 So once we understood this distribution, E t versus T, something like this, we have to now talk about the mean and variance of this. For any distribution you have area under the curve and also the mean, mean is given by the first movement, correct no, moment and also you have the variance, second movement. So those things also will come here. The reason is using this T bar and also variance we can find out the non-ideal parameters. What are the non-ideal parameters? For example axial mixing, that is one parameter. How much axial mixing is there quantification? How do you quantify axial mixing? You have to quantify axial mixing only in terms of dispersion number, okay. That dispersion number we can calculate using this variance and first movement, second movement and all that. And most of the time we will not go for F curve but we will go for the E curve, this kind of distribution, right. So the general definition of this distribution is that, okay, M k, kth movement is 0 to infinity T to the power of k E t D t divided by 0 to infinity E t D t, okay. That is the general one, good, okay. So if I want to take 0th movement, what do I do? k equal to 0, M 0 will be what? 1. So what is the meaning of that? What does 0th movement give? Okay, what is initial? What is 1st movement? 1st movement is mean, okay. 2nd movement is variance, but 0th movement is area under the curve. So M 0 when we put here 1, what is the total area under the curve? We have normalized this, that is why you have C divided by integral 0 to infinity C D t and all that is there, okay. So that area under the curve, this area M 0 is equal to 1 area, okay. That is 1. So M 1 is 0 to infinity T E t 0 to infinity E t D t, right. That is 0, I mean T bar and okay M 1 is nothing but for us, it is T bar, right. It is nothing but T bar for us, yeah. Because this average will be somewhere here depending on the shape of the curve. So this is T bar, right. And this also can be written as what is 0 to infinity E t D t? 1. So that is why this is also equivalent to 0 to infinity T E t D t, right. Okay, good. So now the, yeah, okay. Variance, I think no, the difference between variance and what is the other one? Standard deviation, okay, yeah. What is this one? Sigma square is M 2 minus M 1 square. M 2 is T square, right. Standard deviation, okay, yeah. So this will be for us, M 2 is of course you have to, same thing. So this is, yeah, M 2 please see, please see, M 2 means T k equal to 2, so T square and E t D t divided by this one, that is 1. That is why I am not writing that. But I will simply write here 0 to infinity T square E t D t minus T bar square, minus T bar square, correct, no? Yeah, T bar square or otherwise of course I have to put this one and then put whole square. So these are the, beyond this normally we do not go. Third movement generally we do not go, right. And there are some cases where people have gone to third movement also and most of the time for single parameter models using this and using this we can estimate for example D by U L values, D by U L is quantifying mixing and also tanks in series, right. I have 10 tanks in series. I will conduct R T D experiment and then I will find out whether these 10 tanks are behaving really as 10 tanks or not because ideal tanks 10 must behave as 10 tanks, okay. So and if it is non-ideal that may be equivalent to 8 tanks equivalent. That you can get again, from R T D you have to have that curve and then you find out variance of that curve and you know very beautiful expression there is one, very simple expression in chemical engineering, sigma theta is square equal to 1 by n, heard of it, Swami? Never, yeah. In R T D it comes, those who have got I think some good R T D exposure in their B Tech and M Tech, they know this. And also there is another equation also, sigma theta square equal to 2 D by U L, where sigma theta square equal to you know that is dimensionless variance, okay, dimensionless variance. So those things, those things we have to find out, okay, good. So this is one and yeah and there is another one, Levenspiel simplifies this 2 in terms of even concentrations. So that means if I have simply C versus T will the shape changes, shape is exactly same but only thing is scaling will change, okay. So that is why distribution will not change. That means again my sigma square and also that M 1, you know T bar, that will not change. How do I find out T bar from here? So yeah, for this one T bar will be integral 0 to infinity T C T D T divided by, no this is not 1, so I cannot ignore that. That is not 1. So this also can be written as sigma of T C delta T divided by sigma of C delta T. If you take equal intervals of delta T, then this also will get cancelled. So this will be simply sigma T C by sigma C equal to T bar. This is what is the problem he has also solved as an example in his book, Levenspiel book, right. So these things are a must for us because for every distribution you are going to find out what will be the T bar and sigma square. These two are a must for us, okay. Good. So this is the one and like this you have, I think even using same thing for this distribution, sigma square also you can calculate as, okay. I will write here to differentiate, okay, this T is okay T bar, this T bar is okay. Yeah, this is T square equal to, we have, yeah, I am giving the final expression divided by sigma of what? C delta T minus T bar square. If delta T is again if you are taking same maybe one minute, one minute and you should take as small as possible. If you take bigger one then you will get more error, okay, yeah, depending on the data. So this also can be written as sigma of T i C i C i also by sigma C i minus C i square. So these are the equations what we have and most of the time in the examination also we will be asking this, right, yeah. So from the data we will give concentration versus time data and these distributions we are talking about only for this kind of increasing and decreasing. F curve also has some distributions which are valid both and that we are not doing that, okay, good, that is fine. So now I think this is enough for us as far as this is concerned and there are other things now which also we need before going further to develop the equations that is, okay, direct delta function we have to do one thing and other thing is, okay, let me first do dimension, okay, the relationship between E T and F T. It is obvious relationship E T and F T, okay. So we know that F T equal to 0 to T, there is no infinity here, E T D T. So I told you E T will, you know pulse input will give you E T and step input will give you F T but once you conduct let us say E T that means pulse input, can you convert that into F T? Easy that is straight forward. The reverse also, that means you conduct F T that is step input experiment and then try to find out E T. How do you do that? Differentiate this, this is D F T by D T equal to E T. What is the meaning of this? Meaning is I have F T curve versus T, yeah, like this, yeah, what do I do? You have to find slopes at various points, okay. So when you take the slopes at various points, right, so this shape, what kind of shape? E T I will get here, E T versus T. Initially what is the slope? Almost zero. And at sometime it goes to increasing, increasing and afterwards again decreasing. Now what shape you get? Initially it is zero slope and then slowly it is increasing, reaches maximum and then afterwards comes no, zero, this slope is zero, no, so yeah, you will get like this one again. So you should get no, see the same thing, okay. So this is that means one is possible to see the other one, that is all, okay, good. So the other one which I have to tell you now is, because all these things are required for us, okay, I think later I can tell, because I mean all chemical engineers are very famous for dimensionless distributions, dimensionless parameters, so that also I have to write, because E T we are talking, what are the units of E T? Time limit, time limit. So I will try to convert that into E theta where there are no dimensions, that will come a little bit later. But I think, yeah, before that let us complete with T everything and there is one more thing which we have to discuss here, it is direct delta function, direct delta function which is normally shown as like this, infinity, this is time, this is whatever that function, okay, yeah, please take this, direct delta function is defined, this is required, I think this is why Eleansville book you know many people like it, because he knows that none of us are capable of remembering all this, so that is why he gives in his book also, these definitions again, so that we do not lose, okay, we are not lost, okay. Direct delta function is defined to deal with discontinuous calculus, no, no, direct delta function is defined to deal with discontinuous functions in calculus, discontinuous functions in calculus, okay, good, thus, please write thus, delta T minus, this is our representing, T minus T naught is a distribution curve which is zero everywhere, which is zero everywhere except T equal to T naught, except T equal to T naught where it is infinity, okay, where it is infinite, right, okay, next line I think, you know these things are required because in an ideal P f r what is the output you are getting, ideal P f r, E t, E t is defined only for ideal P f r in terms of direct delta function, that is why and using that also we can calculate, you know the, from R T D you can calculate volume and also conversion, given volume conversion, given conversion volume from R T D side, okay, so that is why this, some idea about this direct delta functions are required, good, so next line you just write, the area under the curve is unity and the width and width of the pulse is zero, in symbols, in symbols you can write this, delta T minus T naught equal to infinity at T equal to T naught, T equal to T naught and delta T minus T naught equal to zero elsewhere, okay, such that, such that minus infinity to plus infinity delta T minus T naught D T equal to one, you are able to see my handwriting are becoming worse and worse, such that, that is the area under the curve, minus infinity to plus infinity but that function is valid only at T equal to T naught but only thing is it should be somewhere between minus infinity to plus infinity that T naught must be there, that is all that is the requirement, okay, so that is one, you know if you really see mathematics and history of mathematics these functions are defined to solve some physical problems otherwise it is not just for fun they have defined, so they started solving you know particularly how a universe started and the mathematics they started deriving under some conditions they have to define some functions, all these functions are like that, so that is why most of the people talking about astronomy and also you know this universe started and all that they must be an excellent mathematicians otherwise they cannot and those are the people who define many many new functions and of course later in engineering these functions also are used to represent you know direct delta function, impulse function, all that, so that is why if you read the history that is why you know we do not have good syllabus itself, so the good syllabus should also contain stories in chemical engineering, you are really stories in mathematics, stories in physics, stories in physics and mathematics are thousands, okay, so how this idea started and all that, how chemical engineering started I have a story, I will tell sometime later you know in the department or outside I think I have to give that story also otherwise people are forgetting about that stories, how chemical engineering, what is the starting point, the real starting point for chemical engineering is the industries, sulphuric acid industry, calcium carbonate industry, you know those are the washing soda, washing soda is calcium carbonate, you know sodium carbonate, yes, yes, yes, those industries which were a must at that time, I am talking about 1600, 1700, 1800 and you know I was worrying you know recently I am not worrying actually I felt also very bad about the way we are now teaching, the way we are behaving as faculty also that we never tell you anything about industry, correct no, just happily coming and then writing some very nice beautiful equations, right and then you nod your head as if you understood and I will take it that as if you understood and then exam is conducted over, where is chemical industry, this is what is the problem with people who are doing simulations, without knowing rotameter they simulate rotameter, okay I mean just simplest example I am telling, so that is why you simulate, simulate is required once you have thorough physical concepts, then simulation, then understanding automatically comes, not the other way, first simulate and then you know without knowing why you are simulating and then trying to find out something out of it and you know when I say that all of you will be thinking that this man does not know simulation that is why he is telling, okay I mean this is logical conclusion, okay all of you will be telling or otherwise they may be thinking this guy is against that simulator, so that is why I think he is you know all these idiotic things you know mind always goes in wrong direction only, if I tell you do not go in this direction your curiosity will increase why did he tell you know this go, he asked me not to go in that direction, so you finally go only in that direction, okay and if I ask you okay go in this direction you do not care, do not go means you know it is the thing, so that is why all these functions have really beautiful stories and those stories are unfortunately we are missing it, I do not know the schools also they are not able to tell and this planet lacks wonderful teachers know that is what wonderful teachers who knows the stories more, history is very important I say you may not care because you also do not know history you know only your father or maximum his father beyond that who is his father where they stayed I think 90 percent of the story when this generation we do not know that correct know because now the single families and mother-in-law will not accept father-in-law father-in-law will not accept daughter-in-law daughter-in-law or in-laws only okay one in-law I mean actually you know I like the best word in English is this brother-in-law sister-in-law daughter-in-law okay when I have daughter-in-law you know what is the meaning of that in-law she is my daughter when you have brother-in-law in-law he is your brother in-law okay that means as per law is concerned he or she you know correspondingly daughter or sister or brother it is a wonderful expression know in other languages we do not have that kind of beauty only that English language only this word only I really like it okay but in Telugu we will call if you have brother what I think Anna if he is elder brother okay I think Tamil also Anna that is all but I think you know whoever it is brother brother is universal and if you go to some other family you are still brother but in-law okay if your daughter goes to some other family she is still daughter there but in-law wonderful like that there are many many things if you are thinking that this fellow doesn't have any other work really I don't have any other work I don't have really any other work thinking like this only and I don't know how to pay tax and I don't know how to go in fact I cannot even manage TT if I don't have a seat in the train I don't know how to ask him you know can you take money and then give and I don't know I am afraid to ask so that is why travel I think either standing or sitting whatever is available that's all I don't know how to manage and now young people will go with that pant and then something hanging here something hanging here correct no so all wide and then they just simply go that hundred rupees take it give me seat we are not able to do that I think you know I am very useless fellow that's why to tell you the secret always my wife scores me because I don't know what to do and I don't know even to go and manage with this gas fellow now they started giving only limited gas and all that so you know like that so that's why I am I started thinking I am enjoying it I don't know okay okay good so that's why these these functions also have it's not simply mathematical functions without any physics to solve some problems only all these functions have come into picture okay many many functions good so then we will have some more properties okay please write this the useful property of this function is the useful property of this function is a to b delta t minus t not f of t d t yeah equal to you said that how do you know you already to wear with subject here you did it already okay f t not okay but only thing is what is the requirement the next requirement is yeah exactly e not must lie between those two integrals yeah limits that is very important otherwise if it somewhere else we cannot do anything so otherwise it is 0 if t not lies outside a to b not a minus b okay so that is why so now I will just give you one one one I think all of you may be easily getting this so to calculate no no okay 2 to 4 delta t minus 3 t square d t what is the value Olga is very seriously thinking Olga I am waiting for you why 9 what is the function corresponding to this this f t it is t square right so it is inside the interval t minus 3 between 2 and 4 right so this is t not this is 3 this is f t this is t square then what is must be good so one more example 0 to 2 delta t minus 3 p square 0 because I think that is okay yeah now one more one more is this minus infinity plus infinity delta t e power minus k t dt by one that is t minus 0 okay so this will be e power minus 0 so that is one so this is e power minus 0 equal to 1 okay so like that you have all the functions that is good I think this is one and yeah all other things are fine so these things are also required for us after sometime you know when you are actually solving the problems and now let us see the dimensionless e t f t e theta f theta etc so next one is dimensionless e t f t or e theta and f theta so we have to make e t and f t dimensionless so we call them e theta and f theta so first we define everything we will write dimensionless we will write here e theta f theta so that means we would like to express that in terms of time also dimensionless okay so that is why okay the first definition is theta equal to t by logically t bar okay yeah so that is the one so now can you make e t into e theta there are many ways of making anyway I think instead of asking you I will tell you so I have here e t versus t right so the shape may be general shape is like this good so what is the area Abdul one so now first thing is I have to change the t into theta so that is why I divide this scale by t by t bar okay so now I will leave e t as it is now what will be the area under the car the same thing if I divide by t bar what will be area under the car one by t bar but my requirement is area equal to one so what should I do multiply by t bar so that is why you have here t bar e t which is nothing but e theta so then again you get almost same shape area equal to area equal to one okay so that is e theta e theta now doesn't have units yeah you can also do this you know mathematically that is 0 to infinity e t dt how do you convert that into e theta again t bar equal to dt because t definition I know dt equal to t bar so this is 0 to infinity yeah this is e t t bar t bar theta right dt if I said no t bar d theta so this is now e theta so this is equal to 0 to infinity e theta okay reshmi no problems okay so now f t now f t someone was telling that when I write f t f t is dimensionless okay but we want to convert that into f theta right yeah so f t equal to 0 to t please remember t e t dt can you convert this one into f theta again same fund as I said substitute for t dt t bar huh t bar what is that huh 0 to t minus 0 to t theta we cannot have d there we have d theta that should be converted into theta otherwise reshmi will get angry that is right what she said is right that is why entire class got educated otherwise we do not know I think what limit you are putting if she has not asked that question we should have accepted I also should have not mentioned things would have gone so our ignorance would have been there all the time okay it is theta 0 to theta so this will be 0 to theta e theta d theta which is nothing but f theta right no so f t equal to f t equal to f theta that is all f t equal to f theta f t yeah you will get that I think you know because you see here we have e theta theta right this is equal to theta right t by t bar so what is f theta this must be this area equal to f theta equal to integral 0 to theta e theta d theta correct no so this area again is exactly same as this area area wise because but only this is theta value and this is t value that is all so that is why f t equal to f theta right good so why we are doing that is when you are actually writing models for for example axial dispersion model or CSTR with dead space CSTR with bypass or CSTR with bypass and dead space both when we are actually doing all that so many times we will derive the equation in terms of area e theta or f theta which is dimensionless by the way why do you use dimensionless equations in chemical engineering I think not only chemical engineering in engineering civil engineering also they use but I think chemical engineers use lot of dimensionless equations because we have mass transfer heat transfer and also fluid and also reaction for example dam caller number is a dimensionless number what analogy not every time but I think except you know friction factor only has that relationship no f by 2 equal to yeah mostly let me come back to it yeah f by 2 equal to you heard of JD,JH nowadays many young teachers are not dealing JD and JH that is low ponder I have got okay yeah total ponderance on all them yeah so JD equal to JH equal to JH takes care of heat JD takes care of diffusion mass and f by 2 takes care of yeah that is a beautiful analogy between mass momentum and heat that is 1 but not that is mainly mainly it is only to take care of the dimensions because I can convert always my system into 0 to 1 okay or in terms of 1 2 5 10 like that units so that is why we have so many unit you know dimensionless modeling equations where I do not have to talk now what is exact length you see when I am talking about length of the plug flow I do not have to tell whether it is 10 feet or 20 feet or 3 meters or 10 meters I can only simple that dimensionless length is 0.5 so whatever length you put there if it is 5 meters it will be 2.5 meters if it is 5 kilometers if it is 2.5 kilometers so that is one of the main advantages in dealing with dimensionless parameters but you have to choose correctly what are the parameters you have to use so that you will get logical dimensionless quantity that is one of the main important things in chemical engineering why do you do that so may be some of you in your projects also do that make it dimensionless we say in the beginning and then solve that in terms of differential equations or whatever good okay good so this is the one and yeah so now when I have this kind of distribution E theta versus theta this kind of distribution E theta versus theta this also will have variance and also it will have yeah I think you know the theta bar okay so what is now equation for theta bar here for this case that is like T bar here I have theta bar that may be one okay but what is the equation for that you know already so that is why I am telling it is exactly like the definition 0 to infinity theta E theta D theta divided by 0 to infinity E theta D theta which is equal to 0.5 theta so this is one no this is one so this becomes 0 to okay now sigma theta square equal to same thing 0 to 0 to infinity theta square E theta D theta 0 to infinity E theta D theta which is minus 1 okay minus 1 minus 1 or you know actually this is equal to 1 right okay this is 1 square is equal to so this is also equal to you can take only this because this is equal to 1 so 0 to infinity theta square E theta D theta minus 1 which is also equal to sigma of theta square E theta delta theta this theta bar equal to 1 otherwise theta bar square equal to otherwise theta bar square equal to theta bar square okay okay this is also theta bar square okay theta bar square right you know see what you have to do is that when you conduct the experiment and you will have first C versus T data so that can be converted into E T then it also can be converted into E theta if someone gives you directly E theta versus theta data and ask you what is sigma theta square let us say sigma theta square after doing all this I will get 0.1 example okay so from my tanks in series model that this sigma theta square equal to 1 by N where N equal to number of tanks now from the data I conducted the experiment on that setup and then I found out sigma theta square equal to 0.1 so how many number of tanks we have 10 that is what is the advantage 0.1 N equal to 10 that is the answer that means you know that the tanks in series model can also be used for plug flow theoretically if you have infinity you will get plug flow but still you may not have that kind of infinite number of tanks so you conduct the experiments on suppose to be plug flow reactor it is not plug flow you can now convert that into equivalent number of how many tanks why because if I know number of tanks 10 if I have first order reaction what is the equation for conversion 1 by 1 plus k tau X i equal to 1 minus 1 by 1 plus k tau tau to the power of this is 10 tanks I say 10 so then I can calculate very easily the actual conversion that is coming from the system I have the packed bed but you know I do not want to go to dispersion model and then do it because I think tanks in series model also can represent the flow inside through some number I do not know what is the number so I will conduct RTD experiment I will get E theta versus theta curve C versus T and all that convert this, right and in fact you do not have to go even for that so this is also nothing but sigma T square by T bar square that is the definition of sigma theta square sigma theta square so even T curve concentration versus time curve we can calculate sigma T square which is let us say 0.1 what you got from the experiment so now use that to calculate what will be the number of tanks in series and from there I know if it is first order because first order is easy to remember that is why I am telling so first order equation is C A by C A not equal to 1 by 1 plus k tau to the power of n that tau is for each tank, tau i to the power of n, n I know 10 now I can calculate C A by C A not which is nothing but 1 minus okay so conversion I can calculate that is the reason why we are interested in this sigma theta square sigma T and all that similarly D by U L okay axial mixing I am trying to find out axial mix in terms of D by U L now D by U L may be 0.1 if I have that much D by U L how do I calculate conversion that is not that easy to the way 1 by 1 plus k tau to the power of n there is a more complicated equation which I tell you later so that is the equation which you have to use for calculating conversion that is the connection between RTD and conversion equations right so first to find out whether the reactor is ideal if it is not ideal you do not know what is the axial mixing conduct RTD experiment find out what is the dispersion and to find out what is that D by U L you need this equation right so once you know this equation then you can go back estimate what is D by U L go to that equation which equation is the conversion equation that conversion equation has D by U L because you are assuming it is axial mixing model substitute there and then calculate that will come a little bit later but now itself I am giving you the introduction okay good so these are the things now so what is left now is the ideal I think let us now derive the RTD functions for the ideal reactors ideal reactors to have plug plug flow so these two RTD functions for ideal reactors ideal reactors first let us take P F plug flow reactor plug flow reactor this is now ideal plug flow reactor good so we have here volumetric flow rate V volumetric flow rate V coming out so yeah this is P F R okay let me write here P F R first let me conduct the pulse input experiment pulse input so pulse input is a direct delta function which is simply entering here this is at time t equal to 0 okay how do I inject that because I have to very carefully take some without any disturbance some syringe or something and then just inject will go and then sit down throughout the cross section that means your disc that is what is the ideal one in reality it is not so easy so that is why you have to have without any disturbance simply this pulse ideal pulse that is 0 thickness and infinite height and area under the curve equal to 1 the same pulse you have to just inject at time t equal to 0 and now here yeah okay this is t bar equal to volume by volumetric flow rate this is volume V so at this point it just comes out just for representation this is at t bar because it is ideal pulse disc disc you cannot imagine as pulse because thickness should be infinity and all that so there is a very very very very very thin colour slide just inside okay nano nano nanometre thick nanometre okay nanometre thick disc which just entered and we know that because it is ideal without any disturbance that nice beautiful very thin red colour will be moving without any disturbance till here and that also exactly t bar equal to volume by volumetric time it will just come out if that equal to 10 minutes or 10 seconds it will exactly come out at 10 seconds now how do I represent this as an equation because there is no derivation here for this right so here the direct delta function is entering something like this but this is t equal to 0 how do I rate an equation for this this is delta t minus 0 if I can call this one as ET this is at 0 time ET normally we have to only write here okay ET only we have to write there but that is why I think I will simply mention that the mathematical function is delta t minus 0 what is the meaning of this Abhijit it is a pulse input definition valid only at time t equal to 0 so now what is the equation here so the same output that is here it comes out at t bar this we can call now as ET because that is the exact age distribution function actually it is a distribution function but only mathematical definition is thickness equal to 0 correct no and it equal to infinity and area under the curve equal to 1 so that is why here we have ET at the exact delta t minus t bar what you have to remember here is this is valid only at t equal to t bar and then all other times it is 0 right so now I can imagine the same thing with theta theta equal to t by t bar so now how do I write that here this is theta delta theta minus theta minus 0 right and here ET equal to excellent theta minus 1 so this is delta theta minus 1 because we know in this case t by t bar equal to 1 right because all the material is coming exactly and this is valid this function is valid only at theta equal to 1 in all other times what is ET 0 right so that is the one because E theta now definition is that direct delta function where it has got area equal to 1 thickness equal to 0, height equal to infinity so that is why this is very easy for us to discuss no problem at all so next one is step input right for step input how do I represent step input in pictorially here I have normally we put like this no this is at t equal to 0 at t equal to 0 and now what will happen so now I switched over from white fluid to red fluid right so then actually if you have ideal plug flow doing that experiment will be really beautiful and if you like colors okay white fluid and red fluid and like piston the white fluid will be simply displaced till this point and then finally here you will have only red fluid and white fluid you will not have any trace of it right yeah so when I see at t bar by 2 t equal to t bar by 2 this one 50 percent all this side will be red all this side will be beautiful white right so after sometime if you go to 0.75 then this will be red and that will be white so that is the imagination for that is why we call piston flow also plug flow piston flow equation I mean that name has come because of that it just simply pushes like a piston you know you would have seen how the piston is moving in some machinery so there it beautifully goes forward and without very smoothly without any sound and all that particularly old rail engines okay I think new rail engines also have the piston I think all engines all engines have that okay so that is what piston flow and the piston the the ending point must be exactly like flat velocity profile then only it is moving like this of course we are only talking one forward direction so it moves like a piston exactly that is what is happening here exactly at 50 percent of the time I have a piston red piston simply pushing the piston outside so now what is the equation we can write here if I am writing in terms of time ft what is ft okay I will give the clue ft less than t bar what is ft 0 so this is ft equal to 0 for t for t less than t bar so now ft equal to 1 for t for greater than or equal to similarly now ft what is ft 0 for less than 1 and the other one is 1 for theta that is all the equations it is very simple okay but only thing is understanding delta function now let us take mixed flow mf okay so for mf conducting experiment for mf also is very easy okay so if I take let me take step input here and write the equation step input equation and I will ask you to derive for pulse input okay you have to write the material balance and then not differentiating you know f you have f you will have an equation so if you differentiate that equation you will get also et not that you have to write the material balance for tracer and then only derive it is not that easy so that is why I am telling you but I am taking these also both are equally complicated both are equally simple so that is why ft let me take that is step input so this is volumetric flow rate volumetric flow rate so this is here I have tank here it is c not yah so both are anyway connected here so once this is open that is closed this is closed this is closed so at time t equal to 0 c equal to c not it is entering and immediately after that you will see some concentration coming out right good so now at time t it is a step input that means continuously tracer is entering whereas that is the difference between this and the other one so when pulse input only at time t equal to 0 you add 10 grams of tracer suddenly here anywhere you can add because if it is ideal perfect mixing everywhere it will mix instantaneously right yah so then onwards no more tracer input but here in step input we have continuously entering so now let us write tracer bar ends input equal to output the universal equation plus accumulation plus reaction okay what are the terms I can remove there reaction reaction that is what it is reaction reaction again only reaction but why not accumulation as far as flow is concerned it is steady but as far as tracer is concerned it is unsteady tracer is unsteady state it will become steady state once all the entire thing is removed then finally you will have only c0 okay so that is what what you are writing for the balance but the tracer tracer is non ideal sorry unsteady that is the reason so now this is 0 what is input in terms of grams per time let us say grams per second this is v into c0 continuously entered correct no what is coming out after time t equal to 0 so v into c continuously coming out and what is accumulation v into dc by dt we are writing simply c because there is no a and all that there is no reaction we are not talking about reaction of certain components okay good so now this is what we have to solve so it is a first order differential equation and it needs at least one boundary condition what is that logical boundary condition you take t equal to 0 b c c equal to 0 in the outside outside only okay ya and those people who do not believe this you try with a t equal to 0 as initial condition as c equal to c0 you will not get a solution that is why mathematics and sometimes physics will not accept okay ya because mathematics and physics needs a chemistry between them okay so that is why now I think I will ask you to solve this right or one more step shall I give you can solve it no excellent very good ya what do you get here someone already solved it c by c not equal to 1 minus t by 1 minus t power minus t by minus t by now here we are using t power 1 and by definition of c by c not what is that f t because we already proved that for f t so this is also equal to f t good ya so now what is f theta we know that f t equal to f theta ya so this is f theta equal to 1 minus e power minus okay from this what is e theta theta e raised to minus theta e theta equal to Rahul e power minus okay ya and I feel you know this is one of the simplest model equations in chemical engineering simplest model equation it is representing a model what is that model mixed flow reactor okay RTD function so beautiful very simple need not be always complicated equation when you say mathematical model right this also a mathematical model for CSTR or mixed flow reactor good ya now so if I plot this you will get this kind of response here like you know this one okay here also I should have plotted for f t for f t you should have got equal to 1 so this is f theta this is 1 okay and this also of course t also you will get at t equal to t bar that also equal to this is how you have to get the response from the outlet similarly here how do I plot this response f t because now I am plotting this equation f t ya okay e t what is e t exponentially decreasing t by t bar by t bar e t except or do not accept e t e t will be e power minus t by t bar by t bar this is what is e t I think Sushmita looks so tired most of you look tired except me ya ya but you know that is not excuse I have some more one more thing moment I saw slight consideration so immediately I think ya no no this is one more last part that also takes quick time because I want you to see 12th I can take the class and 13th is holiday unfortunately unfortunately because I do not have any other work except this so then 14th you do not have class 14th we have exam so I have to take only class and 15th there 14th so that is why I am trying to finish as quickly as possible next week itself otherwise maximum Monday right so that is the reason why I am just pulling and I think today I think I am on dot I think whatever I thought I have to tell because 12 o clock because other one I think you know once you know all these I will have a nice way of finding out like doctor I will give you some curves you have to tell me the answer what is the disease with this particular equipment that is all it is only diagnostics right okay that will come later so now let us plot this one ft when I am plotting versus t only so it is an exponential curve 1 minus 1 minus e power minus t by t bar so it will be something like this okay Abdul when it reaches infinity when not when it reaches when ft reaches 1 infinity also I have to do this question paper answer out before you know examination okay yeah so when t equal to infinity only then it becomes 0 then you will have ft equal to 1 right okay good so I will ask you another question within 1 t bar t equal to t bar what is ft 63.2 63.2 very seriously started calculating so this is t bar by t bar so this will be may not be exactly scale but anyway 63.632 okay so what is the meaning of that time reaches what is i-funda what is decay what is initial rate totally I think 3D collaborative is always high brain I think not easy to understand tell me so what rate you are talking I am simple my question is I am simple fellow my question is very simple what is the meaning of this that 0.632 at t bar that 0.63 is ft not time that is definition of ft what is the definition of ft fraction of material which has stayed at time t that means what is the meaning state forward meaning is 63 percent of the material is spending only one mean residence time correct no so remaining 30 percent what is happening more than that okay yeah it is taking the you know so that is the one of the reasons again because of the residence time distributions and whereas what is this t bar how much material come in a plug flow reactor what is ft ft for a plug flow ideal plug flow 1 means all the molecules are exactly spending same time within you know that t bar and the conversion in each and every particle I am repeating many time same thing okay so it is each and every particle is exactly converted so the final conversion average conversion also is same for all the packets are particles okay in a plug flow reactor whereas here the 63 percent is coming in the first one mean residence time not only that initially something is coming and afterwards sometime some more material is coming after sometime some more material is coming right so that means I can now calculate what is the fraction of material coming between time t plus delta t or otherwise between 0th minute and first minute 0th minute and first minute what is the material that is coming how do you calculate that okay this is et versus t how do you plot this how do you plot this et versus t how do you plot decrease that is very important so at t equal to 0 this is nothing but 1 by t bar and then exponential decay and Abdul what time it reaches 0 infinite okay now I can calculate from here the t bar is somewhere here okay what is the fraction of material between sometime t and t plus delta t yeah okay this is what is the advantage of plotting this one as e theta e theta versus theta now what is the starting point for e theta versus theta so one starts and then goes like this so this is theta equal to 1 okay now can you tell me when between 0.2 to 0.3 theta equal to 0.2 to 0.3 what is the fraction coming up you have equation I say e theta e theta 0.2 and 0.3 what did I say between 0.2 to and 0.3 0.25 theta equal to 0.25 delta theta equal to 0.5 0.5 0.5 you have to calculate e power minus 2.5 into 0.1 exactly into 0.1 0.5 0.5 0.5 0.077 0.077 0.077 that is the percentage of the material that is coming between this time okay so now I do not know that you are appreciating you know why you have to make dimension less your camera so I do not have to mention now any timing okay so theta equal to 0.1 to 0.2 that is 0.1 interval and anyway the average I have to take out something like this that is why that average you know this equal to this okay so this value is how much you said 7 percent okay 7.8 percent almost 8 percent of the material is spending between 0.22 0.3 theta okay so like that I think we can calculate at whatever fraction and correspondingly we can calculate what is the conversion in that and take the average of all that then you will get the average conversion from that okay good so now ya this is the laminar flow the next one is laminar flow ideal reactor is P f and also M f I think to deviate you I think I will dictate something for laminar flow laminar flow I think Levenspiel has done so beautifully okay please take this when a vessel is long enough then the dispersion series model well describes its flow behavior but how long is long enough question mark these vague words you know always we use long enough okay but how long is long enough okay question mark ends then next for packed beds and for turbulent flow in pipes just about any vessel length is long enough however for laminar flow in pipes we may be in other flow regimes but of the pure convection model pure convection model convection okay pure convection model okay pull stop let us first look at the extremes for laminar flow okay good one I think you can put one say that if the tube is long enough we are talking about only laminar flow now then the molecular diffusion will have enough time to distort the parabolic velocity profile so that the dispersion model applies now second below next point if the tube is short enough and the flow rate is high and the flow rate is high then molecular diffusion has not enough time to act so all we need to consider as causing a spread in residence time fluid is the velocity profile instead of we are in pure convection regime third point is the flow is so slow that the main movement of fluid is by molecular diffusion not by bulk flow then we enter the pure diffusion regime instead of we rarely meet the situation in chemical engineering we rarely meet the situation in chemical engineering good I think next para you write liquid is likely to be in dispersion regime gases are likely to be in dispersion regime not in the pure convection regime liquids can well be in one regime or other liquids can well be in one regime or other very viscous liquids such as polymers are likely to be in the pure convection regime Assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. So thus the spread in residence times is caused only by velocity variations. This is in fact beautiful information I am sure none of you would have seen this, okay. Yeah, I do not know at this point of time you may not appreciate in the class but I think you know this is wonderful information. What is entire thing told is that we always assume that we have beautiful parabolic velocity profile. You know the life of this beautiful parabolic velocity profile is only in seconds. If it is gases immediately after the formation of that profile even at low flow rates molecule starts moving up and down, radially, laterally, laterally and also you know axially all that. So that means it is no more a parabolic profile. If you do not have parabolic profile you do not have laminar flow. That is why laminar flow is called pure convection model because there is no molecular movement. Once that laminar flow starts like the basic definition everyone tells in fluid mechanics classes imagine that sliding of pack, you know pack of cords. That is what we say. That is also told here. So that means without any disturbance with the pack below or the cord below it is simply sliding without any disturbance that will happen only for high viscous fluids just for sometime. So when you have that kind of clear definition for laminar which is called also pure convection model, the residence time distribution is only because of profile. Otherwise if there is a molecular movement, so that also will disturb this profile and your residence time distribution will be disturbed. That is why this is a very restricted case which is called pure convection model. This I think you have not heard. Anyone heard of this? I do not think anyone of you heard of this. That pure convection model is only possible with parabolic velocity profile and when we are talking this one as you know in polymer this comes as a reactor. In polymer processing laminar flow reactors can be used. So that is why we need to know what kind of residence time distributions we get and those residence times are purely by velocity variations if you have parabolic velocity profile like this, right? That is why for gases it is not possible and for liquids with less viscosity and all that also that will be disturbed very very quickly, right? And what happens is if you allow that for some more time, if you have sufficient length you can use normal dispersion mode because by the time molecules arrange it themselves it is flow, it is convection. It is not pure convection. By convective flow material is moving and molecules are getting dispersed up and down because molecules are moving up and down laterally, radially and all that. So you will get this kind of distribution, this kind of normal distribution, right? So how do we now develop an equation, RTD equation, ET for this one, ET and FT, okay? If you are not able to see now, okay, show you what can we do with, can we change the 5 minutes? They will also relax, okay, yeah. Because let me complete this I think again before that I think we can also draw another figure which will nicely represent the same thing, you can draw this figure that is actually the meaning of laminar flow. It is like one pipe inside, over that another pipe, over that another pipe and all of them are moving like that without disturbing other pipes. That is what what we say this pack of cards, okay? And I think this is very important information, none of you definitely would have known this kind of information till now and the centre is somewhere here. So that is what is this profile meaning, parabolic profile, okay? So the material which is there on the top tip of this will go much faster, okay, yeah. So now when you want to derive an equation for this, we need some information about fluid mechanics. So this total length is L, okay, shows you can continue? Okay, yeah, changed already, okay. So this is L and of course this is R equal to 0, this is total, okay, this is capital R, yeah. So we have an equation, I mean all of us know about this is that we have U R equal to U M maximum velocity, yeah, 1 minus R by R whole square, this is equation 1, let me put the equations for this. So we also know U bar equal to average velocity equal to U M by 2, that is equation 2, okay, these things from fluid mechanics and we also know that T time is nothing but length by U R, correct, no, velocity, T time, length by velocity and this is also equal to L by U R is this, U M 1 minus R by R whole square, correct, yeah, this is equation number 3, yeah. Now substituting this equation here, equation 2 in 3, equation through, equation 2 in 3, what you get is T equal to T bar 2 1 minus R by R whole square, yeah, of course where T bar equal to mean residence time, T bar equal to mean residence time, yeah, how T bar is defined, this is equation 4, yeah, so T bar equal to L by U bar, that is the equation, so now this important thing what we have to understand afterwards is only mathematics, the fraction of material between R and R plus delta R, R plus delta R corresponding to times T and T plus delta T is E T delta T given by volume flow of fluid between R and R plus delta R divided by total volume of fluid, okay, let me explain this, the definition of E T delta T we know, that is the fraction of material spending a time between T and T plus delta T, now this pipe will, this diagram will give you a nice picture there because when I have some R here, no, from centre, okay, this is R and maybe slightly delta R is somewhere here, okay, so between, okay, you can also imagine this R and delta R is a small pipe with a thickness of delta R, right, so depending on its position here, so that is coming, no, if you take this one as the delta R, that is coming in the first test, right, so this also corresponding to sometime, that means when this is coming earlier, so that is also corresponding to sometime earlier, sometime T, that means each, each, that concentric pipe which is coming at different Rs will come here at different times, this one will come in the shortest time, what is that T bar by 2, somewhere middle if I take, that will take some other time and somewhere only small delta R if I take near the wall, that takes lot of time, right, so that is what is the simple definition here, that means there is a relationship between R and T and R plus delta R and T plus delta T, so in this flow when I am imagining that these are all the concentric tubes, this, this tube and minus this tube, so that is the fluid, this is one tube and around that we have another pipe, imaginary pipe and that volume of that one will come in a different time corresponding to its own velocity, why velocity is changing along the radius, correct, no, different velocities at different radial positions, that is why they spend different times, so that is what is this and that means if I take one annulus that is the volume of fluid between R and R plus delta R which is automatically connected with some T and delta T, how do you get that T and R connection we will deal later, this is the basic definition, okay, so now if I write an equation, I mean the equation for this, this E T delta T will be, what is volume flow of fluid between R and delta plus R, if you remember your fluid mechanics earlier, it will be U R 2 pi R into D R, okay or delta R, right, I can also write delta R, so divided by what is the total, total flow rate, total flow rate is simply U bar pi R square, correct, that is the volume flow rate, cross sectional area multiplied by the U bar, this is U bar, right, okay, so this is equation number 5, okay, of course pi pi can get cancelled, good, so now another thing that is left to understand is, yes, this equation 4 is now actually, actually we have already the relation, the equation 4 relates R and corresponding time, correct, no? Yes, R and corresponding time, so now we, you can, you can differentiate this, differentiate equation 4, differentiate equation 4 with respect to, with respect to T, so to get D T, okay, I think I know, with respect to T I do not have to write, okay, yeah, okay, D R by D T or D T by D R also we can write, so differentiate equation 4 like this D T equal to, can someone differentiate quickly, D T by D R, D R you can bring that side, at least one person if he differentiates and tells me, race me quickly, sorry, T bar by R T bar by 2, okay, T bar R by R, excellent, so many people would have not tried I know, because you are tried, so what you get for D T is T bar, no I am just writing this one, T bar by 2, so that later for me it is a bit easier, actually what is said is right and also I am writing T bar, 2 R by R square, correct, no? What is said is this 2, this 2 will get cancelled, okay, that is what you said and then D R divided by 1 minus R by R square and whole square, that is the one, that is D T, okay, yeah and from equation 4, this equation what is T by T bar from equation 4, T by T bar equal to, not T, okay, 2 T bar by 2 equal to 1 minus R by R whole square, correct, then I try, 2 T bar by 2 equal to, this is one, this is equation, T bar by, T bar by 2, yeah, right, equal to this one, 1 minus R by R square, so this is equation 7, okay, so using now equations 7 and also 6, yeah and substitute here in this equation, in this equation, oh, yeah, 6, this is 6, this is 6, yeah, okay, E T delta T you have to use, yeah, I am just thinking about your timing, you know, whether you are able to take, otherwise I have, okay, I think let me do and so once we start doing that, okay, okay, please take, write this, substituting equation 7 and 6 and rearranging, substituting equation 7 and 6 and rearranging, rearranging, what you get is R D R or delta R, okay, I think, let me write D R only, delta R D R same here, so I think here also I will write D R only, R D R equal to capital R square T bar by 4 into D T by T square, so can someone check, this is equation number 8, correct? So now another thing is, now substitute equation 8 in 5 in this equation, substituting equation 8 in 5, equation 8 in 5, what you get is E T D T equal to U by U bar T bar by 2, so this is equation number 9, okay, okay, so we also have equation from 1 and 2, from equation 1 and 2 what you have is U R by U bar equal to 2 into 1 minus R by R whole square, so this is equation 10, U by U bar equal to, yeah, okay, good. So now 7 and 10 can I combine what you get, U R by U bar equal to T bar by T, so from equations 10 and 7 what you get is U R by U bar equal to T bar by S, so this is only just for converting one from the other, okay. So now substitute 11 in, this is equation 11, can you tell me substituting 11 in 9, yeah, so Ramakrishna what did you get? E T D T equal to T bar square by T square, D T by 2 T, D T by T square, 2 T cube, now what is E T? T bar square by 2, T cube, so this is the equation for laminar flow E T exit as distribution function and yeah, so this is equation 13 but I think before writing 13 there, is there any limitation for this? When it is valid, when it is not valid, when is this valid? If I answer that I think that, okay, you are still alive, huh? It is valid only between 0 to T bar by 2, no, you start seeing the fluid only after T bar by 2, because you know from fluid mechanics the average velocity, twice of the maximum velocity, right, correct, half of the maximum velocity, yeah, okay, U bar equal to half the maximum velocity, so that is why T bar by 2 is the first time what you see the trace are coming out, okay, so that is why the limitation here is this is valid only for T greater than or equal to T bar by 2 and E T will be 0 for T less than T bar by 2, okay. Now tell me E theta, E theta is again one of the very beautiful equations, E theta equal to, oh this, this is 14, 1 by, excellent, 1 by 2 theta cube, this is another equation which I like, the first equation is E theta equal to E power minus theta for a mixture flow reactor, okay, so this will be good, so this will be again validity theta greater than or equal to half, 1 by 2, so this is equation 15 and E theta equal to 0 for theta less than, okay, so we can also, of course just, we can integrate this and then try to put the limits, because F theta equal to integral 0 to theta E theta d theta, so this will be 1 minus 1 by 4 theta square, this will be F theta which is also equal to F T, so this equation is 17, 17, 16, 16, that is equal to 16, okay, so this derivation is not you know may be at this point of time you are not able to observe but the derivation is not at all difficult, okay, but only algebra comes after that definition of writing volume of the fluid between r plus delta r by total volume, that is all, afterwards it is only mathematical manipulation where you can easily find out but only I just want to ask you how do you draw this E theta versus T or E theta versus theta, you are very good it is not 0, okay, it is valid only from half theta, half to theta, okay, upper limit, then only you get that otherwise you do not get that, okay, good, so how do I plot this E theta versus theta, where does it start? Theta equals half, E theta versus theta, theta direction is part 3, half, half, half, yeah, so till what point I have to go, what is this starting point? Excellent, till some people are alive, okay, okay, this is 4, yeah, then afterwards how does that go? Decreases as 1 by x cube, yeah, 1 by x square, 1 by x cube, 1 by x we have no, 1 by x cube no, 1 by theta cube, so this falls very fast and goes like this, this is not exponential, why I am drawing now is that this is not exponential like, that is also decreasing but that, yeah, mixture flow is decreasing exponentially down, e power is simply minus theta but this is 1 by 2 theta cubed, okay, so that is why it comes out here, so somewhere here you will have theta equal to 1, what is the value of E theta for theta equal to 1? Yeah, somewhere here, 0.5, okay, so now you see on the same thing if I just show you like, okay, for plug flow, on the same graph I just want to plot plug flow, right, E theta, how do I plot? What, Swami, still this fellow is not leaving here, yeah, so F theta, sorry, yeah, E theta for plug flow, theta equal to 1, this is for P f, okay, at theta equal to 1 and for mixture flow, yeah, at theta equal to 1, somewhere here it goes exponential decay, so this is M f r, that is P f r, okay and this is, see all three ideal equations, I mean ideal things are there, okay, yeah, only one thing is left I think what I thought today doing, you know now the diagnostics may be I think we will do in the next class because I think may be it is too much for you, this also I do not know how many people have absorbed that but God knows it may be in third quiz.