 so do you remember anything about recycle reactors now the expression which we have derived of course there are lot of discussion and then also I think I could clear as much as possible the doubts you still have some doubts or I know you would not put your brain at all on this because I think focus is only till 5th chapter now single reactors so multiple reactors even though this is a single reactor okay so the expression what we got finally was that v by f not equal to r plus 1 integral x a 1 equal to r by r plus 1 x a f this is x a f d x a minus r a so this is the derivation design expression for recycle reactor okay good and you should start doing that you know those people I do not know whether all of you have done well in the examination and in spite of solving problems if you have not still got the correct answers are not able to do all the problems means that I think still it requires more training more practice okay so this mental solving again I am repeating please do not do that mental solving means you know looking at the problem reading so what is given in the problem only volume so you try to find out whether conversion is given f a not is given rate is given oh everything is given I can calculate so you cannot say that you know you have solved that problem unless you put on the paper definitely you cannot solve a problem particularly in engineering that is true because specific answers specific type of you know the procedures all that is also required this is a training for you I say to go to industry and also solve the problems it is not just to get the degree or just to get the you know every time an examination passed not that ultimate aim is that this training goes to industry where you should be able to design a reactor or distillation column or whatever equipment without any mistakes okay so that is why even this one still you will have lots of doubts right and I am sure you would have never discussed about this with yourself okay so again you know you will postpone all this till next examination that is the problem we use recycled reactor when the mixing is required between PFR and MFR but when I look at the side it looks like PFR only why do we say where is PFR then just for example we use PFR yeah we are using PFR and then we recycle it recycling it is only a PFR okay there is no action mixing ideal PFR only ideal PFR but why do you say it is PFR just for example if you take it is a PFR but with recycle means you know what is the concentration that is entering where is this lower limit for PFR where was this equation in the PFR it was not there why because recycling is important that is the difference that is why it is a recycle reactor how can you say that is PFR no but you said that recycle that was used when mixing is required between PFR yeah this is what is mixing it is you take the product inside and then you know all the product whatever happened that one again you are recycling back and then you are putting along with the fresh feed it is not outside you know it is once the fresh feed comes then only it joins okay so these two are mixing and then entering so that means when the stream is entering the plug flow reactor you have products as well as reactants is it there in the other thing in the plug flow it is not there okay so that is why and intermediate mixing will come because depending on R okay yeah I also told you again unless I give in the examination enough you may not care about this I can give some derivation as a separate test also that show me that as R tends to infinity it is a mixing flow reactor and R tends to zero it is plug flow reactor okay that is a beautiful problem you have to use wonderful concepts otherwise you cannot solve that equation so that is why you can definitely have the mixing between CSTR and MFR when R equal to infinity we can show that this is equal to exactly V by F not equal to XA by minus rA you will get that beautifully you will get that and R equal to zero we know clearly I think you know the straight forward you can get that integral expression so that is why happily one can get and anywhere in between depending on the R okay that is why it is a plug flow element but I am now trying to take some of the product and put it there in the beginning then the rate that is entering is not zero in the normal plug flow reactor the rate that is entering is zero even there is nothing so mixing comes that means this rate should have happened because already some kind of reaction there right so that is the reason that is the reason why it is called recycle reactor even though as you said correctly it is ideal PFR element with products mixing there and when you add products then temperatures will not be there that means you will control the rate rate of reaction will not be very high so temperature control is important means you can use that and when you come to multiple reactions I will show you when you are also should be able to use recycle reactor okay under some conditions you need intermediate mixing it is not always somewhere between the extremes it is not always the extreme somewhere in between also you have to use it okay particularly for multiple reactions so that also I will just show that one to you okay so now this one graphically how do we plot this I think it is not that easy for you to imagine this but let me try and with little bit of clarity in the mind it is not difficult also to understand this normally we are plotting because in this element you see here in the integration you have only 1 by minus r a verses x a so that one okay so normal reactions we may say something like this you know rate 1 by 1 by minus r a verses x a then I will have here this is my final conversion x a f so now what area I have to take here if it is only r equal to 0 what is the area you take area under the curve okay area under the curve if I have v by f a not equal to x a by minus r a that is for ideal plug flow then I take this entire thing okay but now it is not that it is somewhere in between those two correct no the two extremes we can easily imagine because already you have done but this is mixing you know this recycle reactor creates some kind of the products and reactants mixing together and then entering so definitely you will have a value less than mixing flow and of course greater than plug flow because there is this element coming okay this and then this lower limit coming so now let me identify this one as only this part that means I can split this equation into two r into this integral plus 1 into this one so that 1 is here this is x a 1 okay so because this is 1 into simple this one this is the lower limit so this is the area which I have to take that I understood no so this v by f a not I will if I split this equation as r into this plus 1 into this okay let me write r into this integral plus 1 into this integral okay this is okay so this is what first I plotted because this one starting from x 1 to x a f but only this part because r is before here this one so now what I do is we will take the equal areas that means I will just make this one as a rectangle not exactly like here slightly above because this area and this area it is like our normal way of finding out area under the curve okay easily make rectangles and count the rectangles so that this width and height if I know what is the area I can calculate okay so now this is the one the other one yeah this is 1 this is 1 and the other one must be r this is this integral r times this so now how do I get that I have to so this is the total area so what we are trying to do is you know anyway by drawing this what we are trying to do is that we are averaging the rate correct no here I have one rate here I have one rate 1 by minus r a 1 by minus r a so now somewhere in between averaging this area so this is the average rate so now the total area graphically if I do then this is the area which we have to take for v by f a naught so now you see again I split this equation so I have one time this integral that I know the other one must be r times of that so this one alone I take and then I will average and make a rectangle and this must be r times of because this area I think it is not difficult if you are able to really catch the point but a catching point is important okay it is not difficult the reason is that you know I have taken this area separately only with this component and then made it as a rectangle okay the other component is how much of that because it is same r times of that but you can also prove it beautifully I just leave it to you again you can also prove with equations so that means here from here to here I have r times this 1 by r a average here at this point I have minus 1 by r a average at this point right because it is not the actual rate here actual rate here it is the average rate I have taken in between right I mean that is why I am able to take the rectangle area simplifying that this area and this area is equal so now we will get an 1 by minus r a average right so mathematics cannot go wrong this one now says that the other thing must be r times of this area now prove that you see this is only graph you can also prove that using that equation using this graph and then you can say that means this area plus this area must be equal to x a f correct now this is 1 by minus average into x a f total area total area equal to x a f by minus 1 by average or otherwise if you are getting confused with that first I write x f all l k g this is x this is y this is x this is y okay otherwise if I write x by f i in this one you may get confused again this one why do we go above that why do you go above that where is the rectangle how can I make that rectangle if I make this one as rectangle so then what happens all this area this part of the integral may be again I think probably I have to write so this one x a f d x a by minus r a plus x a 1 x a f d x a by minus r a okay so what is this that is this if I do not take this this is the entire area that is the entire area so for me it is easier to have this average rate and then extend further because once I average this rate the other one must be r times of as you said if I leave it how do I average how do I say that what area I have to take what is this it is not that it is not a CSTR separate no you cannot simplify things no if I draw here only one line then that represents that that is separate CSTR which is not CSTR I mean very CSTR there so the combination of CSTR mixing combination of mixed flow reactor both together is there in this area okay so that is the reason why this if you understand this must be r times of this now what we have to do the thing what I am trying to tell you to solve is see I am giving you this kind of problems only to make you think in your room and rich people are getting richer and richer poor people are getting poorer and poorer I think two extremes we are having plug flow mixed flow one is most efficient another one is very very efficient that way I am telling okay this gap is increasing we need something like recycle reactor because right amount of mixing where everyone has equal wealth correct no I think equal wealth means I am not telling that you know you share the wealth atleast food you share that is all so this is what is the general one I think this is understood no and you can now prove beautifully plus this area you take only the rectangles this area you first take and you know the limits what is the area it is a it is a graphical thing it is a figure so then this must be r times of that add those two and show that this is xa f by or xa f into 1 by minus r average average is this that is the average why is that r times this one equation itself in this equation itself it is there this r plus 1 into integral of this entire thing I just split that into r times of this plus 1 time of that that comes from the equation itself from the basic expression itself in the derivation that equation comes you are there on that day I think you are there equation is r plus 1 equation is correct no which one xa f into xa f in the graph xa f into xa f you have made it identical one unit no no no I am just plotting only this see area under the curve between xa f1 and xa f because my area simple calculus simple calculus okay like this you have and when I ask you to calculate what is this area how do you calculate if it is x1 and x2 one thing is to find out all this that is okay total area under the curve another easiest thing is make this so this area equal to this area so this height multiplied by this width this is x1 and x2 x2 minus x1 into this height y so the area is y into because you are starting with 0 x2 minus x1 and the other one is r times of simply this if I take this one as some kind of alpha this is r time alpha area this is alpha and accept that this is r plus alpha but how do you know that is r okay that is what I am asking you to prove that is what I am asking you because you know in this case this is the total equation v by f not okay Abhishek this is the total equation now I have this part as simply one time of this area so this overall thing should be because it is starting from 0 x also starting from 0 that is what this equation tells because when you take the overall mixed flow reactor in the beginning you do not have only at the time of entering because of the stream coming back you have a different x1 that is xa1 right so that is why the coordinates are from 0 to xaf right so now the total area when I take this as one unit yeah but this is definitely then r equal to xa by xa this equation this equation automatically tells you know this area automatically this must be r times of that which is the total area because x equal to 0 onwards it is starting okay so that is the one yeah you do not have a choice no no I think you should be able to do that also you do on your own if you are able to do I will tell you okay later but you do it you try to do it it is not difficult it is not difficult please do not think that it is a difficult concept not difficult at all because we do not know how to use our calculus properly that is all very simple things you know integral y dx I know the moment I put our normal plug flow equation as dx a by minus r a we think that it is totally different that is simple calculus area under the curve that is the reason many students also are not able to answer why should I take total rectangle for mixed flow reactor you know I am talking about single mixed flow reactor why should we take that rectangle I can tell you now you know you may be knowing some of you earlier but my experience with many students is that they do not know why they are taking it is simply nothing but that is x this is y but the x is I mean y is not y y is 1 by minus r a that is all that is so simple that is the area of a rectangle x y y is written in terms of 1 by minus r a the moment we say that that is a problem for us dca if I give dca by dt equal to some right side some c and then ask you to integrate you will have lot of difficult oh my god what is this totally new but if I convert that exactly same thing as dy by dx equal to x you will easily do it because beyond our schooling we are not learning anything okay and the dy by dx it is told that that y can be anything that is what is also here here of course x is x a and y is 1 by minus r a that is all what we have there and we will say always x and y because those are the general variables it can be extended to anything and our mind is always compartmentalized okay okay x means x y means y and I cannot use anywhere else if ca comes and t comes I do not know how to use it and I am not teaching anything else extra except that you know simple calculus where under the curve I have simplified as a rectangle instead of this shape now this equation if you believe that it is correct that is correct only what we have written material balance and all that okay so then this total area must be r times of this then only I will get total area this one that is x a f into 1 by minus r a must be equal to this area plus this area okay I know this area this 1 by minus r a average into x a f minus x a 1 and what is this area 1 by minus r a average into yeah into 0 to x a 1 simply x a 1 and that too r times of that you prove that those two will give you this entire thing that is all and I tell you it is LKG problem because just graphs only we are drawing and then trying to find out that these two areas must be that area good okay so now if I have very large small recycle ratios small r what kind of graph I get this is x a and this is minus r a so like this I have okay how do I draw now my x a f is known to me very small recycle ratio yeah somewhere this side yeah why r is your r is small r is small means area under the curve that means r equal to very very small means r equal to 0 means what total area under the curve right total area so now we are saying that I have small r so that means it may be slightly something there so when typically when r equal to 0 very small this one and then total area under the curve will be there right okay good so now when I have r is small means you should have somewhere x a 1 so this actually this entire thing will be 1 and this will be r that is very small and now you see I have to do the same thing here I have to you know this is the one small one that means when I am coming here less and less r that means the total area under the curve will be coming almost you know like this here this entire thing so yeah this is for small r okay for large r anyway the other one so here what you have to do is that what you have to imagine is that this entire area under the curve is coming that means it is moving towards plug flow area under the curve this is the extremes okay what you are taking for large r again we have this one this is 1 by minus r a versus x a same thing x a f yeah so now I do not have to ask you now only this much I will take yeah so that means of course here I can also this average now what is happening here yeah this is v by f a not okay v by f a not for and here v by f a not for small r for this small r if you average the thing like that so in that case how will you prove that small r where do you average that when I am coming towards this sir you can oh because I cannot average this because when I have like this this equal to this but here itself I have right here yeah so I cannot go above this you know got above that good that is good question yeah so now I can average this this area it will be a rectangle fine it is a rectangle but definitely we are now coming down but how will you say that this area is r times that area which one yeah this area equal to yeah you can prove that again this area equal to so because r is small do not think that r is always large it can be 0.01 okay yeah but who do you think always what are the limits of r 0 to infinity that is 0.01 okay so I think if I move still further then I have to have much more you know it comes like this it comes like this then finally become like that exactly we can we have to why it would not match with the equation because r is very small it should become pf yeah see still there is some r okay so that means still there is some product which is going and mixing in the reactor similarly the combination of mixed flow as well as plug flow this is correct so when you are going further smaller and smaller smaller you go next here you go next here you go almost till here so entire thing will come this entire area is coming see here this this is the area this is the area so now this is reducing reducing reducing and then it goes area under the curve 0 when r equal to truly 0 and please do not think that r should have always larger values it can be 0.001 right that means small amount only so that is why when you are using this definite I think very good what you have asked because this also I should explain then it comes less it comes less what is here when r is large it is almost like you know total rectangle but here this rectangle area will be decreasing decreasing when I go to r equal to 0 right when r equal to 0 still further less means it will come here okay so like that slowly it goes to 0 in fact that also depends on what kind of 1 by minus area we have and all that if I take till this point if it is only same question I think woman was asking you asked me why do you do not take only area under the curve why we do not take area under the curve Avinash is asking why we cannot take only area under the curve even here also you could have asked me the same question okay if I take only area under the curve it is simple P F R huh yeah but it is not P F R how can I take this one which see no no how can I take this one as rectangle afterwards because it is not P F R separate M F R separate see I think you are imagining that there are 2 reactors there are no 2 reactors there is only 1 reactor what he says is I will take till here the area under the curve from here I take rectangle how can I take I do not have separately a mixture flow reactor and then afterwards I do not have separately a plug flow reactor it is the combination of both so that is why I have to take only this average rate and then take the average area where it represents this part also it represents the other part that means these 2 parts you take the average and obviously the value will be distorted first of all I do not understand why distorted this is one method where I know I can average this and then make the entire thing as the rate so that this entire thing will give me that equation V by F N R and if it is so small if you are not losing much area take area under the curve if it is so small that means you are only approximating you are not actually solving the V by F N R you are only saying that even if I take this there will not be much change in the area so I am taking that that is all even this when you are doing this what you are doing essentially you are taking this entire area correct no this entire area only I am taking even if I am drawing this correct no but only thing is instead of taking area under the curve like this I am taking this equal to this and then making as a rectangle so when R is becoming smaller and smaller that comes only area under the curve you are taking 0 definitely it is area under the curve that is plug flow reactor and this is mixture flow because this side when you are moving and you know this area is getting smaller and smaller right so in the limit this becomes only one value and that is the total value that is V by F N R when R equal to infinity that one component is 0 that means this is you can also imagine this in a different way this is plug flow component do not think that this is mixture flow reactor this is plug flow reactor okay this entire design expression because of intermediate mixing has two components one is R component you know with which represents mixing and one is this component one which represents plug flow when R is very very large then plug flow component becomes very small so that is why area under the curve there will be rectangular area R okay this entire area rectangular area when R equal to very large correct no when R equal to very small that means this part will become 0 very small that is why you know this part is very very small and the rest the other part is the area so that is why this two components please remember that equation because you read and then after if you do not read if you do not understand okay close the book that is all okay but here explaining I will be happy to explain that this entire recycle reactor has two components within one reactor right we have the plug flow component as well as mixture flow component and that is reflecting very easily in this equation when R is larger then plug flow component will be smaller right so what is happening when R is very large you are taking almost all that that is reacted small amount you are taking you know this is another thing where you have to imagine that what do you mean by R equal to infinity this is R so this is F A naught this is F A f X A f okay so this is X A naught equal to 0 so what is the meaning of R A equal to infinity mixed mixed no but how do you really maintain that it is still a reactor it is a steady state reactor it is a complete mixing of a reactor that is you know that is not correct actually it is not no output if there is no output what will happen I say if you do not have outlet and then you keep on eating what will happen stomach will burst exactly that is a good answer it will burst right no output only input so that is why if I say that R equal to infinity means there is no nothing is coming out but I am putting this means how I mean that is not a steady state reactor so that is the reason why I asked that question right meaning is that when I take 1 meter cubed 1 meter cubed contains how many centimeter cubed 10 raise to 6 10 raise to 6 10 to the power of 6 10 to the power of 6 ml I am recycling and only 1 ml enters here and 1 ml goes out it is still a steady state reactor only 1 ml going out and 1 ml that means now how do I define my recycle ratio amount withdrawn ratio of amount withdrawn to amount recycle to that of amount recycle to that of amount recycle to that of to that of amount withdrawn so I am now 10 to the power of 6 I mean ml I am recycling back and then 1 ml I am just putting it then what is happening in the reactor if you physically look at that what will be the concentration what will be the concentration here here here here everywhere then if you have concentration throughout the reactor uniform what do you call it I think I do not have to tell really that is the reason why when you say r equal to infinity means then you will have mixed flow reactor meaning is that it is not that r equal to infinity means everything is taken out and then you are recycling then it keep on because you are feeding anyway nothing is going out and then you keep on recycling what will happen to the reactor burst that is why there is output there is input that is coming is 1 ml and 1 ml per second is coming out or for many it is coming out but I am now recycling 10 to the power of 6 ml that means 1 meter cubed of thing is recycled that means practically whatever is happening inside is like mixed flow where after all the conversion I will take it and again put it back conversion take it and study state there is only one conversion okay so that is the reason why everywhere I do not see much change in the conversion everywhere throughout we have only one conversion so that is nothing but your mixed flow reactor that is what is r equal to infinity r equal to 0 is very easy to imagine right when r equal to 0 the stream is not there at all so then even 1 ml is coming out 1 ml is going out but there is change between this place to this place and if I plot x a versus z conversion increases okay this is r equal to 0 when r equal to infinity how do I plot x a versus z that is all constant so throughout the reactor we are talking about reactor please do not again get confused we are talking about reactor where this is almost horizontal because we are taking so much back and then putting it practically inside there is no reaction in the sense that the small amount is coming mixing and then getting reacted that small portion and then recycling back mostly so that is why when r equal to infinity you will get mixed flow because of this reason because everywhere inside the reactor I do not find much change in the concentration temperature and also conversion there is no concentration change conversion also throughout so that is the reason okay good so this is the one and then hope it is clear now so this v by f not minus r a first order reaction then what you have to do is for first order one second I think v by f not equal to r plus 1 x a 1 I will write which is nothing but r by r plus 1 x a f x a f d x a first order only no volume change epsilon equal to 0 epsilon a equal to 0 so what is the equation I have to write here k into c a not into 1 minus x a so this you have to integrate and then substitute the proper limits do not put 0 to x a f proper limit so then what you get here is if I write this in terms of k tau for easy writing k tau equal to r plus 1 in terms of conversion this will be ln 1 plus r into 1 minus x a f divided by r plus 1 into 1 minus x a f please remember 1 plus r into 1 minus x a f this is r plus 1 into 1 minus ya so like that for second order reversible reactions all these things you have epsilon change all that you have so that equation can be used equally well for epsilon there epsilon not there but only thing is mathematics are complicated unless you really solve one or two problems actual integration you cannot do it in the examination good I think I will stop here and Levenspiel has very nice derivations and all that in the book Levenspiel only gave this kind of nice diagrams they may be confusing that is why many teachers may not tell you also about this okay no explanation even in Levenspiel there is no explanation I think you know how much time we have spent now I think hopefully at least for some people it is clear I do not think everyone have that clarity unless they think and also unless they also discuss with others you have not followed still ask me okay good okay you carry on