 Yeah, we will discuss about mixed flow reactor, non-isothermal reactors, mixed flow reactor. So as usual we have to draw the picture here, this is entering leaving, then we have here almost to the top, okay. This is T C in, T C out, this is F A not, V not, X A not, important T not, so this side I have T F A, V X A, C A, here I may have C A not, okay, all that and if it is, if the heat transfer area is not sufficient then we also may have internal cooling coils, okay or both also sometimes you have to use and as usual we have to write here material balance and energy balance, material balance already you know, this is F A not X A equal to, should I write any idea, F A not X A equal to minus R A into V, excellent, very good, minus R A into V, V is the total V, here I have total V, okay, yeah and of course total V is volume by volumetric flow rate that is already we know, this is equation number 1. Then energy balance, for energy balance for the entire, this is only for one species A and energy balance for the entire fluid, so heat inflow plus heat generated equal to heat outflow plus heat removed by external means, yeah, so this will be sigma of F I C P I T not, yeah, plus heat generated is minus delta H R minus R A into V, the other one is again sigma of F I C P I T plus U A T minus T C, yeah, because it is a mixed flow reactor inside also you have T and then C A and X A, okay, C A is enough, those also inside C A, okay, so that is why T minus T C is this one, so this is equation, this is equation 2, this is equation 3, good, so as usual we have to write this one, this minus R A into V I have to convert into F A not X A because we need the relationship between X A and temperature, so that is the reason why we have to substitute there, yeah, so let me write this one as F of C P I T minus T not plus U A T minus T C minus, minus of delta H R, okay, V minus R A into V equal to 0, now substituting, this is 4, that is 4, substituting equation 1 in 4, equation 1 in 4 will give us sigma of C P I then T minus, then T minus T not plus U A T minus T C minus, there are too many minuses, minus delta H R F A not X A equal to 0, so this is equation 5, so this is our working equation along with this, idea is to find out the volume, okay, so now there is no trial and error required and how do you use these two equations, how do you use those two equations, yeah, 3, 3, tell me how do you use, because by this time you should have learnt the game, right, the rules of the game, yeah, this is always the problem, except Harsha everyone will answer, yeah, Harsha is a unique species, only one, yeah, tell me, 2 is X, yeah, 2 is X thereby, 2 is X, 2 is X, okay, he has chosen 0.8, so afterwards what do you do, yeah, 2 is X, why you are not thinking, get, yeah, where is delta T here, yes sir, okay, yeah, I have chosen X A equal to 0.8, then what will happen, yeah, can you directly calculate T from there, yes, there is no trial and error involved in this, why Anupriya, 2 variables X and T and 2 equations, okay, the beauty with mixing assumption is that the time, of course, if it is a batch it is time and if it is mixture, sorry, plug flow, the length dimension disappeared because of perfect mixing throughout the system, only 2 variables, so directly, you know, we are now trying to calculate volume, so that means X you should know, that is what, I think Harsha guess is right, so X you know, then F A not I know, delta H R, delta H R I know, U A I know, T C I should know, cooling system, then this C P I, I mean F I I can calculate depending on what kind of reaction I have, may be A plus B going to R plus S or A plus B going to R only or A plus B going to R plus S, okay, can be constant density or can be variable density, all kinds of things, okay, yeah, so that is why this is easiest one, find out T from this equation 5 directly for a known X A, all other things should be known, then what is next step, yeah, so that means I know X, I know T, now I will go to this minus R A, minus R A contains X and T, right, minus R A equal to K zero if it is first order, okay, all that, then I know F A not X A I know and volume you can calculate, straight forward, absolutely no problem, okay, good, so now only I can tell you a little bit of more complication here, now this sequence, this term, this term can also be written if there is a change of C P I with temperature, then I have to put an integral sign here, so that means what we would normally write is sigma of F I C P I D T, okay, if there is a, if C P I is a function of temperature and you know now C P I is a, because you know in this one it is very easy, because it is an algebraic equation, it is not a differential equation, so there is no delta Z or delta X and all that, so that is why I thought I will mention here, so how complicated it is now, you can just algebraically only, conceptually you do not have to learn anything, I told you long time back, you are not learning anything new conceptually, it is only mathematical details, mathematical details, mathematical details, if you relax assumptions, okay, so that is why and what is the functionality here for C P I, A, okay, so now of course for each compound depending on you know you can go to thermodynamic tables and then find out that what is A value, B value and then C value, now that has to be substituted here, right, then you will get T cubed, correct no, Abdul, then correct or, what is that, you have to integrate then what do you get, T cubed, how do you solve your cubic equation, because you have to get the relationship between X and T, you cannot directly use X and T cubed, right, so then you will have 3 roots, which root, right, is the correct root, so all that will be more and more complications as far as mathematics is concerned, right, because if I tell this one for plug flow and batch reactor it will be not easy to understand but here it is very easy because this is straight algebraic equation, now you can write this, on the top of it is not only, yeah of course here we have taken T minus T not, then you have T cubed there and yeah here I have T, all this I have to take into account and then find out what is X and corresponding T, then only you can go here and then substitute in minus r A, the T and X A and then you can calculate volume, so that is what is the mathematical complications but they are not threatening, it is only you know the time and patience you should require for solving this kind of problems, okay, so that is why one has to be careful and also on the top of it I have here F A is nothing but sigma F A is, sorry, F I, sigma F I equal to F A plus F B plus R plus F S if I have A plus B going to R plus S, it is not simple stoichiometry like 1111, I may have 2 A plus 3 B going to R plus another 2 S, so correspondingly now you have to find out what is F A not, this is where you make mistakes, if you do not want to make mistakes then I mean normally definitely you do not want to make mistakes, right, because you are worried about marks only, right, so that is why if you do not want to make mistakes you have to do all the assignments, I think only, except only 2 JVMs here in the class only are doing the assignments, okay, you know JVM life, only 2, I do not know I think, must be correct because this is unique you know, for given X A, X A is given otherwise volume cannot be formed, you know that volume I mean X A is given, so how do you unique mean this is only unique equation, that is all, this is one algebraic equation and you cannot make anything any mistake there and there is no quadratic equation where you will get 2 numbers or that is why I am telling, you know if you go to this then you will get cubic equation, so it will be unique only because here I have T and here I have T, right and only those 2 are not, I mean not 2, this only T is not known, all other things are known to me, including X A because X A is given to you 0.8 or somewhere else, there also X A will be given, but there is another one you tell me the reason, I told you many times why are you guessing, you tell me why are you guessing there, not algorithm, come to physics first, algorithms you could go later, what was the problem there, I had 2 equations but 3 unknowns, thereby what are the unknowns, now again same thing, except thereby everyone will answer, right now Gopi, that is why you know many times they ask, you tell me what is the fastest travelling thing in the world, not even light, light is supposed to be the fastest, okay, you are faster than light, okay, now you can go to any galaxy, galaxy number G 1250, I do not know what it is, but that is why they give that galaxy numbers, that galaxy number may be, you know may be 100 light years away, you know what is a light year, yeah what is a light year, Harsha, distance, I think that is light travelled per second, how much and then you can calculate all this, okay, but that means even light to travel it may take 10 years, but you will be there in 10 seconds, not even 7 seconds, yeah I see merit is much much faster, one second, so this is the problem, this is the problem with you because I think mentally sitting, physically sitting here, mentally you can be anywhere, okay, even now you may be thinking that something else, even though I am talking all this, exactly, so that is why even though I repeated many times that how this has become only two, two variable problem and the earlier one three variable problem, yeah I see, see you see KK always right, yeah, okay, you must be thinking something, what is happening in Calcutta, okay, whether I think everything is going on, trams are moving properly, I think as if that is most interesting things for you, for us, okay, yeah anyway, even if you do not think things will go there, xa and t values are both known from this equation, see what is the basic thing in a non-acothermal thing, you should know the relationship between x and t and then that x and t you should use in the design expression, design expression is number one, equation number one, so but you know you have to get that x and t relationship, okay, otherwise you cannot do that, in a mixed flow reactor that is straight forward because in this equation there is only x and t nothing else, but there is another thing in this equation, in this equation means it is energy balance equation, in this equation there is another thing in the other two reactors, in one batch reactor it is time that is involved which I do not know and in plug flow reactor it is distance which again I do not know, these two we do not know, so that is the reason why you have to guess and then check that whether your guess is right and then that is how you try to solve that problem whereas here that is not required, right and I also mentioned sometime back in mixed flow by the assumption of perfect mixing the distance is dissolved, okay, distance does not play a role why because anywhere in this reactor I have the same concentration and same temperature whereas in plug flow along the length I have temperature changing, concentration changing and length is there, right, so that is why you have one more variable, that is why you have one more variable z, that z I do not know, there z I do not know, x I do not know, t I do not know, even though x is given the remaining two definitely you do not know and we have two equations and that x corresponding to that x what is z and what is t we do not know, unless we know all that you cannot finally find out what is the total volume or length if it is cross sectional area given, okay, good, so that is the one and another nice thing you know, here you have to remember that it can go to very, very complicated and here I can give even this kind of problems in the examination, okay, to pay new because here somehow you know the idea of taking mathematic courses are only to use here, it is not just to get the degree or just to get the courses just for completion and then some grade, right, this M1, M2, M3, M4, M5, M6, how many M's you had, till 6, 6 mathematical courses, you must be PhD in mathematics by this time, okay, but we do not know how to write the limits, okay, earlier problem we have seen, right, so that is why all that information is given only for using here, that we have forgotten, Savita, okay, how many M's you had? 3 M's, okay, 3 M's, that is I think she belongs to bio-technology, bio-technology basic like all the news she has to have 3, 3 mathematics, yeah, in fact this is another bad thing in our society always we think that you know when you are in the school if you are not good in mathematics, okay, let them go to biology or medicine, okay, and if you are good in mathematics then, okay, let them become engineer as if there is nothing else there in the world, in fact in biology you have the maximum mathematics, you do not know, I think you know there are beautiful mathematics how to calculate the number of spots that will come for a Cheta, you know Cheta, you know Cheta, which runs very fast, okay, Cheta, okay, how do you pronounce it? Cheta, why not Ta? How do you know it is only Ta, it is not Ta, okay, yeah, I think Britishers also call Cheta not Cheta because only in Indian language we have Ta, okay, that is why Tata like that, you know, so that is the reason you know what you have there, okay, so you know the tremendous amount of mathematics and you heard of a person called Rozier Penderos, Gopi, Rozier Penderos uses only his hands and legs, you know, two many grades, but this guy uses only his brain nothing else, yeah, but what is he? Mathematical, yeah, he is also one, you know, he is a friend of Stephen Hawking, you know, both of them together proposed theory about, you know, the universe, how it started and all that, okay, both of them are due for noble price, Rozier Penderos also, okay, these two are really great but you know no one is able to prove that experimentally that their theories are correct, that is why they are not able to get, like Higgs boson also, Higgs is waiting for noble price, okay, Higgs, he told some particles should be there, now it is others, you know, experimentally to find out and say that the particle was there, so for experimentalist you would not get the noble price but he gets the noble price, oh, Higgs gets the noble price, okay, so that is the beauty of using your brain, so that is why which one is given more value, see experiment means again you have to do with your hands, legs, all kinds of, you know, things, right, but idea, first idea, of course, if the experiment is fantastically designed, they will also get, okay, both of them may get together whoever designed that experiment, right, so that is why this, all these people are waiting, Rozier Penderos also, he has a beautiful quotation that he says that, I do not remember exact quotation, he says that nature is embedded with mathematics, if you remove physics and look at that, you will see only mathematics, I think the words are very nicely, if physics is evaporated, then you see only mathematics, that is what I have been also telling you, all the time, that means first learn physics and then it is only mathematics that is left, even telling this also that is what, this is the simplest mathematics we are talking about, okay, this is not real mathematics, there are many many complicated mathematics, that is why in nature it is tremendous amount of mathematics that are involved, in C R E also beyond certain point that physics, if you cross physics and then try to solve, physics is what, like you know defining what is mixing, defining what is plug flow and defining simple concepts like what is recycle, all that if you learn thoroughly and then it is only left with you only mathematics, okay, so that is why that quotation is really fantastic quotation, when I read first time I was so happy to see that quotation because nature is completely embedded with mathematics, that is why you know wrongly we tell that if you do not know mathematics go to biology, biology and geology is nothing but nature, correct no, in fact unnature is only engineering, it is not nature because whatever nature has not done, you are creating those things, nature has not done a flight, flight design, so that is what we are designed, they have not designed train, okay, nature, so that is why we are doing train, car, cycle, all that and we need mathematics, I mean the engineer for this and we need God for that, okay, so that is why appreciate geology and biology, biotechnology and all that, nowadays I do not know all noble rates, most of them are getting, whether it is physics or maths or chemistry, all of them are getting noble prices for which subject, only for biotechnology, it is not biotechnology, I mean something connected with biology and all that, now it has physics and chemistry, there is not much difference, everyone is working only on biological problem, that is why there was an article, now do you want to have this six noble, six or seven were there, no, I think six, how many Gopi, no, not, definitely not five, six I think, piece price also you have to add, piece, economics, piece and all that if you add I think six are there, so out of these six prices now, do you have to go to same prices or do you have to change the names of the prices, that is what recently, you know, this is the season, October is the season for noble rates, right, most of them own three poles also, because anytime they may be woken up and then call, okay you got noble price, because the timings are different no, from Sweden and other places, so that is why, right, yeah, so what is that I was telling, about biology, yeah, so that is why probably only for biology they may have noble rates and physics and chemistry, traditional chemistry, traditional physics gone, okay, so that is why mathematics, mathematics, mathematics are very important, first time do not laugh when I say vectors and tensors, okay, good, so this is the one and now this is very simple problem, you know, mixed flow is a very simple problem, always he is a nice guy, nice guy when you call nice, when he is not giving any problems, okay, this will never give us any problems, okay, now, this fellow also gives you a problem now, right, that problem is if you look at equation five, there are two terms, okay, just before going to that, yeah, adiabatic case, that is usual, so equation five for adiabatic case how it will be, yeah, so this ua will be zero for adiabatic case, then we will have sigma of for adiabatic case, adiabatic case, sigma of f5 cpi t minus t0 equal to delta hr fA0 xA by, yeah, this is something wrong, yeah, Oh, sorry sir, yeah, so that is the one, in our format when we want to write this, t minus t0 equal to minus delta hr fA0 divided by sigma of f5 cpi into xA, which is nothing but beta, oh, yeah, thank you, yeah, which is nothing but beta xA, right, so this is much easier to solve because, yeah, now, what is the procedure for adiabatic system, adiabatic case also I have to find out what is the volume of the reactor, yeah, oh, is it unique or talent error, unique because simply t minus t0 equal to beta xA, of course beta we are taking independent of temperature, so straight away I know what is xA because I am finding out volume, the other way also one can do, you can also find out, I mean volume should have been given, then you can also find out what is xA, okay, that procedure I will not discuss, I think I will just leave it for your imagination, it is just reverse problem, okay, so now this is also xI know, t directly I can calculate, then I can go to this equation, this equation contains minus rA as a function of temperature and conversion, okay, xA, then I can calculate what is volume, right, good, so that is all, that is very simple and again this becomes, there is a special case, if I have A going to r and also CPA equal to CPR, yeah, so what is the equation for adiabatic line, that is called adiabatic line, okay, minus delta HR by CPA, or CPA, CPA, whether it is CPA, yeah CPA, into xA is there, definitely, yeah, okay, CPA or CPR, both are same, so that is the equation, so this is what is beta for us because if you want to look at old literature, new literature also they have, but most of those people use only minus delta HR by CPA as, or CP as beta value because it is easy to discuss and mathematically not complicated, okay, so this is equation number 6, 6, this is 7, this is 8, okay, good, so the next one, next one is, yeah, now we have to look at this equation and then see, there are two terms here, okay, you look here, if I ask you to find out what is the amount of heat that is removed and what is the amount of heat that is generated, we are not writing the balance separately, we are not writing what is entering, what is leaving, what is reacting or generation, all that, so now those two terms, definitely there is a coordination between amount of heat generated, amount of heat removed, so now what is the equation for the amount of heat generated, what is the amount of, I mean what is the equation for amount of heat removed? It generated, I think you do not have any problem, minus delta HR, yeah, RA into V which can also be written as FNR into, okay, yeah, that is actually we will write minus RA into V, minus this one, this one, now then what heat removed? Rho A, this one is heat removed, this one is UA T minus, yeah, so what is this? Yeah, what is that, is it heat removed or heat generated? Yeah, heat removed by what mechanism, correct only, but what is the mechanism, there is a flow, by convection, because flow is there, some amount of heat is entering with the feed and some amount is going out and that is also taking some heat that total is this, right, yeah, the total amount of heat that is carried by the stream and this is one by external means heat exchanger, so that is the total amount of heat removed and this will be the total amount of heat generated, the solution when you are asking, it is actually the cross between these two is the one which we are talking about, that means for a system to be stable, we should have that heat generation is also equal to heat removed, otherwise you know it is not stable, it is not steady state, right, there is something different, okay, so that is why what we write here is this equation 5 can be seen as heat removed must be equal to heat generated, correct no, this is 0, I will simply take this entire thing out, okay, this entire thing out, so then that will be heat generated and, oh, not equation 5, equation 4, sorry, equation 5 also same but here you will have more feeling, minus r A into V and delta H R, that is the heat generated, okay. So now, yeah, these two equations we have to write as sigma F i C P i T minus T naught plus U A T minus T C equal to minus delta H R minus r A into V, yeah, okay, so now this I can write as Q R heat removed, this is Q R, this is Q G, you see and Q G is minus delta H R minus r A into V, okay, so equation numbers if I write, this is 9, this is 10, okay, yeah, okay, this may be 11, this is 12, right. So now equation 12 also, this one, is it a function of temperature? Minus r A is a function of temperature, so now if I want to plot for example how heat is generated with temperature, I have to convert that into an equation in terms of temperature, so now let us do that, how we can do is, let us assume that I have minus r A equal to K into C A first order reaction, that is going on, we are talking about M F R, right, okay, so this equation I will have and substituting this in Q G, you will have minus delta H R V minus r A equal to K into C A, this is okay, I have simple thing like this here, this what is giving here, okay or the reverse, this is what is giving here, this minus r A, okay, good. So now this equation C A, can I write in terms of again tau and all that? What is C A by C A not for first order reaction if I have mixed flow? Because that C A I want to eliminate because everything I want to write in terms of only temperature, so that C A I have to eliminate, C A by C A not equal to 1 by 1 plus K tau, so if I write this one here, this is nothing but minus delta H R V K C A not by 1 plus K tau, now is it a function of temperature? It is only a function of temperature, so this equation is 13, so this can be written as Q G equal to minus delta H R V C A not, this is okay, so this if I, C A not everything is there, okay, so K if I divide, this will be 1 by K plus tau, correct here, okay, so now this can be written as minus delta H R V C A not by, this one is K not into E power minus, so that E power minus I will take to tau, that is E by R T by K not, correct? Plus tau, okay, so now this equation which is equation number 14, right, this equation, equation number 14 is 1 in terms of temperature. Now equation 11, I know that Q R, equation 11 is heat removed, this is heat generated, this is also in some type of temperature, so I will plot Q R and Q G, yes, on Y axis and versus temperature, so first if I plot Q G, how does that look like? If I plot only Q G for that particular reaction, it depends also on reaction, what kind of reaction you have, for that kind of irreversible most order reaction, come on, quickly, anyone has been exposed to this? So I have to now plot first Q G versus T, which curve? Decreasing curve, increasing curve and Rahul says S shaped curve, delta E is fixed, it will not depend exactly on delta E, sine of delta E is exothermic reaction, right, I think delta H R is exothermic reaction, okay, here I will write minus R A exo, then please write exothermic reaction, that is correct, I think as Riva Kramari decide it depends on delta H R sine, okay, so this is exothermic reaction that is positive, yeah, so what shape, I think Rahul, why did you say that it is S shaped? Because he told, if you could have told 0, you could have also put 0, what is your interpretation? Yeah, it is S shaped curve, the reason is that you have this exponential term, you have this exponential term, so and also initially you know that temperature is less, so the heat generated will be less and then we have the right combination of concentration, may be concentration is high initially, temperature is less but total heat generated is less, so then with some conversion heat generated will be more, temperature increases, concentration decreases a little bit but effect of temperature increase is more, you know the rate will be much more now, so then it starts raising and it steeply goes with the correct combination of X and concentration and temperature, steeply increases and afterwards it also flattens because you know why initially slow, right, most of the natural curves are like that, even including microorganisms how they growth also it is similar to that, most there are many many natural processes where you have initially starting because microorganisms when you want to grow, you have to give it food and then you have to put some microorganism there, so they have to first get accumulated, accumulated to the conditions, then it is slow, then after sometime it picks up, so they will be comfortable and then onwards they starts consuming more and more food and that will produce more and more, so it stages and after sometime food is over then almost stationary, but that means sufficient amount of food is there for maintaining that stationary. Similarly here the reaction it is also like that like this, so this is QG curve, okay, this is one where steeply it is going, good, so now if I plot 11, is it a straight line or is it a curve, why it is a straight line? Q R can also be written as I can take this T first out, so that means I have sigma F I C P I plus U A as one term into T, correct no, minus this one, sigma C P I, F I C P I that is one term for minus and also I have another minus here, so this is U A, oh no here T not also is there, plus I have U A, okay, U A T C I will put bracket here, correct no, okay, so this is equation number 15, so in unless we imagine in our normal Y equal to M X plus C we cannot draw the lines, correct no, I do not know who started, I would like to find out that history, who first told that straight line equation is Y equal to M X plus C, this is also, this is Y, okay, all this is M, this is X and this is C, so when I plot that depending on T not where it goes, because that is the T not which is starting here, then you may have a condition something like this, Q R, R you may have like this, this is also Q R or I may have only somewhere here, still it is cutting a little bit there, so this is also Q R, that depends on where you have your T not, so any possibility is there, of course there are many other possibilities also I will just tell you, I think we will stop here, Monday morning first hour we will meet here.