 So we will start now, I think isothermal PFR, non-isothermal PFR, what we have discussed yesterday, there are one or two things still left out, non-isothermal reactor, plug flow reactor, we had the last equation I think, what was that equation? Yeah, material balance and energy balance of course what we had both, let me write those things, m b f a not, d x a equal to, I am writing minus r a, a d z, I think this is two no, yeah. So the other equation, energy balance was, yeah, sigma f of i c p i t minus t not plus you have pi u d t minus t is c into delta z minus f a not d x a minus delta h r equal to zero, correct? I think yesterday I would have written this minus delta h r first I think, this is delta, okay, this is the one, sir now we have also seen what is the procedure to solve that, but unfortunately in all non-isothermal reactors this is the only two equations are available, you have to only do in this way, there is no way how you can design the reactor, without this trial and error you cannot do things, okay, I mean how do you do trial and error is there manually or computer that is left to you, right? But because there are only two equations and three unknowns you have to do only by trial and error, check whether what you guessed is right or wrong, right? And if it is not correct then you have to again guess and then like that, but if you do manually with one or two guesses it is not that terrible, right? With one or two guesses then guess means what? I am talking about you assume delta x equal to ten, for example, this also delta x I have to write, this also delta z I will write, yeah, you are first assuming that you know you have delta x equal to point one, ten percent and what will be the temperature raise or what is the distance, you know, what is the distance that is required for that ten percent conversion, so that you can guess in the beginning, you know just blindly after one or two iterations you will definitely know that what will be that length approximately, okay, like maybe five centimeters, depending on kinetics and everything, right? So like that it is not that desperate, that is why unless you do one or two problems I think you cannot really enjoy that, right? I think those people who have done, I think Ramakrishna you have done, yeah, even the batch reactor which I gave you the other day with UA equal to some value that you can try, that only you tried, yeah. So once you try only then you will know what are the difficulties otherwise you know you will not know definitely, right? Okay, good. So now let us see what will happen to this for adiabatic case, adiabatic case, adiabatic case this is zero, correct no? So then you will have sigma C P I delta T or T minus T not equal to minus delta H R, I am writing again minus delta H R first, F A not D X A or delta X A, okay. And yeah, delta X A I know that it is not X A not equal to zero, you know that X A not equal to zero, so that is why we can write this one as sigma F I C P I T minus T not equal to minus delta H R F A not X A, okay, delta X also you can leave it there but I just want to show you that the equation which I have told you in the beginning for all adiabatic reactors whether it is batch or plug flow or mixed flow you will get the same equation, so that is the reason why I again I want to, so this is the one and numbers if I put what was this number? Nine, nine This was nine, okay, so this is ten, this is eleven, so now I will write here T minus T not equal to minus delta H R F A not by sigma F I C P I into X A, yeah, so this is equation number twelve and we have been taking this as T minus T not equal to beta X A, okay, T minus T not beta X A and where beta equal to this slope minus delta H R and F A not sigma F of I C P I, so now we know that with this slope you have to draw the line if it is adiabatic reactor, we are assuming that that beta is constant that means delta H R not is delta H R is not a function of temperature and C P I also is not a function of temperature, right, so this is the reason why we are taking that as constant and then we can start drawing the straight line but there is some problem here I think actually you have to calculate the F I F I C P I, how do you get that information? In fact sigma of F I is nothing but it is the mass, okay the total sigma I told sigma F I is nothing but the total, okay, total of your molar but anyway if you multiply by correspondingly with that you know molecular weight it is mass flow rate, mass flow rate will not change at any point across the because mass is not changing it is only converted from one to the other that is all, okay, good, so this is why and I can give you one example you can just find out and tell me if I have A going to R can you tell me what is sigma F I sigma, what is this one, yeah, first of all when I write this this is F A plus F R, no still some people may not be knowing this I would write both yeah F A C P A plus F B F R F R C P R, okay, now you tell me what is that value what is F A, F R not is zero, now do you tell me what is the thing how do I because finally you have to write that in terms of conversions because now I have two F A and F R, okay I may also ask you to do this also A plus B going to R, so here A plus B this one is this is equal to F A C P A plus F B C P B plus F R C P R I am sure you will definitely make mistakes in this that is why I am now again telling you once more many people still may not know how do you get that information what is the relationship between because you have to convert everything in terms of F A what is the relationship I mean you take the simplest one first yeah so this F A equal to F A not into 1 minus X A right so F A not into 1 minus X A C P A, okay and then this one yeah if you assume F R not equal to zero this is simply nothing but F A not into X A C P R, okay so like that even you have to write there, okay for the other one also because now again you see why I am telling this one is a different case is that now you have F A not you should also hear F B not that is why when you assume that F R not equal to F B not that means equal equimolar flow rate when you take then only you will have one otherwise that will be in terms of again F B not F B not you know no problem so these are these are the things where you make mistakes in the examination, okay because your interest also is examination yeah so that is the reason, okay so now finally if I have in this equation sigma F I C P I so let me say this is equation 13 this is 14 this is 15, okay if C P A equal to C P R possible no C P A equal to C P R is possible many times we will take equal yeah now what is equation sigma F I C P I equal to F A not C P A F F A not into C P A yeah now substitute this in the in this equation oh I think sorry yeah I think this this this equation number okay that is leave it yeah right so what is beta now C P A minus delta H R by C P A X A minus delta H R by C P A X A oh sorry sorry not beta I am writing the final one sorry yeah beta is okay so T minus T not equal to beta is minus delta H R by C P A into X A so this is the one so now slope is simply minus delta H R by C P A that is why some books you know when they are discussing academic problems most of the time they take minus delta H R by C P A if you take Levenspiel you will get minus delta H R equal to I mean minus delta H R by C P A as this slope because he has taken that all the assumptions you know C P A equal to C P R and all that and also A going to R the moment you do not have that like for example here you have F B not and all that you will not get this kind of equation and in fact you get X A by in the denominator also X A which is not a straight line, okay this I am not I am just leaving to you just to try for you and here also we have the one more one more one more yeah you know you assume everything in the beginning and complicate with mathematics and you do not understand what is the use that is the reason even crude assumptions are made part making the subject clear, okay and the next point is that you know how to use you know all the complications we know now our assumption also is wrong we know because most of the time you may not get this kind of cases but yeah but I think you know if you start with very very complicated things like you know tensors and vexors vectors in the beginning no really I mean because you lose your interest in the moment you say del, okay del del that is all I think God, okay so that is why we give you know these explanations first once you understand everything then you put del so then you have the feeling for del right what are the terms you have right so all the how do you handle them all that will come later so that is the reason why in all these you know academic courses we first assume simple things and then make complicated that is why you know the definition of teaching is that taking students from known to unknown understood no known to unknown that is why we give very simple examples what you see every day or if I give you some complicated example no one has seen then what is the use of that example is also more complicated than you know the actual one so that is why that is the reason why you know one definition of teaching is some of you may become teachers and I tell you teaching job is excellent job absolutely no responsibilities coming to class and going simply come come to the class take class and go home, okay no one is there to particularly in IE it is no one asks what you are doing, okay what you are doing only depends on when you go for promotion at that time, okay at that time they may ask what are what you have done right so that is why I think in universities also I know there are many professors where they take class and then go home and then do side side business, okay so many people have in many universities I know that, okay that is why this is a wonderful job we do not to enjoy life but only thing is you should know the subject and you should teach well and anyway automatically students will be happy there is nothing like you know making students happy I said if you have very dull students and angry students or indifferent students in the class there is no use at all, okay instead of seeing get you I may talk to board that is much better, okay so that is why many faculty members do that they only talk to board most of the time, right, okay good so this is what I think this is the reason why we get minus delta hr by cpa even if you see Aries book which is a very complicated book, yes Rahul Aries book which is very complicated he uses still this one he uses lot of very complicated mathematics but still slope many times he uses this for explanation, yes it will not be linear, right but I think that is another assumption where I think in our range of operation we are always taking that the temperature is not affecting both of them delta hr and cpa and you know these are the lousy temperature functions know cpa is a function of temperature as t square t cube and all that you never get any straight line there, okay but I think you know in most of the time as engineers that assumption will make many our life very very easy and understanding is very easy so that is the reason why we go for that very good so this is one and yes so now the design of adiabatic reactor is so I know I have this equation t okay this equation t equal to t equal to t0 plus beta xa that is the equation I have forgotten to write the number there so may be this number we will write 16, yes so what is the precision now for adiabatic reactor I think I have to finish that adiabatic let us say plug flow reactor you know for mixture flow reactor it is very easy we also get the same equation that we will talk about little bit later but for plug flow what we are discussing now is under plug flow I think I have to underline this, right so plug flow equation what is that I have v by fa not equal to yeah repeat 0 to xa dxa by minus ra which is also equal to if I have first order reaction xa dxa into k0 e power minus e by rt ca0 1 minus xa, right yeah so again this constant things I may take out that is k0 ca0 so then it is 0 xa dxa so here I have minus e by rt into 1 minus xa so and this is not integrable because I think I have t as a variable and also xa as a variable so now I have to substitute for that t, ca0 is I have taken it here, no no it is ca0, okay not is not ca0, okay that is ca0 only this is ca0, okay so yeah now this has to be written in terms of that equation t this this equation and then you have to integrate v by fa not equal to 1 by k0 ca0 0 to xa dxa e power minus e by rt 0 plus beta xa this is 1 term into 1 minus xa that is on that term, okay good so there is one solution that is available for this, okay you can also actually integrate that, right I will give you very simple equation, no then you can appreciate the mathematics much better than what we have been discussing so 16, okay this is 17 so when I integrate what I get is like this v by fa not is not that easily integrable because I have here exponential term and also you have x term here, okay this is e power e power ax into x thing you know number of times it reoccurring, recurring formula something they say, okay so this is k0 ca0 e i, you heard of what is this e i symbol e by rt minus e of i e by rt0 yeah here e by minus alpha again k0 ca0 e i z minus e i z0, okay now we have to define you know that is the equation and where e i is called what exponential e i is exponential e power minus also is exponential what do you mean by exponential? Who said exponential? Somya no no no he is of right yeah what is that exponential there is something else exponential something yeah rahu not heard that till now it is called exponential integral there is a definition of e i, okay yeah I think first we have to draw that exponential integral e of so this is in fact see this derivation you get not the way I told you know this x also you have to convert into actually you can leave this in terms of t I have written in terms of this but if I convert this okay this x in terms of t that means 1 by beta and all that will come there right so that is what so this equation 18, okay you can write the equation 18 is obtained because I cannot write this and then give this one because you may try this some of you who are interested in mathematics, okay equation 18 is obtained when equation 17 is written in terms of temperature that means not only this here yeah here I have temperature e power minus this term and okay we do not write that can you write that or you will understand yeah so x is converted so this one is same term as it is then this 1 minus x is nothing but yeah beta t minus t not by beta of course 1 minus 1 minus you have to write and then substitute there not only that you have to differentiate this for d x a, okay you have to differentiate that d x a and then write convert that into d t so that means everything should be in terms of only, okay so that is why I thought we will better write correct no I know your memory is not that great, okay now tell me puja what is the equation 1 by k not c a not yeah in fact in fact not only that you have to even convert limits limits also you have to convert yeah what is that error function it is exponential integral not error function exponential that I will give you later definition but first let us see from which equation you got this first, okay let us see that one first good yeah so then here I have 0 to yeah so now x a how do I write this one in terms of this yeah this is t minus t not by beta that is the upper limit first of all, okay good so then next one is yeah d t differentiate and tell me no you tell me delta x d x equal to what 1 by d t this goes to 0 that is all what is that all stories you are telling that is what you said okay so only beta will come there that is all right yeah so that beta we can take out constant that is in the numerator or denominator yeah you put in numerator what what is that here is t not there lower limit please tell me what is the lower limit lower limit x equal to 0 simply it is yeah t not okay good so now the other form so now I have d t here e power minus e by r t yeah next tell me for x a t minus t not oh one minus one minus t minus okay in the bracket I will put divided by sir t minus t not by beta I think you know this is coming here right x a so one minus of this so only that much only this much beta that is all that is all yeah so this is what you have to integrate yeah now you can write equation 18 you know this is solution before the actual equation yeah equation otherwise we will change I think you know this one has 18 yeah instead of writing all that that is the simplest one so this is the one now we have to also tell about the functions right so we do not know what is z right and also we do not know what is e i so though we do not know what is alpha those things we have to write here okay yeah first alpha from this equation I am writing okay from equation 19 alpha is defined as lambda e r into 1 minus x a not that can be 0 first lambda t 0 and lambda is lambda is written as m dot c p that means of course here you know it is converted I have taken this one from fromant and Bischoff have you heard of that book fromant and Bischoff yeah what is the title of the book no one has seen limits I am talking about I am talking about book we will come to limits later what Rajasri I mean you are worried about the upper limit so what is that you are talking about this is okay or this is 1 minus t minus t 0 by beta not correct upper limit what is upper limit when in place of x you are just simply putting that x a is converted into t right only yeah I cannot have another x here I say then I will get the solution in terms of x okay I think okay let me write this chemical reactor reactor analysis and design chemical reactor analysis and design and I think this is why li why li yeah tell me now what is the problem mathematical problem see that is what you know why we simplify things is only for that unless oh okay that is the limit I am not saying that you know limits are yeah we use it correct or not I have not checked this correct you check it and afterwards we will see okay good yeah so this is I think okay okay I think that is fine you can check that and then let us see now I have given the book and then you check there whether how you have done it okay yeah I have not looked into the details I just wanted to tell you how complicated this will be if you want to I mean integrate this and that is possible only for first order reaction other orders you cannot do that okay so that is the reason why I just brought it here to let you know what kind of complications we will have m dot c p f a not minus delta h r okay where m dot is the mass flow rate m dot is the mass flow rate then we will have z as u yeah u minus alpha where u equal to e by r t and z 0 is again u 0 minus alpha where u 0 equal to e by r t not okay so 11 this is okay I think this numbers not required then I think that is all all other things we know yeah where exponential integral y is defined as minus infinity y e power t by t into t this is called exponential integral that is called that is the definition of exponential integral right and the values of this that means you have to convert the integrals as minus infinity to some value y okay some value y and now you have to go to the mathematical tables if e of y let us say y equal to 10 or y equal to 1 or y equal to 5 or y equal to 0.1 right those values are listed in mathematical handbook of it is mathematical handbook you have come across any one mathematical handbook mathematical handbook Christy K is not mathematical handbook it is only mathematics book handbook where all the numbers are listed I think someone was telling exponential integral someone was telling another integral error function yeah even error functions are also given in that table okay big tables are there in that one book was abrumovic I do not know whether you heard of or not abrumovic a b r a m o w i t z abrumovic not heard no at least you heard no yeah very I think is not given I think error function they have given but exponential integral I do not know whether they have given or not okay so you see chemical reaction engineering beyond certain point is highly complicated mathematics only for that reason I want to tell you this I have not derived this I do not know I have not converted limits but to make you comfortable I said that you have to have that equation converted into temperatures and then we are just discussing the solution that is why we need both we need this mathematical techniques beyond certain point but before that we need physics only exactly for that reason I wanted to discuss this problem in the class right only for that reason otherwise I think you know I should have gone to some other you know mixed flow reactors already right so now you know your idea is here that how complicated it is the reaction engineering beyond certain point right okay Rahul that is the reason why we always assume things okay some of the assumptions may not be real but still is okay logical assumption only we will take and then try to derive the equations because once we understand the basics then going to mathematics is only mathematical you know algebraic conversions from one to the other and then try to find out the solutions that is the reason why transport phenomena is very complicated for most of us because we always try to have the solution first rather than the formulation of the problem right that means the physics of that problem we do not spend much time simply we take the derivation on that too we will take that Navier-Stokes equation and there also you assume that this term is not valid and this term can be removed and all that then you start solving the equation that that means it becomes simple mathematics exercise and you do not know what we are solving physics wise that is why beyond certain point chemical reaction chemical engineering itself is very complicated okay but you have to appreciate the beauty in chemical engineering even without mathematics also how people beautifully designing the plants okay but not all plants but if you need very very very very few rest form of products then you need this mathematics and then only you can design the plants with one example which I have already told you SMB they call you know simulated moving bed reactor you just go to the website and then just see you know in Google search for simulated moving bed reactors okay how complicated they are they are used for adsorption particularly getting adsorption I know the enzymes from biochemical broth after reaction you have that broth reaction broth so there you will have many enzymes and if there are three four enzymes you have to use three four different adsorbents to separate them the enzymes are so temperature sensitive so if you want to again you know use temperature for separation like distillation distillation you are using energy for separation temperature for separation right so that is the reason why enzymes cannot be used because I think they are very sensitive temperatures so only some physical methods you have to use like you know physical adsorption right you know the enzymes going and sticking to the surface of the adsorbent and later again you wash it that particular adsorbent you take out and wash it and then purify later see that is why chemical industry beyond certain point is highly mathematical okay so most of you may not be knowing that I think macabre Thiele is another extreme without any mathematics we are able to solve distillation problem it is very simple beautiful problem because you do not even that is a mass transfer equipment design what you are making but you never talk about mass transfer coefficient throughout what is information required there entire macabre Thiele method for absorption drying and also for leaching and you know all operation distillation extraction all these operations what do you need there you do not need any mathematics you need material balance where you normally use this material balance every day in your life because what is entering what is leaving what is converted you know that kind of balance and what is the other one you need energy balance also you do not use where did you use energy balance to find out punch holes average that is slightly sophisticated that is why I said one end completely one end okay so in this end you need only equilibrium data and you need this material balance as straight lines you have to really thank you know this original chemical engineers where they made the entire plant look like very simple design but if you go to for example multi-component distillation now you can feel again the problem with mathematics so that is why I thought in this class I will let you know what are the difficulties in mathematics that is why unless you are good in mathematics you also cannot enjoy beyond certain point chemical afterwards you will know that is why sometimes I am not discussing that whether it is right or wrong okay you find out that whether it is right or wrong and you tell me later okay I will also know don't discuss okay you tell me later I will also check with myself okay I will also check myself whether because I have not checked that I simply took this problem as it is from the book and then I wanted to discuss importance of the mathematics in chemical engineering because you may be feeling that whenever I am saying matrix or vectors and tensors you are laughing I don't know why okay but I think it is not a simple laughable matter that is what I want to tell you from the beginning of the class whenever I say matrix and you know the vectors and tensors you are laughing so then I know I don't know the reason I don't want to know also the reason right but I tell you that it is not that kind of simple laughable matter because you need that kind of things also when you are really solving complicated chemical engineering problems that is the reason okay so that equation whether right or wrong we will you know I will also try to do you do and tell me later okay yeah then sure okay yeah you say that upper limit is wrong which is wrong you tell me okay that we will see okay then what is the correct upper limit simply T anyway I think leave it I think you know you also prove that in the next class that why it is T and why it is not you know the other one you can also check that I mean because we are not fighting here and I have not applied my mind so I am sorry for that that is all I should have cleared that which is right which is wrong I have not applied my mind very clearly because I my aim of this class was totally different and in fact I I don't have this equation at all in my notebook so after seeing some faces blank then I thought that we will try to write this equation in terms of T right that is where I went into the problem my problem because I have not thought I mean I have not simply I thought of giving you this final equation and then try to say that exponential integral you see this minus and why and you don't have real analytical expression here for to use this you know this yeah here here if E by RT is let us say 20 I cannot straight away get that value I have to go to the mathematical tables and then get that value and substitute here okay and this one I also got I thought I will also give you apart from giving the complicated problems in chemical engineering I think Sivaram Krishna was asking me sir I think you try to analytically do something in batch reactor yeah yes that is what my next one is how easy to draw the graph that is what you know the closing of this class I thought only that is my you know the closing of the class that is instead of doing all these complications why don't you simply plot this one as yeah this entire thing so not that entire thing X is not there yeah so this is 1 by E power minus E by RT minus into 1 minus XA verses simply XA you will get like this totally area will be V by F A not into all this you know K not yeah K not C not so all this that is what I wanted to tell you you know you always get excited with the graphs so I wanted to tell you that how beautiful this is and how ugly this is correct no I mean for me because beauty always depends on the person's eye right for you it may be looking so beautiful that is why in our on the planet I think you know they may be awkward people but to some eyes that is why you cannot define beauty because if you want to define beauty you have to now go to where eye whose eye you are talking right yeah so for me some person may be looking so beautiful same person will be awkward for someone else correct no possible okay yeah so that is the reason why I say that for me this is much much simpler because I want to be a good engineer and trying to solve the problems as simple as possible and this is very very complicated and where it is not that easy to get the solution right but there may be some people who are always complicated and then they may like this they may really like this and they may awkwardly say that what kind of idiotic diagram that is I don't like it that's what they may say so that is why we have all the people in the society I think you know someone may like if you go to iris you will simply throw this out you may like this one okay so that is why I think as far as I know there is only one book which has given this that is this book I don't know any other book may be of course recently there are so many books have come and at least there are twenty five thirty chemical reaction engineering books I don't know how many of you know how many books are there or if I ask each one how many books you can simply tell as a quiz for example may be I think I have to give you surprise test one okay one test as you know write all the titles of the books okay as many as you know and whoever gets maximum they will get ten okay chemical reaction only chemical reaction engineering okay I think this is very nice okay Rahul next time okay sometime okay so that is the one you know yeah I think we will close here I would also go right I think yeah I am sorry for this equation I did not apply my mind I just simply try to derive on this one here but I think may be I don't know whether this is right or not is it what do you say is simply t t zero plus beta x t zero plus beta x anyway she is again telling then I should allow you final equation is in terms of t so you have to sum up the limit should be t right so t zero plus beta x lower limit t not must be there must be t not there okay anyway I will check I will check and those who have told this you know you also check