 the number of banks in series, you will get plug flow. You will know, again see that from the actual reactions. That is why, you have the calculators, at least a few of you? I think let us solve one problem, then you will know, how beautifully what we said is right. first order reaction, again simple, first order reaction. So then I have to first draw the tanks in series, okay. I have N, this may be I minus 1, sorry, I, so this is 1, 2 and here I have C A not entering, this is C A N and here may be V entering, V, liquid phase reactions, first order, because C S T R most of the time we use liquid phase reactions and to write this I have V 1, V 2, V i, V N where all of them are equal, that means equal volumes you are taking, equal volumes you are taking. So now we will define our tau, tau of i as volume of i divided by volumetric flow rate, okay. V is constant throughout. Actually try to find out what is the total, tau equal to N into, in one tank, I, in one tank I have V i by V and in this many tanks you have total, so this is nothing but N into V i by V, by V. So this N times V i, also I can take total V where V equal to sigma of V i where I equal to 1 to N, okay. So this is what is again convenient, you know writing, right. So this is summation, okay, good. So this is the one just to start with and let us have that we have first order reaction, okay, liquid phase and we would like to write the balance for this, idea is to find out what is C A N by C A naught which is nothing but X A N equal to what, okay, conversion, I would like to find out what is the conversion that is coming out, C A N by C A naught, okay, good. So that means I have to now write this equation as other symbols also, this one is C A naught and coming out is what? C A 1 and now C A N is entering here, this is C A 2 and here I have C A i minus 1 and this is C A i and here I have C A N minus 1, yeah, so those are the all things. Now let us write the material balance for first reactor it is V into C A naught, V is C A 1 leaving, this is input, output, reaction, accumulation and this is steady state, okay, I think I have to also member force in series, first order, I can also write here steady state, okay. So that is why accumulation term will be zero, this is I have minus r here into that is K C A, K C A 1 into V, it is not total V, actually it is, I will write here this is V dash, this is also V dash, now all this equal to V, each one equal to V, okay, so V dash equal to sigma of V i that is okay or sigma of V also I can write all V's or N into V, okay, good. So now this, this I will write for one reactor because now each one has V, all these are equal to, I mean for easy thing. So now can you tell me what is C A 1 by C A naught 1 by 1 plus K tau 1, yeah, so now tau 1 also I will take that you know this is V tau, right, so then I have to write this one as tau dash then, okay, sorry, that one as tau dash, okay, then I will define tau as K tau where tau equal to V by V, I think you make that corrections dash dash and all that, okay. So now this is equation 1, equation 2, this is okay, now equation, yeah similarly for second one this is for N equal to 1, for N equal to 2, can you write on your own now, can you write on your own now or otherwise you tell me I will write V into C A 1 equal to V into C A 2 plus K C A 2 into C A 2 into V again, correct no, so what is this equation in terms of C A 2 by C A 1 equal to 1 upon K tau, so again for C A, for second reactor also you have 1 by 1 plus K tau for this side the concentration, okay similarly now we do not have to write for everything, now this is C A 3 by C A 2 will be again 1 by 1 plus K tau, so similarly when I write all these I have C A N minus 1 by C A N equal to 1 by 1 plus, thank you, thank you, thank you, thank you, very good, N minus 1 equal to 1 by 1 plus K tau, so now I would like to only find out this one C A N, so I can now multiply all these, you know in fact I have C A 1 by C A naught, C A 2 by C A 1, so like that we have C A N by C A N minus 1 and here I will have C A N minus 1, so you can cancel or if you want you can write one more, this is C A 3 by C A 2, so like that you will get this one as C A 0 which is equal to, now I have N number of terms we will have 1 by K tau to the power of N, okay, so if I write the equation this is 4, 5, 6, this is 7, okay, so this equation also C A N by C A naught can also be written in terms of total time, 1 plus K tau dashed by N, correct, tau Y equal to tau and tau dashed equal to N tau, okay, so that is the one, so this equation we can nicely convert into K tau dashed N equal to, yeah, so 1 minus X A to the power of minus 1 by N minus 1, this one I have written here, this is also equal to 1 minus X A N, right, so this is equation 8, this is equation 9, must be X A N only, so now this equation for N equal to 1, what is the equation? So that means I have only just one reactor, N equal to 1, that means all these things are not there, so for N equal to 1 K tau dash equal to 1 by, 1 by 1 minus X A, X A 1, for example, okay, that is all, I think this is minus 1, I am simply substituting that one there, now tell me for 95 percent conversion for X A equal to or X A 1, from this I am talking about, I am talking about one reactor where I have 95 percent conversion, what is the volume of the reactor, that means of course K value I do not have to separately give, I mean assuming that unity also, so I have X A equal to 0.95, what is K tau dashed, that is the measure of volume, 19, oh you said 19, not 990, okay, so this is 9, now for N equal to 2, tell me now, K tau dashed by 2, because N is there, N equal to 2, but now this one is 1 by 1 minus X A 2 to the power of, oh that is all, correct no, minus 1, half, 6 point, yeah, any other things, always Pooja and Diabayan only telling, somewhere that Karna, Ramakrishna you do not have calculator, Manikantan, tell me, 6 sir, 6, 6 sir, you round it off, how did you, I mean mental calculation, approximately, approximately, I think okay good, so this will be K tau dashed is 6.94, now tell me for N equal to 3, K tau dashed for this system is 5.14, okay, 4, N equal to 4, 4.45, okay, so now N equal to 6, excellent, okay, N equal to infinity, yeah, I know this answer, which is wrong, what do you mean by N equal to infinity, you are talking mathematics, I am talking physics, excellent, so what is the conversion in a PFR for first-order reaction, N equal to infinity means you are straight away going to mathematics, only one person sitting there thinking like me physics, right, so N equal to infinity, when N, infinite number of tanks when you put together, we know that that is going to go to plug flow, so now what is the equation for first-order reaction liquid phase, you know equation for plug flow, N equal to, yeah, exactly that is the one, okay for N equal to infinity, we have N equal to 1 minus XA equal to minus K tau, K tau dashed because that is the total volume there, how much is that, 2.996, 2.996 that is 3, so K tau here K tau dash equal to 3, so what can you predict from this, we have all the results, which one is decreasing, K is decreasing, yeah K is constant for you know the entire reaction, you know whether you have, yeah I mean K tau is a measure, K tau when you say by the way what is the name of K tau number, damkohler number, what is the general definition of damkohler number, what is the spelling of damkohler number, damkohler number, in fact there is that umlaut there, kohler, okay, Germans, damkohler number, yeah what is the definition, K tau CA naught N minus 1, so for first-order it will be K tau, for second-order it will be K tau CA naught that again naturally it falls, if you see the second-order equation for plug flow what we have written, you know just before that that is K tau CA naught, for second-order, for first-order it will be K tau, for 0th order, K tau by CA naught, that is damkohler number, that gives a measure, for example what is the meaning of having large K tau, when I say that I have large K tau, that means that I am going to get large conversions, conversions, because you know damkohler number, so I say that this reactor has large damkohler number means in that reactor we have conversion more, okay, good, so now here coming back to this results what do you learn, beyond this you cannot go, okay, if K equal to 1 as you know as for easy thing, and you know here I may take this one is 19 meter cube, as an example if K equal to 1, okay, 19 meter cube here, so that means in the second reactor that means if I use two reactors for the same conversion, one reactor I have used, 95 percent conversion, what is the volume I have to use, 19 meter cube I have to use, the moment now I take two reactors, same conversion 95 percent, you know the total volume itself has become 6.9, and each volume, volume of each reactor only two are there, so 6.9 means, okay, 7, 3.5, engineering approximation, 3.5 meter cube of two reactors if I take, then I will get 95 percent conversion, now next one, 5.14, K tau dash is total, tau dash is N into tau, ya, you already multiplied by 2, because in the denominator this you have, so you multiplied by 2 and then you get, otherwise if you write only K tau dash by 2, that will be 3.5, approximately, then this will be 5 by 3, how many, 1.5, 1.6, 1.6 or so, you see now, so this is 19, individually, so like this with 1.6 or 1.65, I have to use 3 tanks in series, what is happening to the volumes? Decreasing, so when you put an infinite number of tanks like that, what will be the volume? Almost zero, what is the meaning? Not plug flow, each cross section, no, each cross section, cross section thickness we cannot define, almost zero, not that there is no reactor, that is too much approximation, okay, we have a reactor with 3 meter cubed finally, total volume, total volume, in fact this divided by infinity, 3 divided by infinity, so 3 divided by infinity is still some number, it is not zero, okay, so ya, ya, very very small one which is equivalent to our one cross section, see beauty again proved, now the reason, why from 19 to second reactor 6.9, here steep fall, if I plot this K2 versus N, K2 versus N, this is steep fall from 1 to 2, from 2 to 3, not much, from 3 to 4, not much, you see here, in fact after 3, no, after 6, very 6, ya, after 6, I have to go to infinity, after 6 is almost 3.88 total, and for infinity number also we have almost 3, okay, so that is why, why the first reactor has tremendous effect, see the moment you say higher rate, the volume should be smaller, correct no, ya, that is why Abdul, I think we have to really think about that, because when you say high rate, what do you mean by high rate, so volume should be less, is there anything else? Ya, you by definition, this mixture flow reactor has bypassed it, by definition, you are saying that you have perfect mixing, and through RTD you can find out in one mean residence time what will be the, this is what one mean residence time, not tau, tau equal to volume by volumetric flow rate of one reactor, okay, that is only one reactor, so what happens is the moment you put the liquid continuously, there is lot of fresh liquid which is going out with the old liquid, that is what I explained also, what is mixing, perfect mixing, it is not instantaneous reaction, please remove that concept from your mind, instantaneous reaction, continuously you are feeding, mixing is instantaneous, not reaction, mixing is instantaneous, so when mixing is instantaneous, concentrations will come to some steady state concentration which is outlet concentration, but that does not mean that everything converted, on the whole when I look at the outside or inside also same, there are molecules which have just entered, there are molecules or packet of molecules we cannot say, packet of molecules which have entered long time back, average of all that is 95 percent conversion, average of all that, so to maintain that 95 percent conversion I should use 19 meter cube, when compared to plug flow reactor, infinity equal to plug flow, otherwise you may think that there is another reactor called infinity reactor, okay, this infinity reactor equal to plug flow reactor, okay, so in this plug flow we have to use only 3 meter cube, why this is where the beauty of plug flow comes, by definition each and every particle must spend exactly same time, so conversion in every particle, particle means it is containing lot of molecules, so in every packet or particle you have the same conversion, when I take the average and then again mix all that and take the average conversion, also same, whereas in the mixing flow I have a particle which is coming with a lot of molecules, one quickly coming out, almost spending 1 minute or 1 second, when the mean incidence team is 10 minutes, so if I take 1 second, it is true, the moment you put, when it is a continuous mixing, what is continuously entering, for imagination I am telling, continuously you are taking out, you put just one drop of ink, instantaneously it will appear in the outlet, instantaneously you can check that, I think in our lab also when you go transfer operation lab, there is that CSTR, tanks in series 3, 4 together, so you start the starter and then take your ink and then just drop one ink, instantaneously it comes out, what is the meaning, so those molecules which are immediately coming out, they do not have, they have no reaction at all, that is waste for us, that is bypass what we call, so that is why when you take and there may be another one coming in the second, another particle or not one particle, many particles coming in the second second, third second, fourth second, fifth second, like that you have and through RTD studies, we can actually calculate from 0 to 1 how much fraction of the material coming out, that is what is your ET DT, the fraction of material coming between time T and T plus delta T is ET DT, that fraction we can calculate, so I can calculate what is the first fraction, you know 0 to 1 minute or 0 to 1 second, the next one is 1, 2, 2 seconds, 2, 3 seconds, 3 to 4 seconds, all that fractions, there definitely the conversion will not be much, why the time spent by that fraction is only 1 second, 2 seconds, 3 seconds whereas the average residence time is 10 minutes, that also steeply falls, I do not know whether anyone plotted, you have done experiment in RTD, some of you, you have done ideal one single CSTR RTD, if you could have done that is exponential decay, initially it falls very fast and then slowly it comes out, the tracer, okay, so that means the last portion when I take not much between, you know, may be 55th second and 56th second, it is not much or maybe minutes if I take, so that is why it is the residence time distribution which is affecting the performance of a mixed flow reactor when compared to plug flow where theoretically there is no residence time distribution, residence time distribution is equal to 0, right, you understood now because please do not say that, you know, the moment you have the reactant going instantaneous reaction, it is not instantaneous reaction, it is instantaneous mixing and in that mixture what I see is molecules which have entered just now, molecules which have already converted, so all this average what I am seeing as 95 percent conversion, right, so that is why the first reactor has tremendous effect, from first to second you would not see that kind of bypass, I mean it is also perfect mixture, second reactor you cannot, that means if you find out RTD for two reactors, we can calculate, RTD for two reactors, we can again calculate two reactors together, first reactor it entered, second reactor it is coming out, right and what is the fraction that is coming out in two reactors in one minute, it will fall, it will be less than one reactor, so like that you know second, second, third, second, fourth, second, all that I can calculate, so like that I can take six tanks RTD and calculate what is the amount that is coming out of this sixth tank after one minute, how much, you will not see much, so then if I take to infinity all the molecules would have spent exactly same time, why the molecules which have bypassed here may stay here, some other molecules will bypass and from here some other molecules bypass and will go, some other molecules will stay, on the average if I put infinite number of tanks all these bypasses and all these staying inside the each reactor all will be uniform for each and every particle, that is why the residence time of each and every particle if you have infinite number of tanks in series is exactly same as plug flow, then also you can see how this volume is going becoming smaller and smaller and smaller and finally it is going to the size of almost zero, limit tending to zero, where the tending to zero is nothing but volume of your cross section, I think with this you should not forget that, beautiful concept. So that is why we know that it is very difficult to operate plug flow reactor because you need very high velocities, okay what are the Reynolds numbers? Theoretically infinite, but practically when you want to see that flat velocity profile which confirms that all the particles are moving at the same speed that you know flat velocity profile you need around 40,000, 60,000 in an empty tube, you have also mentioned empty tube, if you go to packed bed 500, 600, if you go to 4000, 5000, guaranteed, you know almost plug flow, that kind of more and more Reynolds numbers when I am using, what will happen to my residence time? Less and less. So I cannot use this system for all the reactions where we have more residence time required for conversion. That is why we go for gas phase reactions where gas phase reactions, reacting time is seconds, 10 seconds, 15 seconds, 30 seconds like that. Even then you have there 1 kilometer, 2 kilometer length they putting like this, like this, like this, like this. I am not exaggerating, that kind of length will be there. So if you go to real, you know, may be no seal, that kind of solid chemical engineering industries, you will have this kind of lengthy reactors for plug flow. See now how beautiful it has come out, you know, the series reaction. So that is the reason why because plug flow is not practical most of the time, we try to use 4 reactors, 5 reactors where mixing you can guarantee, temperature control is beautiful, absolutely no problem and not only that, again you know this is what that expansion of the mind should come. You should think also, we talked about isothermal system, right. So that means even though the concentration is high here, okay, high here and when compared to this one, the rate of reaction will be high here because concentrations are high and the average again I am talking and here the concentration has fallen, right. But how do I increase the rate here? See for this alone you can have a separate temperature control. You can supply separate heat for this alone because outside jacket now I will maintain, this is at 60, 60 or may be 75, 90, 120, like that you know you are trying to compensate now through temperature the rate of reaction more. Then again the total volume will be less. But only thing is you have to see, I mean it is not that easy, you have wonderful ideas but practically everything is measured in terms of paisa, money. Temperature means beyond certain things you cannot go. Temperature means what temperature you are talking, 200, 300, 400. What temperature, 40 degrees, you cannot use, that is all simple. That is why C S T R in biochemical they cannot use, right. If they use it will be 30, 35, 36, 37, 38 like that only. Because may be 40, 45, many microorganisms will die, you know. Yeah, if the temperature is 50, 40 and continuously you are exposed there, that is all, Margea. Ok, that is all, finish. Ok, so that is why if the temperature is sensitive, we have to use our brain, Ok, this is temperature sensitive, I cannot use beyond this temperature. That is very simple question to answer. So that is why you can also try to compensate and Wallace book gave that kind of problems. W A L A, Wallace, the kinetics for chemical engineers also, that is the title. You know that he has given, I think, you know he has given I think 4 or 3 reactors, if I remember correctly. So in each reactor the temperature is different, 15, may be 30, 45, like that. So now that means K will change in each and everything. Concentration whatever we lost, through K we are trying to gain Now I think the series one, next one is combination of reactors. I think, you know, it is only some very simple thumb rules are there. That is why if you go to really very, very high level, everything looks very simple for you. In combination of reactors, how many ways you can combine reactors, let us say, I have only plug flow reactors. First of all, why should we combine reactors? It is a stupid thing, you know. To increase conversion. To increase conversion you could have designed in the beginning itself, you know. Why again adding another reactor? Let us say we have P of R's first. Which is the temperature sensitive we are telling, in that case we can, combination of the reactors we can do. In some cases if it is... What is the use? I am getting more volume only, that I can design one reactor. We can control the temperature. Why do you want to combine? I mean even at a controlled temperature only at one temperature, isothermal conditions, all the reactors are there. Why do you want to combine reactors? I think partially our answer is right. If I want to increase conversion, already existing, okay, may be 100 meter length, diameter may be 2 inches, of a plug flow reactor is already existing, right? So that is giving me 90 percent conversion. But someone tells me, no, no, no, I think 90 percent conversion. So another 10 percent waste is there, I do not expect, you know, I do not take by your product. So it should be, may be 3 percent only you should have, the other unconverted reactants and all that. Then what do you do? You do not have to design again a whole new reactor, okay. So then you can add reactor. Where do you, how do you add? Where do you add this? You can put parallel, you can put series. You tell me, I mean, why different, different? What do you mean by different, different? There are only two different. One is either series or other one is parallel. Any other way you can combine, cross current also you can put, but that will not happen here, whereas cross current will come later, I will tell you where, okay. So either series or parallel, okay. So when you have this conversion, if you want more and more conversion, do you put this P of R in series or parallel? Why? Easy to understand our real life. Easy to understand our real life. So that is why you put in series. Yeah, but by increasing length, where do you get maximum conversion? Yeah, I mean, answer is right, but I think correct word. Correct word. Exactly, more residence time. So by putting in series, you are putting more residence time. See that is why, like story only, we can tell this. I think you do not need any, of course, if I tell that, okay, I have this much 90% conversion, how much length is required for going from 95 to 97. That I think I will give you as a problem, but right now I want to discuss all this as a, because you do not need any new equation. I tell you, there are no new equations now onwards. Basic equations are already you learnt. But only thing is, mathematically you have to, you know, integrate or differentiate and then try to find out what will be that extra information, what you require for all this discussion. So to increase conversion, first thing is to increase conversion, so, okay, reactors should be in series, okay. This is general, I think. Even CSTR are also same thing. CSTR are also, okay, parallelly when you discuss that and you are not fixing the conversion, there we fixed conversion and try to find out how the volume is reducing. But I want to increase conversion. I have 19 meter cube in the beginning. So another 19 meter cube if I put, what will happen? Conversion definitely increased. How much and all that we can discuss later, okay. Yeah, so like that. So that is why in general, even though it is PFRs or MFRs, this rule is correct. It means to increase conversion, reactors should be connected in series, okay. When these parallel things will come, what do you mean by capacity? In our language, can you tell what is production capacity? If you have same conversion, if you have got your maximum conversion, you do not have to go with the series. If you want to increase your capacity. Ah, series we have forgotten. I am just asking you, when do you use parallel combination? Is it same time? Like increase conversion or decrease conversion, when do you want to use? What you said is right. But I think some technical words if you use. Production rate. Production rate. Increase the production rate. Yeah. Increase the production rate. Yeah, instead of capacity, production rate can be increased. So that means if you are able to produce 100 tons per day, someone tells that no, no, no demand has increased, you can go to 150 tons per day. So you do not have to design for the entire, you know, new plant. You have to only put now parallel reactor, okay. It is only now 50 tons. So you have originally, when you are talking about PFRs, like this one reactor, which is giving you 90 percent conversion. So this is for 100 tons per day. Now we thought now that we can put another parallel reactor for increasing to 100, okay, parallel. So this will be 150 tons per day. Both together, not one. Both together. Good. So what is the condition I have to maintain here? Because, yeah, so parallelly I may, that means I have to again separately put another flow rate at all that here, right. But is there any condition we require for outlet? What is the conversion? What should be conversion from outlet? Why? Otherwise we cannot mix both the outlets. Why I cannot mix? I can always mix. But what do you get? Different means more or less. That is the condition. That is the condition. So to maintain that condition, no one complain about 90 percent. But only thing is total production rate we should increase. So that is why this also should have 90 percent. This also should have 90 percent. Now you have to calculate back for 90 percent conversion, how much is the volume required and what are the flow rates? Ok. I mean automatically flow rates will come there, right. So this is one. Now Levenspiel says that you divide this, you know, the volumes. I think he gives an example of this is 80 liters maybe. This is 40 liters. How do you now split your flow rates? Because you have only one stream coming here. You remember that? V by F A naught. This is V by F A naught 1. This is V by F A naught 2. So we have to maintain V by F A naught these two same. Why they should maintain same? That is time. Yeah, you know the other side we have integral 0 to X A minus R A. That should be constant. To maintain that constant you again, yeah, you adjust your V by F A naught decide and V by F A naught. This there already. The other V by F A naught so that you will have, if you know V only F A naught has to be changed. Because you are taking 40 liters here. So now you can take that ratio, how much ratio you have to use to maintain same conversion so that this is nothing but you know 0 to X A D X A by minus R A. This is equal to in general V by F A naught. Why they are maintaining same conversion to V reactor? Yeah, let us say if this is 90 percent conversion, this is 70 percent conversion. What will happen? Because I have to tell, sell the product by mixing those two. What will be the average conversion? It depends on weighted average and all that. So it may be between 90 and 90. So that means your conversion is falling that fellow may not accept now. So the minimum thing you have to maintain is 90 percent only. Otherwise any other conversion even if you go to 95 percent here that is 90. So again it is changing, may be higher side but still that is not required. Because unnecessarily you know if it is 90 percent again you have more time required. So that is why whatever fixed conversion you have in the outlet, in the new reactor the same conversion so that there will not be any dilution. That you have to adjust. That V by F A naught is nothing but tau by C A naught. So that is why residence time is automatically adjusted. So to adjust those timings you maintain the flow rates such that because you are taking 40 liters and it is available with you Ok. Or some other thing. So accordingly you maintain flow rates such that you will get the conversion same. Simple thumb rule. It is not only this Yeah. Yeah, no problem, you know. That is true. You do not have to put this common point there Ok. But still this condition is say even if you put parallel without any touch no touch but still what should be the outlet concentration still 90 percent Yeah. Still 90 percent conversion, Ok. So that is what is the condition for all parallel combinations you have to use only same conversion whereas here to increase the conversion you go for I mean in series and to increase production production means production you can increase but you cannot decrease again conversion if you have different conversion of reaction of reactors if you take individual CSTRs and individual you know PFRs now we have another combination can I put a CSTR and mixed flow together under what conditions you do that we know that for all n greater than 0 mixed flow is is a luxury reaction Sir suppose I mean 90 percent conversion should be so we use a PFR but it is giving only 80 percent conversion Sir if you put CSTR in series after PFR I can put another PFR Why CSTR? So V by Yeah but I think that is another constraint which we are not discussing you know temperature control means I should go to the other PFR also only MFR Yes Yeah But later we need higher temperature to continue reaction Yeah So in that case first we do PFR That is not that kind of convincing reason for us because I can also maintain we can also maintain temperatures for PFR also Ok but I think which is convenient is definitely MFR but I know that MFR gives again lower conversion when compared to PFR Correct No I mean I have used extra temperature but in spite of using temperature that cannot compensate PFR except under some conditions it is a wonderful problem in Carberry where he has done that you know under some activation energies adiabatic reactors I will tell that when you come to non isothermal reactors I will also tell that under some conditions you can you know CSTRs also will go beyond mixed flow reactors particularly if you have adiabatic reaction under some activation energy conditions So that is why I think these combinations generally we do not use MFR and MFR followed by CSTR for normal reaction CSTR followed by PFR and PFR followed by CSTR that is not generally followed but in one case you beautifully use that We need high concentration we can go for PFR the later the remaining unreacted things can be mixed in CSTR No there also I can use PFR because there also PFR will give you better conversion Yes sir but unreacted conversion will be in the concentration unreacted means I mean unreacted means so for auto catalytic reactions you want product to mix with the reactants Yes it is for auto catalytic reactions that is a beautiful combination but in spite of that we have to definitely find out whether when do you use CSTR I mean this combination Ok in spite of that still we can discuss this like for example I have this reaction Ok this combination going for N equal to 1 Ok So I have already going on I think for some reason I have used these two Ok and then the reaction is going on and now I have for first order reaction So suddenly I got the idea can I interchange Interchange means instead of pumping this side can I pump this side That is my interchange that is all I am not changing again defabricating and fabricating and all that Ok so will the conversion change these are all academic questions nice and academic questions Ok so this one please derive that means take N equal to 1 constant density system and you know first it is going through PFR and ideal ideal PFR ideal MFR and in the reverse direction it is first going through here CSTR MFR and then PFR Ok so N equal to 1 constant density system minus R equal to KCA I will simply say that A going to R constant density system R even gas phase also same because no volume change Please find out which setup gives which gives more conversion for N equal to 1 Repeat this for N equal to 2 Ok Yeah Again repeat this for N equal N equal to half Half order reaction Will you do that This is fourth assignment You also know that you know some kind of optimization between two reactors If I have two reactors CSTRs for example Plug flow will be very complicated But CSTRs two tanks I have in series Ok So this may be V1 This may be V2 Which are not same V1 is not equal to V2 Right Or otherwise I will say that Another condition V1 is less than V2 For this one How do I optimize That means for given conversion For given conversion XA How to minimize the volumes Ok How do we minimize For given XA All this you have done Ok I will give you a clue for this Which you have done also So this one let us say Residence time is tau 1 and tau 2 Ok And this is given conversion Conversion is fixed For given conversion What will be the minimum volumes Of A and B Let us see only for first order Because easy to do How do you optimize this What is the mathematical condition For minimizing the total volume Total volume minimizing For given conversion D by dV by dT What is V there V1 plus V2 What is dV by dT What is that T Various time of reaction Study state Always d by dT dXA by dT dXA by dV dV by dXA dV by dXA dV by dXA1 or dCA Yeah Why intermediate conversion Because this is fixed So if I am able to optimize this How much is coming out of this and then entering This is already optimal one So that is why over also You will get the optimum That is why total time If it is T1 So if I call this one as T1 And also this is T2 What you have to do is dT1 plus T2 dCA1 That should be minimum I mean minimum or maximum Then you can calculate from second derivative Okay the first derivative should be 0 first So if I have a rate of reaction I can have only rate versus concentration data Because you have gone to kinetics And then you found out concentration versus time And somehow you plotted dCA by dT And then got rate versus CA versus minus rA data You do not want to fit This rate of reaction Equation You do not want to find out any rate Rate equation Now for design Because these are all not new There is nothing new concepts Already what you know I am just trying to ask What is which concentration I think you know As concentration is increasing or decreasing You have correspondingly It is simply n greater than 0 type How do I find out volume of the reaction Plot what CA2 minus CA1 Divided by rA give you tau You are thinking that you are only trying to You know using mixed fluid Right In general I am asking 1 by rA versus dCA Very simple I say Again you know how many times you have done that The simplest thing is 1 by minus rA versus CA Or XA XA is easy for us because our equations are XA and this is again Special case where Consent density system So if I write XA Like this So now for a new reactor I have to fix the conversion So I fixed this conversion This is XAF And now Abhijit Whatever you say So area under the curve alone will give you P f for volume And That one will give you Even two reactors also you can try Right Similarly for Tanks in series also For tanks in series There is a method of plotting this And then finding out what is the Either conversion or Volume Do you remember that method? This you know All design expressions What do you have? V by fA not equal to integral dXA by minus rA So that is why area under the curve dXA by This is the area That is equal to nothing but V by If it is plug flow So then if I take Because these equations already you know But only thing is that connection should come Not able to follow still So now if it is Mixed flow reactor All this because This one is only PFR This is PFR And this one is MFR Okay So this will be representing V by fA not Because that equation itself is Rectangle How do you know it is rectangular? V by fA not Equal to XA by Minus rA So 1 by minus rA I am taking as Y So this is nothing but Y into XA So this is Y And this is XA So this versus this will give me the Area that also gives Tell me Regarding this XA XA I think C A and minus rA data Only given C A and Only C A and minus rA data given Because assignment 1 for example Why you cannot You need C A not See for example You have to go to differential method And then you have to collect the order Where is C A not here No where is C A not here For conversion XA You need for XA XA is equal to C A not 1 minus XA XA you are fixing I said Because When you are finding out area under the car What is the meaning you are finding out volume of the reactor So to find out volume of the reactor You should know Conversion Conversion means C A Either one, either this or that Because for example in assignment 1 Problem number 24 C A and minus rA data is given There order is not mentioned Order is not required But for example if only minus rA What is the problem what you have to find out For example if you are finding this volume What is given in that problem What is the reaction Problem is out of the reaction For example I am correlating with this See in that problem when I gave There must be some objective Find volume, find conversion Nothing is given Minus rA is 0.2 moles per liter Second When C A is 1 mole per liter What is the rate of the reaction When C A is 10 moles per liter I think I am not talking about 1 point You are talking about 1 point I am talking about if you collect data C A versus minus rA How do you use that for reactor design That is what my question So there the answer is that You cannot find out That is all you cannot find out It is only 1 point So answer is known to us Because I think you know Some data of Lounspiel problems You cannot find out, you have to say that For order of reaction Where A plus B going to something What is the order of reaction you ask You cannot say Unless otherwise you mention whether it is Elementary So again you know he wants to provoke you That is why I also want to provoke you I gave the questions and then This 1 point Answer for that is you cannot find out But I am not I mean I thought you are Asking something great about this C A minus rA I was asking unless X A is given How do you find out volume You are talking about 1 point Where I am talking about a process Where how do you find out Volume without even finding out What is the rate of reaction Okay And this is one thing The other thing is that you have The rate versus Again same thing This side you plot rate versus C A It is not 1 by It is simply this Okay So now when you have the same kind of data We can also find out what is the conversion If you have 2 to 3 reactors in series 2 to 3 Tanks in series Or otherwise given conversion What will be the volumes Combinations You know how to do that Again same equations It is only nothing new Expansion of the brain That is how Equal to C A naught minus C A By minus rA Is the equation for M F R Correct? Please check V by F A naught and all that we checked Right So this is the equation what you get The same thing Can also be used to find out You know the volume and also If I have another one this is tau 1 Tau 2 equal to this is C A 1 C A 1 minus C A 2 Is rA 1 rA 2 Okay What we do is We try to arrange this equation 1 by tau 2 Equal to minus rA 1 By C A naught By yeah Yeah this is the one So that means I have to draw I have to start here C A naught Yeah 1 by tau 1 Oh sorry Tau 1 Yeah so this is the one This is C A naught Okay now if I draw a line here This is C A 1 Okay This is negative slope Yeah this is negative slope only That is Okay minus That is okay I mean shape itself Will tell you it is negative If it is positive like that it will be That is fine So this is C A 2 This is C A 3 So this slope is nothing but I have you know this minus r This is minus r that is this Minus rA 1 And This one this is the slope This is the slope Right so again you know What I am trying to say there is That we have not learnt any new equations Using the same kind of equations How are you manipulating So that you can get this kind of thing But it is very easy If I know volume of the reactor first And then to find out conversions But it is very difficult for me To go the other way that means If I know C A 3 I have to now find out 3 reactors So that means by trial and error You have to draw these slopes By trial and error Then only you have to Come here Coinciding with C A 3 Or not So easy problem is Knowing volume That means knowing tau And finding out conversion That is easy Knowing tau Getting conversion is easy In this problem But difficulties Knowing conversion Finding out tau By trial and error you have to go To change So that finally you end up here With C A 3 I think it is full We will discuss later