 the two catalytic reaction and recycle reactant. Okay yesterday, we have written an equation, very nice condition. So what is the condition for optimal recycle ratio? For optimal r, what you have to follow is 1 by minus rA at x1 must be equal to xA1 xA f dxA by minus ground tube, divorced zone, displaced zone, all divided by, yeah, that is the condition, right? And actually, how do you get this optimal recycle ratio, if I ask you to find out, ok for 90 percent conversion you get me the optimal recycle ratio. How do you do it? What is a normal procedure? A normal procedure is an analytical procedure electrical procedure where we have V by F A not equal to 0 to no not 0 X A 1 X A 1 is R equal to that you remember d X A by minus r A okay so this minus r A now depending on the right right very good ya like that you have to catch the teachers ya so this one this is the one and now if I have a scheme autocatalytic reaction no autocatalytic reaction A plus R go into R plus R the rate expression for this is K into C A C R that has to be substituted here you cannot integrate straight away this so then you have to write this R in terms of C A what is the relationship generally the relationship is total concentration C both together C A not plus C R not also equal to that is simple material balance okay ya so now either you take this C not you have C R equal to C not minus C A so that you substitute here so now this you have to optimize that means V by F A not you have to integrate this one first so you have to integrate and that integrated expression you have to take first derivative equal to here integrated expression right so that means this you will have some integrated expression I am not giving there because I think even though I give more work I think you are not working at all I think this is the analytical procedure where you first substitute it need not be always K C A C R it may be K C A square C R okay that depends on what is the order with respect to A and what is the order with respect to R and this also possible that this may be reversible reaction then you will have here minus C R C R C R square okay I mean with respect to each first order if you take good ya so that is why this is one of the biggest headaches in chemical reaction engineering no I think the moment I change either one or two order you know this is A plus A plus R going to this right so if I say elementary only then it is C A C A and C R in the rate otherwise if I do not know if I say that this may be second order then it will be C A square so then you will have again lot of problems in the integration right so that is why integration integration differentiation most of the time in this course you have to do it right I think that unsteady state also I think she is trying anyone has done that unsteady state C A C R no one no is the problem unfortunately okay so this is what is the analytical procedure is then the moment you have this derivative equal to 0 that equation will be only in terms of R and X A F okay so now for given conversion of may be 90 percent or 80 percent or whatever which I ask or with the problem demands then you will solve that R and you will not get simple analytical expression it is mainly trial and error so examination do not again scold me that you know the question paper is too long and all that if your speed is 0 you need infinite time correct no if you are not able to write at all your speed is 0 how much time you write infinite time you have to write where is infinite time in IIT you cannot wait no you cannot be allowed here to stay infinite time so that is why I think you have to practice quickly good so this is one method now I will go to Levenspiel beatful method what is the beatful method graph okay excellent this is really very very nice method so graphically this is very in fact easy easy to do it instead of doing that but you should be loving to draw the graphs right okay yeah so for autogatalytic reaction how do I show the curve 1 by minus r A versus X A curve like this right good something like this my X A F is here because we fix X A F then only we are trying to find out R okay optimal R we are only talking about optimal R this is X A F okay so now for example without knowing if I say that okay this is my X A 1 what my condition says the condition says that the rate that is entering at X equal to 1 that is at entry okay X A 1 must be equal to the average rate throughout the reactor okay so here how did you get average rate graphically in recycle reactor earlier I have to give connection no recycle reactor earlier Anupriya I think you have solved that problem two areas we have taken and hit like this sometimes loose connections will be connected the area that is left out is compensated by yeah so we will take you know average rate will be yeah something approximately may be slightly above I have to put yeah so this area when it is equal to this area this is the average rate okay that is the average rate for the entire reactor then what is the rate that is entering entering the element entering the plug flow element in general I am asking not the condition you have not come to optimal at all okay I am just simply asking what is the rate corresponding to X A 1 where is X A 1 here this one is X A 1 correspondingly this one is the rate that is entering now this if I say I mean as per the yeah so this is point K and this is L and this is this rate P this is no no no sorry this is P and Q both are equal here P Q is greater than P Q is greater than K L okay so that means recycle is large or recycle is small I mean compared to optimum you have optimum let us say optimum equal to 2 will this be more than 2 or less than 2 you are moving towards plug flow here or towards mixed flow yeah towards mixed flow that means R must be yeah R must be more than optimal value okay so here this is R large so that means this is not optimal we are all discussing optimal optimality because yesterday class was continuing okay good yeah so this is the one so the other possibility is that we have 1 by minus R A versus X A again same thing okay so you may have somewhere here X AF so I will go somewhere here this is X A 1 so average rate again I can now get yeah so again here idea is this rate this area this area must be equal to this area this area and this area I see and this area then when I have that then I can have here again average rate okay so this average rate again we call this is P Q you know average rate average rate is you are trying to find out what is the area of this and what is the area of this so that I will get an average line okay but here do not get confused because it is too much it has gone this side because of autocatalytic nature right okay so now this is P Q where is K L for me X A 1 this is L where is K this one now what is the condition P Q is less than K L K L that means what is R value it will be small that is R is small small R okay so now we will come to the now optimal okay how do you get that optimal that means this rate P Q also equal to K L K L is the rate which is entering the plug flow element okay if you have forgotten about that plug flow element we are talking about okay this one here this is X A 1 corresponding this when it is just entering what is the rate that rate is this rate plug flow element right and all this one is overall from here to here what is happening okay so now my optimal ratio will tell me that these two must be same now you can plot the other graph again X A versus 1 by minus r A then you will have like this X A f is fixed right so now we will move like this slowly okay this area you know this area equal to this area if I take at this point K L is not equal to K P right so then I will draw this X A 1 somewhere here and again take the average line okay so like that when I adjust my areas you will get a condition where both P and Q will be same okay P Q equal to K L condition you will get so that will be okay something like this yeah so where this area equal to what you have to do is you have to calculate you know imagine imagine a small line you have to draw and find out this area and then afterwards move again find out this area afterwards move find out this area and you will have a condition where this area will be equal to this area and then this will be the X A 1 and this one this this is yeah this is P Q and this one is K L those who love graphs they can try this method and those who like equations you will take two hours okay because differentiation and particularly that equation always you will get one side logarithmic term other side you will have a normal term so then you know to find out by trial and error you have to do assume r equal to something calculate both the terms by trial and error matching or otherwise you can use some other technique you know to find out what is the intersection value okay yeah so good that is nice so this is one this is what is the optimal ratio when K L equal to P Q we have optimal r okay so now how do I get optimal r from this graph now okay I managed this area equal to that area and P Q equal to K L that condition is there so then how do I get now r yeah excellent X A 1 I thought you have forgotten but you remember this is r by r plus 1 X AF X AF I know already because I am fixing X AF okay otherwise that is not possible so easily one can estimate the optimal r this is very good particularly for biochemical reactions and you know Monod's equation is an autocatalytic reaction not Michaelis-Menten equation Monod's equation why because initially you have an how do you produce by the by alcohol biologically fermentation what is really happening in fermentation yeah you take the sugar source right and then you grow the microorganisms in that so initially you seed like curd only you add a small amount of previous curd that is the microorganisms you are putting so that will multiply multiply multiply and then it will make a solid curd right that means you are also taking you know lot of those microorganisms you are not separating them again from curd are you separating no actually you should not separate because those are good for the stomach yeah so that is why I think what is what they are called probiotic probiotic okay so antibiotics only we do not like but anyway we have to take most of the time so this is why initially when you put that small amount of curd that means small amount of microorganism it takes some time that is why that 8 hours and all that is required okay it first gets acclimatized to that environment and then once it starts eating food once the environment is good I told you know like human beings once you have good food good house and all that only production of children that is all nothing else no because that is the nature of this planet we should we do not have to really take it as laughing matter really because any life definition of life is propagating in your own species that is one of the definitions of life on this planet otherwise you would have not been here I would have not been here there is no CR3 class no really I think you know if that would have not happened right yeah so that is why I think you know initially slowly it will grow and then the rate of reaction in fermentation if you come if you forget about curd and fermentation if you come then what happens initially small amount of microorganisms are there then it will get acclimatized and then it is ready now to produce and afterwards it starts producing more and more now what happens if it is a batch process batch fermentation then you have more and more microorganisms growing and the sugar source that total amount of food available is only limited so that is why after some time the population is more there is no food then people start dying exactly same thing our planet also is going to go to only the direction because we have so many people 7 billion by the time class is over another 2 millions would have been added yeah really because throughout the world so that is why initially when I write the rate of reaction it is the concentration of glucose and concentration of microorganism initially microorganism concentration is small CR okay then after some time CR will be slowly increasing CR will be decreasing so that is why it goes to maximum rate and then again falls okay now on this planet we are in this region no falling region because more people less food many people are dying so that is why it is a beautiful example for auto catalytic reaction fermentation okay what you produce from fermentation alcohol what do you do with that yeah we drink we drink all the time if I tell you you know those at least happy to drink or those who are thinking of drinking you cannot drink at all you know what you drink there it is the waste material of microorganism what do you call that in English S H I T okay yeah that is what what what we drink happily and then do whatever we want I think you know if you just imagine that exactly could also you say yeah you know the microorganisms take the glucose and then they produce the waste product okay that waste product is alcohol even antibiotics and all that what you produce through microorganisms all of it is only waste product see that waste is so great you say saving lives and all that right but alcohol taking is not good I think that is useless material really useless material do not drink that okay good so that is what is the auto catalytic reaction one of the very good examples for us okay there are many other auto catalytic reactions particularly in organic chemistry reactions so all of them in they need only this procedure if you want to produce if you want to use a recycle reactor and then if you want to produce very very optimal design optimal reactor or for a given volume produce maximum X A okay yeah conversion so this is the one good so what else we have now I think we have now we have to do a problem very quickly we will do that okay and there are one or two things before doing that problem you have the recycle reactor just forget about auto catalytic reaction you have the normal reaction right you have an equation I also gave the equation for first order second order I am sure you would have not derived and in the examination if I give again you know only thing is you only scold me by scolding me you do not get marks you know because I think he is giving very tough question paper and all that you know what you have to do is you have to solve the problems not once many times so what I was telling was if you come to recycle reactor now there are three things where for simple orders you can have that equivalence right plug flow I have and that can be replaced by infinite number of mixed flow reactor that is one equivalence now the same plug flow can be replaced by zero recycle okay or infinite r will be mixed flow these are the two extremes but in between any time if I have three tanks can I imagine that that is equivalent to a recycle reactor yes definitely that is what is the beauty in mixing that is what all the time we are discussing about recycle reactors where I can control my mixing one mixing equal to infinity you know single CSTR if I have then if I have two I am reducing mixing or increasing mixing because are two times you are stirring so mixing is increased when I take two tanks two mixed reactors and then put in series I am a decreasing mixing or increasing mixing I am decreasing mixing so three some more decrease four some more infinity zero mix okay so similarly now I can imagine recycle infinity means perfect mixing recycle zero means no mixing zero mixing so in between I have various hours which are now also equivalent to n try to find out that and it is very good for me to give a nice simple separate test every time I am blackmailing with tests only okay I may ask for example okay two tanks n equal to two means equivalent to how much is the recycle ratio or n equal to three means how much equivalence in terms of recycle ratio and when you come to RTD I hope you will come to RTD so at that time that is also replaced by what is called a dispersion number because some of you know already that is why I am telling you dispersion number zero means which reactor dispersion number infinity means so now any intermediate dispersion number means equivalent to some r equivalent to some n so those are the three equivalences so wonderful you can also always find out what is the dispersion number and normally should be a plug flow reactor if it is a single CSTR then I do not have that chance but I can also imagine I have five tanks in series now I can imagine this is equivalent to some recycle ratio recycle reactor with some recycle ratio or plug flow reactor with some dispersion number you see the beautiful connections at that level all talking we are not at all unsteady state people here only one problem I have given unsteady state okay how can you say that zero dispersion is sorry some dispersion equivalent to steady unsteady state that means what will you call yeah but what will what will you call mixed flow reactor when I have infinite dispersion is it called unsteady state then why are you not calling there only when you say the PFR with dispersion how can you say that it is unsteady state ideal is different and you know what she is asking is different unsteady state steady state is different it is not ideal PFR that's all it is non ideal PFR with dispersion because our definition of dispersion is I mean yeah reactor is ideal reactor is dispersion equal to zero right yeah I think you know I told you all Indians are brothers and sisters only she asked most of you also may be having the same doubt okay so when you have this dispersion or when you have some kind of dead space or bypassing they are not unsteady state by the way what is the difference between unsteady state and steady state then is it equilibrium then no no no no then then what is the exact difference between steady state and unsteady state no gradient no gradient no gradient yeah yeah at any position when you look at any position you look you know that concentration if I is same but not gradient zero there is gradient gradient is there that means you know along the plug flow reactor I am introducing CA not and I am getting CA outside but everywhere inside I have decreasing decreasing decreasing till that outlet concentration but at any point if I look even 100 years if it is steady state that concentration is same that outlet it is same that inlet it is same that is what is steady state and equilibrium is gradient zero that is thermodynamic equilibrium like you know you come and spray some perfume here right so after the five hours if you come then the entire room will be smelling uniformly same that is what is equilibrium there are no gradients otherwise the moment I spray here only I will get maximum smell next puja may get that's all but gopi will not get gopi gets only after five years after how if the diffusion is slow slowly it is diffusing and then you know all the gradients died down so finally it occupies uniform concentration throughout this room that is what is equilibrium please remember that is one of the good questions that will be asked in good interviews good interview there are many lousy interviews where they ask how many fathers how many mothers if you say two mothers two fathers two jobs you will get because this is a great innovation okay yeah so there are many many lousy interviews also I have seen I think as we are talking about decent interviews in engineering you will definitely learn a lot okay through interviews also good so that is why please remember this equivalence if I have two tanks how can I replace that by with singular or if I have recycle ratio of five how do I replace that with yeah tanks in series equal number of course equally everything this is very good problem you see that is what you know if you had sufficient if my zero test was really successful test zero test that means all of you would have answered all these things I have really pushed you okay but now I think we have gone to LKG and then from there we started that is the problem otherwise you know all these wonderful things you will have lots of time to discuss if we had that is why I am taking more classes and all that to cover that loss okay good so that is one of the very good problems you have to please try okay good and also it is by trial and error and all that so please try unless you try in your room you can never solve that in the examination hall good now take this problem we will solve a problem first we will define the problem I think afterwards we will see tomorrow we will solve the problem we explore various reactor setups for converting A into R full stop the feed contains 99 percent A 1 percent R and the desired product is to consist of 10 percent A and 90 percent R full stop the transformation takes place by means of an elementary reaction A plus R going to R plus R with rate constant K equal to 1 liter per mole per minute is autocatalytic reaction know very good is elementary so we know the exact one yeah so the concentration of active materials is CA0 plus CR0 equal to CA plus CR equal to C0 equal to 1 mole per liter yeah yeah throughout you can also write there throughout that means throughout the reactor this material balance is valid okay next one what reactor holding time will yield will yield a product in which CR equal to in which CR equal to 0.9 moles per liter okay what reactor holding time that means tau we have to calculate right what reactor holding time will yield a product in which CR equal to 0.9 mole per liter what is the conversion 90 percent okay because it is 1 we have to now find out what is tau for a P F R okay P F R and B M F R that means one one reactor and C minimum size setup without without recycle minimum size please see the order minimum size setup so it is not one reactor it can be two reactors also minimum size setup without yeah without recycle without R without recycle okay next one is D please take this if a separator is used to separate 90 percent R and 10 percent A from the reactor what is the minimum reactor space time or holding time what is the minimum reactor space time so I think you understood that no D D what I am asking is this this is 90 percent this is 10 percent okay that is separator only you have to find out reactor volume now separator is we are not asking the other one is two reactors you know till maximum rate and then P F R these are of course individual rates I think all these tau right this is a very nice problem I think now we do not have time and let me see how many of you will come tomorrow with this