 you will be somewhere in between goes maximum here and then takes a turn so this is R equal to 1.5 draw that okay thereby what do you predict from these two graphs nothing else no I think graphs you can predict really a lot so that means unless you understand this graph means when you are drawing this graph you have chosen these two alpha 1 because those are the parameters okay those are the parameters alpha 1 equal to 005 K3 that means what T undecided that means you know parallel reaction is there but it is K3 by K3 CA0 by K1 is it small one K3 or is it larger one small one so that means what is the rule because that is ignorable for example p of r now you see when I have p of r p of r giving me then next one what is R equal to 0.1 very near to p of r very near to p of r so that is why it is almost yield also is almost like p of r yield and then definitely we are going to get low m of r yields because rule says that use high concentration but m of r uses low concentrations so yield reduces definitely and also you can see the yield is only around 0.3 0.4 here and whereas here it is 0.6 that is what is the meaning of maintaining low concentrations high concentrations always you cannot get 90 percent yield every day but whatever yield you get using these rules you will have more of the desired product the overall yield itself may be 5 percent out of that 5 percent you may get 4 percent as desired one the another 1 percent may be only undesired one so that is what is the meaning of that so this is very clear because rule tells me that you have the a going to r r going to s series p of r is the best and this follows as it is all that he has shown in this paper in this paper so now the other one is here alpha 1 equal to 10 and alpha 2 equal to 1 what is the meaning now that means the side reaction unwanted reaction has more rate means more k3 so what is our rule says use CSTR we have used CSTR but that is why it is giving more yield the overall yield itself is reduced it may be around 0.1 if I take here and this also is conversion this is actually 1 minus xA so at this point I may have 60 percent conversion 60 percent will not be here I am plotting 1 minus xA 60 percent conversion maximum yield itself is 0.1 in a mixed flow reactor because we know at low concentrations you will not have that kind of high rates so the overall yield is suppressed a little bit but out of that if you look again MFR gives me 0.1 and which is p of r this takes a turn here this crosses actually this also I have to write 1.5 okay good so p of r gives only till here p of r there is no use that means only at low conversions if you use definitely p of r will give you more conversion more you know here less conversion because you are starting here x equal to 1 here and x equal to 0 here no no no the other way 1 minus x I am talking so that is the reason why xA by CA not values p of r will give you definitely more yield so now you have to choose is it better to go for that kind of low yields that means low conversions low conversions and then use p of r and then use a separate mass transfer equipment to again increase the concentration of the product which you want see how many open ended problems are there in chemical engineering unfortunately this is not only reaction there are like this thousands of reactions are there multiple reactions so every time you have to go to the corresponding I mean corresponding reaction find out the basic rules if they are not following then you have to choose writing these equations then solving them and finally plotting and then deciding which react is the best and if I want only conversion between this and this recycle ratio is the best if I am maintaining my conversion only between this recycle reactor will give me the highest yield now how do we decide that which one is better which one is better how do we decide is only by economic analysis at the end now I will take this conversion is only 20 percent now I use mass transfer equipment and what is the cost for mass transfer equipment what is the cost for this 20 percent conversion what is the overall cost whether I am making profit or not the other one what I said you know attainable region I have been telling you attainable region there is a the website I told South America South Africa South African website so what they did was you see what is the maximum attainable maximum attainable region is they take for example all this and then they take this and then from here they take this okay that means till here it is MFR which is giving this is what is the maximum attainable region understood so till this concentration if I am able to maintain MFR I will get maximum then beyond that I should be able to maintain recycle ratio then I will take PFR so that is the maximum attainable region it is not that simple the way I told you but you know there are some rules to follow depending on its gradient and all that there are many things but this is the simplest one which I can give you what do you mean by that attainable region now this red line is the attainable region tomorrow I will start temperatures effects