 this one PFR and MFR. Yeah, our idea is to find out under unsteady state conditions if I start PFR under unsteady state condition or MFR under unsteady state condition, when do you get steady state? Because we should know, approximately we should have that idea. Ok, let us take first PFR, easiest one. When do you get a steady state if I take a PFR? Unsteady state? Residence type, yeah, then. Why? Input to take the output. Yeah, very good, yeah. Now if you go to CSTR, mixed flow reactor, MFR, CSTR, that also no problem, that is also one only. Ok, one mean is next time. In the PFR, I think steady state is reached after twice of the space time in 2 tau after 2 tau. Why? That you do not mean same. You have to tell. Yeah, so in a plug flow reactor when do they pass and in a mixed flow reactor when do they pass? Zero. Zero, that means even before starting you will get steady state. What do you mean by zero? Zero time? Zero time means you have not started. Ok, what is the time required for PFR, all the particles to come out? One batch. Ok, one batch is one volume. It has to come out. So that means one T bar or tau, right? Ok, so that is why as far as PFR is concerned steady state is no problem. If it is really ideal steady state, you are starting, Ok. So then the material will start coming, entered and then slowly coming out, right? So once it fills up and then comes out, then we know that is equivalent to one mean steady state, Ok. So then afterwards next batch will enter and then exactly it also comes out in next T bar. So that is why once first T bar comes out, Ok, so then onwards you have always the steady state. That means just one mean residence time is enough for unsteady state PFR to become steady state. From then onwards it should not change, Ok. And mixed flow reactor? That is also one tau. Then what is the difference between these and these? All the particles will come out in a mixed flow reactor in one tau. What has happened to your RTD funda? Theoretically it is infinite. Because theoretically it is infinite and you cannot wait for infinite time because you never get the product. Correct, no? Theoretically if you want to infinite till infinite time. So that is why we have to now find out that what is the practical time? So the practical time even for mixed flow, sorry plug flow, the theoretical time one mean residence time. Practically you know you have the slight disturbances near the walls and also you have that small fluctuations, Ok. If you give that allowance normally 1 point to 1 point 5 times you should be able to get steady state in a plug flow reactor. So plug flow is no problem, Ok. So ideal plug flow PFR needs needs one tau, Ok, one tau for steady state. We are talking about ideal. Practically it may be you know one time 1 point 5 or sometimes only if you have slightly more disturbance then you will have twice, Ok. So like that I think one can wait that much time and we have that much time to wait and then we can get it, no problem. But for MFR, unsteady state MFR we cannot say so easily. So that is why now let us write the equation. So here I have V CA0, this is V CA, I have volumetric flow rate V and of course I may also have FA and all that, FA0, FA, XA, here XA0 equal to 0 all that we have. So now we have to write the balance for unsteady state, Ok. So unsteady state here we have input equal to output plus reaction plus accumulation. Now this we cannot ignore. So that is why we write easily in terms of concentration that is easy. So V into CA0 is input, what are the units of this moles per time, Ok, good. So this is V CA outlet, this one is minus R A into V, so I can write if it is first order, Ok, I will take it is first order plus accumulation is V DCA by DT. So now this, now let me say that I have first order reaction, so if I have minus R A somewhere here I will write, minus R A equal to K into CA, liquid phase reaction. So now we will substitute that and then we will arrange this equation, I will give the final expression, V CA0 V CA plus K CA into V, Ok, plus V DCA by DT, that is equation number 3, so divide the whole thing by volumetric flow rate then you will get in terms of tau, Ok, so that is CA0 equal to CA plus K CA into tau plus tau into DCA by DT, so this is equation 4. So now I will also write this one in terms of a differential equation, this is, Ok, I will write here, DCA by DT as plus tau CA, you can take, this is minus CA0 by tau equal to 0, so this is the differential equation, Ok, so this needs the boundary conditions, Ok, good, so now the boundary condition is that at time t equal to 0, one boundary condition, this is a first order differential equation, we need one boundary condition, so at tau at t equal to 0 because this is unsteady state, this is t, Ok, so at time t equal to 0, we have CA equal to 0, that means in the outlet at that instant of time the moment you put t0 is 0, I mean that is one logical explanation, Ok, good, so that is the one and substituting this boundary conditions and if you solve this equation, Ok, so what do you get, I am not giving the solution, you have to do it, right, so what do you get is CA equal to CA0 1 plus k tau 1 minus exponential minus, Ok, 1 plus k tau by tau into t, Outlet concentration because this equation is CA, CA outlet, Ok, so that is why, good, yeah, boundary conditions, mathematical boundary conditions sometimes confusing with physics, so that is why you have to be careful, so this is equation number 6, Ok, yeah, so theoretically speaking that is what Ramakrishna was telling that you know if I take t equal to infinity only, then this becomes 0, then this becomes, yeah, CA0 by 1 plus k tau which is nothing but steady state concentration, Ok, but you do not have to wait till you know infinity, practically we have to also think, Ok, so that is why what we do is let t s, t small s be the time, be the time necessary to reach 99 percent of the steady state concentration that is CA s, steady state concentration CA s, where CA s equal to CA0 by 1 plus k tau, I am assuming, not infinity, but I am assuming that I have 99 percent, Ok, good, so now if I substitute equation 6, you know because I want to write this one as steady state, so I have, this is equation 7, equation 7 I will substitute here, so that will be CA equal to CA s 1 minus exponential minus 1 plus k tau by tau into t, Ok, yeah, so this is the equation after substituting 7 in 6, Ok, 7 in 6, Ok, good, yeah, so now here my definition is that, you know, this CA, I will get the steady state when this is equal to 0.99 CA s, 0.99 CA s, now please remember 0.99 CA s, so when I write here 0.99 CA s, that is what is given here, Ok, 0.99 CA s, so then I have here CA s, Ok, into 1 minus exponential minus 1 plus k tau, all this stuff, here t, Ok, good, so now this, this I can cancel out, now can you find out the t equal to how much, from that equation, exponential you separate, 4 point, how fast you have done this, so yeah, I think what is that is right, I think you have to do that, so that is t s is, t s is, so 4 point 6, 4 point 6 tau by 1 plus k tau, this is the time, so that means the steady state time for 99 percent, you know, reaching steady state value depends on the kinetics as well as residence time, Ok, kinetics as well as the space time, Ok, we will now assume that we have k tau very small, that means k may be very very small and k tau also is very very small, so then we can neglect k tau, Ok, so you can neglect k tau, so that means when I say that I have, for reactions where k tau is approximately 0, then what you have here, so t s is 4 point 6 tau, excellent, that is the equation, so this will tell you, Ok, volume by volumetric flow rate, now you see it is not 1, minimum 4 point 6, Ok, so if I have 1 minute for example, 1 minute is the tau, then I have to wait, minimum 4 point 6 to reach 99 percent, yeah, that steady state conversion, Ok, if it is 10 minutes, 46, Ok, so this is the practical time, generally we say 5 to 6 mean residence time, there is a thumb rule in the laboratory, 5 to 6 mean residence time, that has come only from this, but that strictly valid for slow reactions, Ok, I think I have to also write for reactions, this is slow reactions, otherwise you do not know what is that, up to sometime, minimum 4 point 6, maximum minimum is same here, because to get 0.99, 0.99 C A S, Ok, steady state value, I have to wait this much time, there is no question of maximum minimum there, so K tau is approximately 0 for very very slow reactions, because K value is very very small, Ok, so then this is 0, this is 1, I have 4 point 6 tau, do not know, I think you know I do not, I do not want to call that as maximum or minimum, that is the time I want to wait, to get 0.99 C A S, yeah, if the first reaction is there, so that means K tau is very large for first reaction, K tau is far greater than 1, Ok, so now what you have there from this equation, ah, numbers, this is 8, 9, 10, 11, Ok, so then of course 1 is neglected, then you will have T S is 4 point 6 by K V, see now kinetics also come there, and life is not that easy, why, because I have done only for first order reaction, in the examination I can ask you, second order reaction, second order reaction, so that is why the steady state value for a C S T R is not that kind of simple thing, Ok, but in industry they wait for you know 10 times, Ok, 7 times, 8 times, 10 times, 12 times like that, and then finally measure the concentration for may be 1 hour, 2 hour, 3 hour, 4 hours, and if almost all that is same, then we will say that we have steady state, you see, very innocent looking C S T R, where happily we ignored that, we do not have steady state, we have only, sorry, we do not have one steady state, we have only steady state, how much thing is involved there, Ok, so this is the information, but I think you know plug flow is a nice guy, because ideal plug flow, it will simply sweeps the entire volume out once, and then onwards you will have steady state, Ok, good, so now, I think story is not yet over here, if I plot this concentration C A versus time, Ok, that means this equation, which equation? 6, when I plot that, what do you get is for, of course, you have to plot that for a given tau, the K tau I have taken as 1, right, so what do you get here is C A by C A naught, because C A by C, C A naught I can take here, so then it is only in terms of K tau, right, so when K tau equal to 1, you will get like this, Ok, there will be slight increase, there will be slight increase, that depends on the conversion, conversion depends on K tau, now I have not told you the real story behind this, and now we have, we may operate this in various ways, right, there are 3 ways of operating C S T R for getting steady state, Ok, so this one is, somehow that we filled up and then the outlet concentration at time t equal to 0, Ok, so how can I do that, I can do that by filling up the entire reactor with inert, and then fill up, entire volume is occupying by some inert liquid, right, so then when I start the actual reaction, my reaction, then at that time only time t equal to 0, C A naught equal to C A equal to 0, because only first inert comes out, in it means fraction of second, 0 plus, we are saying instantaneous mixing, so 0 plus if I take, you know these are all mathematical boundary conditions, 0 plus if I take, I may get some concentration, exactly at 0, I may not, I do not see that concentration, that means not even one molecule coming out, but 0 plus means because of mixing, some molecules may come out, Ok, so that is one thinking, right, that means mathematics and physics you have to really try to match in your mind and then only write the boundary conditions, at that time this is fantastic, this is fine, I will now, you know, inert we have filled up, the second option is, Ok, first option is inert, filling up with inert and starting reaction, that is 1, here only you will get C A equal to C A naught at time t equal to 0, because inert you have, I take water and then I actually put my reactants later, so that is inert, I mean because what is not reacting with that at all, so that some inert, so at time t equal to 0, exactly at that time when I start outlet concentration of this reactant will be 0, this is one condition, there is another way of operating, you tell me, what is the other way of operating? Yeah, that is one, Ok, so that means I have C S T R, I have C S T R, nothing is there inside and then slowly you fill up, what is that method of filling up you call, I mean in our continuous system, batch system, semi-continuous, you are only feeding, nothing is only input, it is semi-continuous, that is much more difficult, that will give you in the examination, Ok, you think about that, but Ok, that is one, other one, right, what is the other operation, this is one, semi-patch only you told, semi-patch only, you start putting both the reactants, slowly it builds up, during the time reaction also will happen, correct no, like batch exactly, but batch with variable volume, variable volume, you have to think all this I say, think, think, think, think, right, yeah, variable volume and then when it comes to overflow, from then onwards it comes out, from then onwards how much time it takes for steady state, this time to find out is very easy, can you tell me, how, raise me, till it comes to overflow, that is all very simple, Ok, volumetric flow rate, that will give you, that is straight forward calculation, if I have 10 liters and I am sending 1 liter per minute, so 10 minutes it takes for to fill up, that is what everyday we can also see when we are pouring water in the glass or in the bottle, all that, you know, how much time it takes is depending on how much time, you know, rate of flow into the bottle, that is what everyday you are doing for drinking water and all that, that is very simple, so then once it fills up, reaction also is happening, during that, that is complicated, semi batch, so from then onwards it overflows and comes out, then it reaches steady state, that is the actual practical way of doing things, Ok, right, that is what, empty, but need not be, I think if I am an intelligent fellow, I will say that no, no, no, why should I put that, other way is to fill up the entire reactor with CA0 itself, you fill up the entire reactor with CA0 and then start reaction conditions, may be temperature suddenly coming to 100 degrees centigrade, there are so many ways of, I mean conducting this steady state operation for C S T R, I do not have product, no, if I have product no problem, ya, somehow I think, you know, in industry you go and tell sir, somehow we will make the product sir, I think that somehow will never work, Ok, so that is not the practical way, you know, because reactants, but CA0 I have, I will fill it up and then I will start the reaction, reaction conditions, that means there is no reaction, that means in the beginning it is almost inert and then it overflows and you know, at that time I can start the reaction also, reaction temperature, ya, that is also, but I think steady state, again I have to wait, because I think you know, what he says is, but steady state, all these are the possible things, I mean at least I am happy you started thinking about this, and if I have two reactants, A and B, ya, when, again you have so many possibilities, why I have to put only CA0 and then start CA, CB, Ok, I can put both and then start, I can put one, this way, that way and then start, see how many possibilities are there to start the reaction and you will get real award from your boss in the industry, if you go to chemical industry, Ok, if you are able to tell what in the simplest method what you can do, simplest method means time saving and also, ya, I mean that is quick steady state, without losing much reactants, because unsteady state you lose the reactants, because you are not getting the product what you want actually, so that is why you have the so many possibilities, I will give you just another possibility that you know that filling up with CA0, that is option 1, option 2, fill the reactor with CA0 and start flow with reaction conditions, this is one of the, this is one of the ways only, I am not saying that I am right, Ok, this is one of the methods I have chosen, right, where the differential equation is exactly same, equation 5, only boundary condition is, BC for this is, ya, at T equal to 0, CA equal to CA0, Ok, the other things are not that is you have to solve, you know, like for example slowly filling up and then it overflows but the reaction is going on inside that, Ok, so after, by the time it reaches to the top, what is the conversion, from then onwards how much time it takes to reach the steady state, all that one can do it and all the time it is only first order differential equation, you do not get more, right, so corresponding boundary conditions you have to take and then solve the problem. The solution what you get for this is, the differential equation is same, this only, and then we have 1 minus k tau by 1 plus k tau, 1 minus exponential minus 1 plus k tau by tau into T. Can you tell me what kind of graph I get when I plot, concentrate CA by CA0 versus CA by CA0 versus T, starts with 1, no, I mean maximum is 1, I do not say it will start with 1 or end with 1, start with 1 and then very good. I think all of you should be able to imagine this, ya, then it reaches almost steady state, so this must be the steady state value, Ok, good. So there are so many other possibilities which you can do, so these are the boundary conditions I told you know, so when I explained first this one, the first method, he was not very happy because that was not in his mind, he was thinking something else. So then I try to convince him with the first method and then he said finally SR and all that but I know that his eyes are not convinced, he had his convincing, Ok, SR, SR. So then I went home and then started seeing this, thinking, that is why, till then I also not thought, that is why when you ask questions or when you show unhappy face, I have to think. Then I thought and then I went and again saw, you know, transport phenomena book by our grandfather, BSL, so grandfathers, Ok, so then I, you know, he has solved this in a different way. So like that when I thought and then, sit down and then think there are so many ways of operating this, so many ways of operating this, Ok. But PFR is not a big problem because in one residence time, everything happens and normally what is the residence times we use in PFR? Very small, gas phase reactions, very small, you know, maybe seconds, so that is why you can wait 10 minutes, you will get beautiful steady state, no problem, industry has that much time and then you can go but here you have the problems. So this is the one and this is only just to give a sample of, you know, how you have to operate the CSTR and I have done only for first order, this equation will be very, very complicated the moment you go for the second order reaction, that is how I like you to do in the examination. So that is why you have to have all, you have to work more and more and more, I tell you. So now this is the one I think with steady state and, you know, unsteady state reactors, uhh, yeah, these are the simple things only what you have done. Now let me go to recycle reactor. So when do you use recycle reactor to increase the conversion, which is wrong answer, Ok, any other answer? No, I am asking the reason, you are telling about an example. You are giving an example where it is used but what I asked is, when do you use? Output is low or output low, there are thousands of reactors, Ok, including us. Our output, our output is always less. To increase the yield, in case of reversible reaction, who told that? 3rd culprit, yeah, who told that? When mixing is required in PFR or when you want to have mixing between PFR and MFR, what is mixing in PFR? 0. What is mixing in MFR? Infinity. Infinity, somewhere in between when you require, when you require, Ok, so when you require, you told the answer, so that is why I am asking, how can you increase conversion, you know, by putting mixing, we know that no mixing reactor is the best for normal reactions n greater than 0, right. So that is why how can you increase conversion, this is a bad thing to tell, you know, at this point of time, recycle reactor is never used for increasing conversion because plug flow is the best and now here in mixing, in recycle reactor, the mixing is there, some mixing is there where that is bad for the reaction. So that is why maximum mixing in CSTR gives you lowest conversion, Ok, it dilutes the concentration, so that is the reason. So intermediate mixing is required sometimes for multiple reactions, multiple reactions A going to R, R going to S, again this fellow A going to R, R going to S, that is series. So again A go to some other product, B also go to some other product. Under those conditions, sometimes under some conditions, you have to control the mixing, right. So it is not ideal, plug flow will give you 100 percent yield, Ok, yield is the required product, Ok, not the conversion yield. So to improve the yield in multiple reactions that can be used. Then another example what Rahul told was that autocatalytic reaction. Autocatalytic reaction is beautiful one. Why? Because I need some mix, Ok, what do you mean by mixing? So that means the products as well as reactants are mixing. In recycle reactor what I am doing, I am taking out the products and then bringing them and then putting into the inlet. So autocatalytic reactor, for autocatalytic reaction, this is one of the best reactors but there are other arrangement of reactors also for autocatalytic reaction best, Ok. It is not the only method. But if I want to use a single reactor, then the best reactor is recycle reactor but again there are many conditions. That is why autocatalytic happily we cannot accept. If the conversion low is somewhere intermediate, 80 percent conversion, 70 percent conversion. If I have conversions low around 0.5, 0.4, CSTR is the best. How I am able to tell all this is if you plot 1 by minus r A versus C A or X A for autocatalytic reaction by looking at that you will know which reactor is the best, Ok. So that is why best way of looking at the uhh you know which reactor is the best, Ok. The best way is to first plot, I think I think let me tell this one before going to recycle. So 1 by minus r A versus X A plot will give me lot of information, Ok. I know that if it is monotonically increasing something like this which reactor is the best because the area is just area under the curve. So here this area only P F R and this area will be P F R M F R. That is more area. So if I have uhh 1 by minus r A some crazy versus X A, so I have here like this goes up like this which reactor is the best. So now we have to divide this as parts. Yeah, so 1 reactor may not give me any idea here. So here when it is decreasing if I use a reactor till here like plug flow reactor for example area under the curve will be this entire thing same symbols this is P F R. If I use mixed flow it is full region. From here this is the outlet, this is the X equal to 0, Ok. Yeah, so this is the one for M F R same symbols. Next one this is P F R, Ok. If I use here mixed flow this is the one, the outlet concentration. So this is the one. Again next one, yeah. So that depends on what conversion I need. I need conversion here, right. So that means this entire thing is P F R and this one is X F R best way is that. That is why we know our people told that one uhh uhh picture can speak thousand words, Ok. Yeah, so that is why pictures are best. Good. So that is the one. Now recycle reactor. First you have to write when the recycle reactor is used. Whenever we have intermediate mixing between M F R and P F R, Ok, used if mixing required is between P F R and M F R, Ok. Conversion and all that will not come into picture, Ok. That is one. And another time, one more thing is there. Anyone can say that? Sometimes we do that apart from this, of course uhh this automatically tells me that this mixing is required to increase yield and all that. That we are not talking. I think everything will come there. Other than that there is another use. All those things will come under whenever you require uhh different mixing than these ideal mixing. Yeah, how do you control the reaction? Yeah, when do you do that? Why should I dilute the reactor? Correct. Both are stilling almost the same. Controlling the reaction you said you are still diluting. By diluting I am controlling the reactor, right? Yeah, gopī. Yeah, if you want to control the temperature, Ok, sometimes the products are used and then you can control the temperature inside the reactor. That is the other one, Ok. You control temperature inside the reactor, Ok, under these conditions it is done. Good, very good. So now we will try to, we have to derive an equation for this and the derivation must be somewhere between these two only because it is not either mixer flow reactor completely or it is not completely plug flow reactor. So the derivation also must be in between, the equation must be in between that, Ok. So now first we, we imagine that we have a plug flow, ideal plug flow element, Ok. Recycle reactor consists of ideal plug flow element and now we will send the reactants, they come out as products and you have to take out some, yeah, product and then recycle. This is ideally PFR. Good. So how do you derive an equation? Here there is a slightly confusing thing. Leven spear also, I do not know how many, all of you have done it in your B-tech? Do you remember anything? Absolutely no. All files are cleaned, Ok, good, right. So that is very good. Starting with clean sheet, Ok. So now we have, yeah, this PFR and the confusion comes there that how do you first of all operate this recycle? My imagination is, I think, you know, I have not seen, I have drawn my own diagram sometime, long time back because I also had this confusion to tell clearly to the student but as far as possible I am now trying, still I do not know how many of you really convinced in what I am trying to tell all those things, Ok. So what I do is, we first introduce FA not here, no reaction, no reaction. So if there is no reaction, at the outlet also for steady state I should have FA not, right, yeah, now. But initially, this is under steady state but now I have to take out something and then recycle, right, Ok. So now that much, what I put here is that I have R into FA not will be recycled. That means if I have FA not as 1 liter per minute, 1 liter per minute will come out, right, then I am now recycling, depending on recycle ratio, may be let us say 5, recycle ratio 5 I have taken. So how much I have to recycle now, 1 liter, 1 liter, how much I have to recycle if R is 5, see LKG question I say, yeah, you should call me, sir, do not ask stupid questions. But I think you are thinking very, very carefully whether even though I think it is, you are before Supreme Court where if you tell one word more I think judge may sue you and all that, just to open your mouth and then say, Ok, yeah, so that is what I think in molar flow rates also it can be done, that is no problem, right. So that is how I operate otherwise if I send 1 liter and 1 liter and then if I take it back what will happen here, it is not steady. So that is why initially we fill up and then we try to recirculate that much all the time, this is before reaction, there is no reaction, here there is no reaction, right. And now under steady state, hydrodynamic steady state condition, you know what is hydrodynamic steady state condition? Yeah, flow is constant, excellent, flow is constant. How do you measure that? Normally we measure in terms of pressure drop for example. Your pressure drop is not changing, constant, then we will say flow also is constant. I think pressure drop we had a lot of discussion earlier, right. So the moment you have the velocity changing then pressure drop will be different, so that may be unsteady, right. So that is why you have to wait that time hydrodynamic steady state and then suddenly bring this entire thing into a reacting condition. Then what will happen to this entire diagram, okay. That I will draw now, you have same thing, okay. Nothing will happen to F A not because that is entering, right. Yeah and now I have, this is the recycle, yeah by the by. So how do I define now recycle R? This I have to define before going there. In terms of volumetric flow rates you can write the amount of liquid recycled back divided by volume of liquid, not amount, I think volume of liquid coming out, okay. Volume of liquid, okay, of liquid recycled divided by volume of liquid withdrawn from the reactor. So that is what is recycle ratio R, okay, good. So then what happens? I have here reaction condition started. So that means some reaction would have occurred here, good, okay. So again we are now talking about steady state conditions after the reaction conditions started, steady state. This will be F A F and the corresponding conversion will be X A F. This is what what I wanted earlier, okay, in the beginning before designing. This will be V F, good. So now this point, at this point I have, yeah, so this is already converted one in the reactor, something converted and I am now taking back and then again mixing here. Originally I have this R into F A not also moving, please remember, R into F A not also moving. Otherwise I cannot operate recycle, okay, recycle 3, 4, 5, whatever. So now this entire thing which was inside and then recycling, that also will get converted, right, okay. And before, okay, before that reacting conditions, what will be the input here, total input? This is F A not plus, yeah, so that we call it as F A not dash, as F A not into 1 plus R, 1 plus R or R plus 1, okay, good. So that is the one which is there, that is what is actually entering there, that is what is being converted inside and because you are taking out under steady state conditions, it will be F A F, X A F, V F and here now the conditions will change, that condition is, yeah, F A not is originally was there, that I will write, F A not dashed equal to 1 plus R F A not, then I will also have F A 1 here, F A 1, here also X A 1 and also of course V 1, all that and at this point I call it as X A 2 and R plus 1 V F, if I write in terms of V F also, this is, no, V 1 only, this is V 1 and at this point I have R into V F, V F is coming here, sorry, V is coming here, V also is entering there, right, so this is R into V F and we also call this one as F A 3 which is R into F A F, either in terms of moles or volumetric flow rate, okay. So now, yeah, I think all these things are okay, so what we have to do is, if you want to develop an equation, I have to develop only in terms of this and then this because inside I do not know what is happening, right, but that automatically will include this X A 1 because actual reaction is taking place from here only, right, from here only and this is again P F R, yeah. So because it is a P F R, I will take a small volume and then write the material balance entering, leaving all that, in that, okay, yeah. So now I have to also define my rate, I mean the conversion here, the conversion is defined as F A equal to F A not dashed, please remember, F A not dashed into 1 minus X A, F A not dashed, what is dashed? Originally, yeah, it is 1 plus R, so that is what is entering originally all the time, okay, yeah, because you are recycling that, you are recycling here, so this is the 1 and now this equation, you have this equation as, okay, minus D F A equal to F A not dashed will come from this here, the other one, minus D F A equal to, I am talking about balance, material balance here, you are only trying to tell what is the meaning of F A, but this is equivalent to what, in that volume, reaction, okay, yeah, what is that, exactly, minus R A into D V, that you cannot forget, I say, should not forget, okay. So now D F A, we can calculate from this, this is equation 1, let me say, this is equation 2, now substitute that D F A here and then tell me what is the equation, because this I have to simply differentiate, right, yeah, F A not dashed D X A, I will write in a different way, so that means minus R A into D V equal to F A not dashed into D X A, so now how do I write this one as an equation in our, because it is a plug flow, V by F A not, so this is V by F A not dash equal to integral D X A by minus R A, but now what are the boundary conditions, yeah, it is X A 1 here to X A F, so this is what is the, the design expression for recycle, but still it is not completed in the sense that 3, 4, because I do not know what is F A not dashed, but anyway I know here, F A not dashed and also I do not know what is X A 1, how do you find out X A 1, F A not dashed is easy, that I think you understood, but can I define X A 1 in terms of, how do I define for continuous systems, for flow systems, conversion, I am asking again stupid question, you should quickly react, what, initial minuses, donkeys or horses or what, which flow rate, that is what, initial molar flow rate F A not, here it is F A not dashed, okay and minus F A 1, that is here, okay, if you are defining X A 1, right, so that is what we do now, X A 1 equal to F A not dashed minus F A 1 divided by F A not dashed, good, okay, good, so now we have to find out, to find out X A 1, what is F A 1 and F A not dashed, F A not dashed I know, this is equation number 5, what is F A not dashed, 1 plus R, okay, 1 plus R into F A not, that is no problem, this is no equation given number, okay, 6 and we have, yeah, so F A 1 I have to calculate now, what is F A 1, F A 1 is, you see the material balance here, this is F A 1, yeah, this is L, this is K, yeah, so I think very good, what you understood, I think that is fantastic, very good, so this one is, because all of you, I think you know, Gopi able to follow now, at this junction, this is coming and also this is coming, this is coming F A not, this is coming F A 3, F A 3 is nothing but again R into F A 1, so that is why it is F A not plus F A 3 equal to F A not plus R into F A F, okay, that is F A 1, now substitute this equation 7 and 6 in 5 and tell me what is X A 1, 1 equal to R A, R A by X A F, very good, X A 1 equal to R by R plus 1, yeah, R by R plus 1 into X A F, that is all the derivation, so now the equation for you is that V by F A not equal to R plus 1 integral X A 1 equal to R by R plus 1 X A F to X A F d X A by minus R A, so this is the design expression for not dash, so because dash is inside, I do not want to go to the dash there, okay, yeah, so finally I told you know we have to express only in terms of what is this point and what is this point, so all the things I know, V by F A not dash which I am continuously putting, I know that F A not dash comes before starting the reaction where you have recycle coming and you have so much volume which has to be recycled and that effect of all that is only this F A 1, what is entering, because of some extra F A not was there, that R into F A not, all that together getting reacting and then giving me F A not, F A 1 inside the, at the start of the plug flow element, good, so this is the one and yeah, so initially most of the time you have to take some recycled ratio, you have to say that okay my recycle ratio equal to 1, okay, so and conversion is 90%, then you can calculate volume, not volumetric volume or otherwise if I know volume you can calculate X AF, but again R should be known in the beginning too, okay, yeah, in the beginning you have to choose the R, right, but mathematically, yeah, how do you choose, okay, good, how do you choose, so what will happen when R equal to infinity to this equation, R equal to 0, and then why did you say mixed flow, I said infinity, oh sorry, okay, R equal to infinity mixed flow, right, I am sorry, yeah, so R equal to 0 is plug flow, now how do I choose, between 0 I think one life, one life is enough to choose or not enough to choose, because you have to now decide whether you want mixing very close to P F R or mixing close to M F R, right, so when you have close mixing to M F R then do not go to infinity, even 500 is too long, too big, 20, 30, 15, 16 also is very high, okay, like exactly, you know you do not have to go to infinite number of tanks to get plug flow, right, so you have seen, no, 6 reactors, after 6 reactors, 6 to infinity number of reactors when you use, then only you will get that small change, I think it is 3.3, yeah, 31 and may be 3.5 or something, to compensate only 0.5 you will go to 6 to infinity, number of tanks, so here also recycle ratios also are same, okay, beyond 2025 you almost reach, at 2025 so large value I am telling, you can calculate that actually, right, so you will almost get mixed flow, so that is why what you have to choose is, okay, now I am very, I want to be very close to plug flow, then R can be 0.1, 0.5, 1, 1 also is large, when you want to be very close to P F R and that comes when you have the sufficient experience, okay, what kind of mixing may be required for me to control my temperature, right, and to get some yield, what kind of mixing we need, so that is why we have computers, we have brains, so we have to simulate and then try to find out what is the best recycle ratio and there is another way of finding also optimal recycle ratio and all that, that also I will tell you, okay, that is all, but only thing is P F R is not receiving fresh reactant but it is mixed with product, so that means some mixing is happening, what is happening in mixed flow? You are putting fresh reactant but it is now perfectly mixed with the products, old ones, okay, there it is perfect mixing between products and fresh liquid, that is why you have the minimum dilution, maximum dilution possible but here we can control, if I send only very, very small quantity then I am operating almost P F R, if I send very large quantities, this will behave as mixed flow, I think that is another my favorite question, this equation can be beautifully simplified when R equal to 0, R equal to infinity, two extremes we can beautifully simplify that equation, that also please check that, okay, so the next thing will be how do we plot this, plotting this is really big headache because I really like graphs, right, so that is why, how do I put this information on graphs? That means again I have to plot 1 by minus r A versus X A, so what is the volume equivalent to area? See like if it is ideal single plug flow, you draw like this, if it is monotonically increasing, take area under the curve, this area under the curve equivalent to some volume, right, so similarly here also what areas I have to take and how do you calculate what is the volume? From the graph, that we will discuss in the next class.