 this variable volume what we have been discussing yesterday and you know for constant density is how do you write concentration definition how do you write concentration definition for variable I mean for constant density system c a equal to c a 0 1 minus actually what is the definition of c a moles per moles per meter unit volume so what you write is c a equal to n a by v that v is constant so that is why straight away you can take it as c a so but when you when you have variable volume even the concentration definitions also will change okay so that is why we have to write now c a equal to n a by v and when I express this in terms of conversions and this v is v not yeah v not into 1 plus epsilon a x a and what are the assumptions in that there are assumptions in that it is truly valid for variable volume is okay but I think it still it is valid valid for ideal gas constant temperature and constant pressure so that p by p not equal to 1 and variable I mean the what the other one I said t by t that is isothermal system and also ideal gas so please remember that these are the things assumptions are very important there so now this is the definition what we write and I have to now substitute if I want to write this one in terms of conversions so this will be n a not into 1 minus x a by v not 1 plus epsilon a x a so normally this will be again c a not c a not into 1 minus x a by 1 plus epsilon a x a you see now the problem because this definitely increases you know our algebraic difficulties you know when you are actually solving so this is equation 1 and of course from this you can also get from equation 1 you can also write x a in terms of c a the same equation I simply solve for x a so when I solve that I will get minus 1 by 1 minus c a by c a not divided by yeah 1 plus epsilon a c a by c a not this is equation 2 these equations will be helpful to you and even rate definition also changes rate is simply minus r a equal to minus 1 by v d n a by d t this is the basic definition okay but this v is also changing right so if I substitute v equal to v not into 1 plus epsilon x a and also if I write for d n a because conversion definition is always moles initially entered minus moles at any time divided by initial moles so that I think you remember that is why I am not writing that moles wise but c a definition is wrong again you cannot use c a in terms of you know c a for x a all the time we also write x a equal to c a not minus c a by c a not which is not correct all the time it is only strictly valid for constant volume okay constant density system this can be written as c a not 1 plus epsilon a x a into d x a by d t this is actually this is the definition of now rate for variable volume okay yeah so now of course d x a also we know what we have written there and all that so x a in terms of volumes this definition how do you get this this is strictly for isothermal and ideal ideal gases yeah and also constant pressure sorry very constant pressure yeah this this you know how do you get v equal to v not into that one okay this is good why I am writing all this is that you are trying to use the technique of measuring the volume and trying to find out the kinetics measuring the volume this is the total volume so at any time t the volume expands right so that volume if you are able to measure now can you get the kinetics okay so again you can use here either integral method or differential method so integral method when you are using so there is another thing also which I have to tell you here yeah so okay what is d x a here or okay d x by d t okay d v by epsilon a v not right so for this yeah d x a by d t we can substitute this equation okay in terms of volumes right yeah we can substitute that equation yeah okay good so now now I will just give you with all the this is also equation 5 I will just now try to give you for 0 order reaction how do you find out volume change that is the simplest one see our idea why I am writing all this is that you are measuring our idea of finding out kinetics is either you can change the pressure for a constant volume right so you will get total pressure change that total pressure you are able to convert to partial pressure yesterday's last equation was that so once you know partial pressure of a and partial pressure of b what you assume is that whether 1st order or 2nd order integrate and you have all that information now so similarly when I have constant pressure and variable volume right so there you are measuring pressure change total pressure change you cannot measure individual component change P a P b P c you cannot change okay so that is why total pressure even here it is total volume what you are measuring once you measure the total volume how do you now convert that into rate equation so this is the definition of rate now minus r a right so if I have a 0 order reaction okay what is minus r a k so this must be equal to k right or for 0 order reaction integral method integral method for variable volume I think very nice problems are given in 11th sphere very beautiful problems I think you know I lost a few problems of third chapter very very nice problems of you know in terms of volume change yeah so our idea is to find out through variable volume what is the kinetics for that we should have an integral expression if I am using integral method okay so if I am not using integral method then differential equation that is the yeah this equation I have to use and then this is equal to k into c a to the power of n where n again you know n is okay but c a again written in terms of yeah c a is nothing but c a not into 1 minus x a by yeah this is where you make the mistakes okay so that is what you have to do that so then you can plot as far as possible in terms of straight lines whatever information you know there epsilon will be knowing there x a you are measuring there so c a not initially you know and with the time you are trying to find out what is the conversion so you can do that right so similarly here for integral method for zero order reaction minus r a equal to this one c a not 1 plus epsilon x a dx a by dt equal to k this is for zero order reaction right I think as an exercise can you convert this x a in terms of v and then try to find out what is the volume change I think you have here what is dx a by dt can you tell me the equation because this I think simplest one which I can give you without losing much time if I give slightly complicated equation you will have more problems what I am asking you is forget about this one these two you have to take so this is five this is six now get me an equation in terms of volume v by v not v by v not equal to what see me not able to get not able to get see for x a you have an equation here know yeah so dx a by dt you substitute and then try to integrate what do you get can you tell in terms of log v by v not let us see some more let us see some more people very good but all of you should do this all of you should do this I think I do not want to yeah those who have not done definitely I think they have to practice more right they have to practice more so what you get here is log v by v not equal to epsilon by yeah c a not into t this is the equation after integration okay so now you know I am using an integral method so what is the thing so I have now I do not measure x directly now because that is the idea because how I take is Lounspiel method is that is you know I mean this he has used a wonderful techniques here he just takes a tube where he fills up somewhere here the gas batch system okay that is v not for us then he will put a graduated scale and then he puts a small ball here okay otherwise I will put the color yeah you know you would have seen rotometer no there are some rotometers for very low flow rates that ball has spherical it need not be you know that cone shape cone shape and all that will give more stability but this is a bad design but still at very low flow rates this sphere is it can be used so similarly here this will this can easily float this can easily move the moment reaction started let us say all this is the reaction actually this is the reactor for you batch reactor so this entire reactor you put in hot water let us say the reaction is occurring around 80 degree centigrade right so then reaction starts and this is variable volume that means there must be more change otherwise if there is no more change this fellow will only sit there no you won't move so because there is volume change so it will try to expand and this will easily move then it goes here in one minute here okay another minute may be here another minute may be here okay another minute may be here that means now you are trying to find out what is the length you know cross-sectional area of the tube and then you can calculate what is what a wonderful beautiful method as you know for engineers this is what what you have to imagine when you are now trying to do any experiment I am not talking about only kinetics of any you know in any chemical engineering problems when you want to solve experimentally you have to think all this what is the easiest and best method what you can do with the simplest one as far as possible with without costing much right so I think this kind of problems are there in Levenspiel second edition also wonderful book all of you may be having only third edition go to library and then try to find out second edition okay so actually he has given this kind of problems also I don't know whether it is third edition it is given there or not okay so it simply expands then we will record V versus time that is what what we are trying to do now V versus time so when I want to have my integral method what I plot now yeah T versus so I want to see that this is as a straight line so this entire thing as Y this is M this is X Y equal to MX so that means there is no C so you should not go through I mean you should go through origin right so then you will have you are like this this is the one here I have to ln V by V not versus T so what is the slope now epsilon A by KC you know epsilon A in the beginning itself how do you know yeah either you can use your age old process like you know V at 100 minus V 0 otherwise you can use the other equation where you know delta delta A versus Y A not if it is pure Y A not equal to 1 right but only that delta difference will be there delta is products stoichiometric coefficients minus reactants stoichiometric coefficients if product is let us say 4 1 mole going to 4 moles 1 mole okay using the this yesterday's technique how do you find out it is pure A yeah so Y A not equal to 1 and you have here 4 minus 1 by 4 by 1 by 1 A no delta N by A yeah you have to see you have to be familiar with the equation right so A is 1 so that is how one one can easily get the moment I say I have 50 percent innards then 2 point how much you get okay so it is 3 into point 5 1 point 5 right so that is why it is also easy for us to find out that is the reason now we have taken the simplest example and then I asked you to solve that now I will explain this one to first order reaction right let us say that we have first order reaction this is oh my god I have not written here 0 order somewhere you write there okay first one is yeah next one is first order reaction for first order reaction again we will go to this definition okay so when I go to this definition I have C A not because I have already ready made otherwise you have to write every time minus r A equal to this yeah this one and write for V what is this and DNA by DT in terms of DX by DT but already we have done so I am straight away taking C A not 1 plus Y A X A and DX A by DT equal to first order K into C A which is nothing but K into C A not 1 minus X A by 1 plus Y A X A okay good cancelled you can cancel out and C A not also goes correct no see it also okay now tell me what is the equation integral equation for this in terms of again volumes only thing is correspondingly you have to substitute what is DX by DT and all that and this said also you have to convert this 1 minus X A in terms of the I mean the volumes so X X equation is this so you have to substitute for X and also DX by DT then integrate and someone tell me quickly those who could not do here I am sure they cannot do in the examination right that means you do not have the touch I am not saying whether you are intelligent or unilligent but only thing is experience the touch is not there practice is not there that is why we have a beautiful saying practice makes yeah makes a person perfect L N V by B not equal to not so complicated X A cannot come there okay X A come because X A you are totally replacing because X A should be in terms of only V by V anyone else a minute tell me in terms of lawn you have to get the logarithm there is another simple way of very simple way of being also that is the correct okay I will write here you check minus L N 1 minus V minus V not by epsilon A V not equal to KT that is the equation you would have made some mistakes please check this why I am telling you practice practice practice that is the equation and I can tell you in a very simple way also you know this equation but I want you to learn through hard way okay yeah what is the first order expression as it is if I want to yeah so minus yeah minus lawn 1 minus X A equal to KT what is X A that is all you see one minute that is why I give my surprise test sir yeah so the another example is second order reaction which I do not ask you because three hours it may take so that is why this is second order reaction we just give the expression in terms of differential equation this is A plus B going to products products where C A not equal to C B not okay that is again simplest case I am taking in fact that means one more one more C A not equal to C B not that is the simplest case so now what is the rate equation here minus r A equal to K into C A square that is the equation right so this equation equal to K into C A not square 1 minus X A square divided by 1 plus epsilon A X A square it is a batch reactor we are talking all the time so this is also equal to this is 8 this is also equal to what yeah that C A not 1 plus epsilon A X A into D X A by DT okay so what are the things I can cancel here one C A not I can cancel and this side one epsilon A I can 1 plus epsilon A X A also I can cancel so now what is the differential equation I have to now solve the differential equation will be D X by DT equal to yeah this is equal to K C A not 1 minus X A whole square by 1 plus epsilon A X A so this is equation number 10 this is equation number 11 now this is what you have to solve in terms of volumes in terms of volumes right yeah so now definitely if I give this one may be next week examination also I do not think you can do unless you practice practice practice now you have some techniques you know either substitution techniques this technique that technique you know when you are solving this when you are integrating okay so that is why you have to use your think really all the techniques what you know till now and that is the idea of giving you mathematics course no not for only just getting great okay so all that knowledge we have to also use in the real problems but I will give you the final expression final expression or just before final expression I will also give you one or that integral equation how in terms of V by V not please check me sometimes I may make mistake also equal to 1 minus V minus V not I think you know some people can write V minus V not as delta V del V I mean for simply avoid more writing so this is I am writing everything because so that you can remember what is happening here okay so whole square this is square equal to because this one no equal to k C A not epsilon A V not into integral d t so at time t equal to 0 it is V not at time t equal to t this is some V V this is what you have to integrate when you integrate you also get this kind of equation but please check again V minus V not by V not epsilon A minus V minus V not plus epsilon A long 1 minus V minus V not divided by epsilon A V not equal to k C A not t so this is equation number 12 this is equation number 13 okay this is really complicated I do not think I am not saying anything bad about you but only thing I am telling you is you need practice to solve this kind of problems once you have this equation integral method when I am using how do I plot this again because what do I measure I measure V by V not or V so V by V not I know so that means this side epsilon I know in the beginning itself so that is why I know all this side so this entire thing I take as straight lines are always better curves are always dangerous okay that you have to always remember curves are when you can turn curves are dangerous because you do not know how the curves are turned so straight lines straight lines you can easily see right so that is why you have to try to plot as far as possible this has entire thing as Y and this one as M T as X again Y equal to M X so that you will go through origin and then you have to put in all the you have to put the points and then draw the line and you know if it is falling on the line then you are right your second order assumption is right so then you will have this one as the order of reaction in terms of variable value okay good so I think I have given you the real difficulties in this you know integral method because integral method is the best method but what is the difficulty because there is no free lunch anywhere you have to work hard that is why you know there is no shortcut for knowledge most of you won't realize that there is no shortcut except you believe in matrix movie so there I think I already told you also oh yeah loading load C R E 1 just go to I think a computer load and then all C R E 1 problem solutions everything is there with theory everything is there in your mind if that is the world I think I also would have liked it it is really fantastic but right now we are not in that kind of world okay so that is why practice practice practice I am requesting you many many times this practice and that was the reason why I also gave you a lot of problem if I give one problem no practice okay so that is why if I give 100 problems at least 50 problems if you do that is practice for me if you do 100 problem that is excellent practice if you ask me 1000 problems very very excellent you know for me because that that means you are real master and that mastery comes only by practice you may be very very very very intelligent person okay very very to the n very n tends to infinity you may be so intelligent but still without practice you cannot do anything this is life lesson so that is why you have to I think humans have not come to that level of just looking at that and then you know again that robo movie so this is what now I want to sum up with the kinetics is that kinetics can be found out using either integral method or differential method those are not only two methods and the remaining methods depend on your smartness like for example we also have what is called off-life method okay this is very smart method but is a differential method or integral method both can be used in fact yeah right both can be used but the differential is very easy one because you assume now you have nth order reaction and then you will have yeah for constant density system and that method is not that easy for variable method right for constant density method we have dca by dt equal to kca to the power of n can either use this differential method or integral method that is fine but I think yeah we are talking about only integral method for this right why integral method I will integrate this equation and convert this equation in terms of only initial concentrations how so ca is equal to ca0 by 2 yeah that is the definition of off-life in fact it need not use off-life you can use any length it may be 0.75 life it may be 0.8 life okay need not be but off-life is very easy for us and also off-life started because of this radio radio activity so that is why off-life we use very frequently so then you will have only ca0 and time just ca0 and time so that equation you have to plot and I am not discussing all that because once you are experts in this then your knowledge will increase and then you can use all these methods and your smartness will come out what is the method you choose so that you can solve very accurately and very quickly so this is the method and then now at this end of this discussion what we have now is a minus r a whatever method you use okay whether variable volume or constant volume right so even though gas phase reaction the number of moles are changing I may take a constant density reactor batch reactor with closed volume is not increasing or decreasing no pushing like that so then it that becomes constant volume variable pressure reactor so pressure is the easiest one to measure only thing is you should know what is pressure gauge okay if you do not know that also you cannot run that so take that and then measure pressure and then go to that equation which I gave you know last class and use that partial pressures and again come back here to write this kind of equations where minus r a this c i is nothing but p a right but only thing is when I write the c i in terms of p a then k by r t is a new constant for me so that is why I think that you have to take care of with correct units and you know correct dimensions all that you have to take care of otherwise the method is same everywhere but again I tell you it depends on your smartness particularly those people I think m tech people may do coursework and then some project and leave but I would like to see if I am here I think my next June I will be out but I think before that if you are able to do something very smart please tell me I will be the first man to be very very happy to see that yeah this is a very smart idea right that smart ideas that is what what we see from the student okay yeah the smart idea is nothing but how simply you can do within short test time that is what the smartest idea right so that is why I think you know please remember that at the end of all discussion we have what is called minus r a this minus r a is simply can be a function of simply c a and t it can be as simple as k equal to I mean minus r a equal to k c a square or it can be as complicated as k c a square by another capital I think capital k if I write you may take it as equilibrium so m plus c a plus c a square now just imagine this is the equation which I have to use here for finding out the rate also somehow I found out and then this is the kind of equation what we have right so even this equation also what I can write here is this is minus r a I get the minus r a data right that means imagine that we have constant density system I am telling all simple things because easy for you to visualize next one is only difficult things where you have to you do not have to use your brain there but only you have to be careful in using mathematics that is all mathematics come there only but conceptually it is same right now again I have to plot this one as a straight line how can I do that if it is possible otherwise if it is not possible then what I do is I turn this way 1 by minus r a equal to this by this now I will have each one dividing so that means I have 1 something plus something plus something now that is easy for me to solve instead of this equation where it is as it is so that is why all these smart things will come by practicing so this is the one what you have it can be one thing what you have to remember is this minus r a is no more order of reaction you are very very lucky only you will have first order second order so that kind of specified orders right otherwise it can have any form that is what that is why do not say that minus r a equal to k c a to the power of n but say that minus r a is a function of c a and temperature and there is no other function I mean there is no other variable which is changing rate I mean of course sometimes catalyst may change but we are not talking about catalytic reaction that is postponed to next semester good okay so now using this information what is our next thing what we have to do so now I will start with my diagram so now of course reactor design we go back again information required is this output input kinetics contacting so here I have chemical chemical physical or physical I have written first order okay so this is one starts batch continuous and in continuous we have P f M f so we are very happy to have only three here all three together so then we have output performance equation no this is a nice word performance equation okay sometimes you feel oh very nice word but I think the performance equation is nothing but the design equation for that for us okay what is the final design the thing what we are trying to do so output is a function of input kinetics and contacting yeah that is the equation now do you have all these term that means output is a function of input kinetics contacting like here rate as a function of concentration and temperature right so this you understood what is the functionality of rate with respect to T temperature right that is all only one you have very easy but most problematic parameter is function of it can be anything you do not have any choice with you but whereas with temperature you have at least one choice it is I think proved till now that is only the best equation that can be used for finding out change of rate with respect to temperature okay similarly here I have now output that means if I want to design a new reactor okay I should be able to find what volume if I have a old reactor and then I want to find out again output what is the output from a old reactor existing one conversion that is all the design problem so now we assume that we have a new plant and then I am going to design the new volume okay for the reactor so that means I should know output that is how much you know we have to send to the reactor then I should know kinetics kinetics is nothing but minus r a some form of that f of c a t some form if you are lucky that may be simple first order reaction k into c a then the other thing is yeah contacting so contacting also you have an idea right contacting will tell you contacting will tell you which design expression you have to use please remember that okay contacting will tell you which design expression there are three design expressions so one is the batch reactor you know when you have to choose and the other two are continuous reactors you also know when to choose so if I say that okay I need design a plant for me for 10,000 tons per day very large okay so our mind will automatically go to continuous so now continuous I have again two so what is that I mean where mind has to go in the next step yeah why plug flow reactors by definition plug flow reactor is always efficient for n greater than one zero zero all positive order reactions n greater than zero okay and if you are not believing that just draw one by minus r a versus x a one by minus r a versus x a and then see which one will give you lowest volume area lowest area okay that is one of the easiest methods to find out the arrangement of reactors and all that okay instead of worrying about remembering n greater than one n normally all n greater than zero the plug flow reactor is the best reactor but we do not use all the time why because if it is exothermic reaction then temperature is crucial if the temperature is crucial I will go to mixed flow in spite of in spite of you get more conversion or small volume for the reactor in spite of that you know so this is always the correct thing when I when when you know that you know this guy may be highly intelligent super intelligent but he is always a troublemaker will you hire him in your organization merit will you take him and live with him with all the problems so you do not so that is what you know that he is an excellent even his brain is super but till we do not the reason is we want to compromise you want to have smooth operations in your plant that is what exactly the choice between plug flow and mixed flow because I know plug flow for all the reactions n greater than zero definitely plug flow will be best but still I will see that whether I have to live first okay to operate the reactor so if I take the plug flow and if the temperature is uncontrollable explodes we will run the reactor there is no reactor once it explodes so that is the reason why we go for the mixed flow which is very easy to control and temperature one temperature you can maintain by definition so that means contacting also you know how to choose kinetics also you have to choose I mean you know how to get and input you know that is the simplest one what you get this is what is entire reaction engineer that is all now you have clear idea what is minus r a and how do you get it okay including heterogeneous systems because I gave some examples if it is coal burning how do you get the rate if it is catalytic reaction simplest one okay I have not done Haugen Watson model okay but that I will do next semester so but that is also method is exactly same approach is exactly same you will get minus r a slightly complicated like this I am talking about Haugen Watson models for heterogeneous catalytic reactions okay you remember that form minus r a equal to P A minus P B into K divided by in the denominator you have one plus something one plus something you know like that you will have that is a complicated equation but approach is same exactly like slurry reactor slurry reactor you have so many steps the other two at least we have taken only two steps because irreversible and all that so but you also know at this point of time how do you get a rate for heterogeneous system so that means you have all that information I am not exaggerating that is the information basic information that everyone should know in any reaction engineering course now next onwards it is only the details details details and slight combinations that is all what you have to learn otherwise you do not have to learn any new concepts right but something you know even if I say recycle reactor it is a combination of mixture flow and plug flow because it is intermediate mixing between plug flow and mixture flow right you I mean other thing I can tell you is okay I take now five CSTRs in series or two CSTRs in parallel and one CSTR in parallel and another CSTR sorry CSTR is MFR okay okay one MFR in one branch and another PFR is another branch okay how do you get what all kinds of manipulations only what you have to learn and it is not easy also to learn because it involves mathematics and mathematics require practice okay if it is a practice like this only you have so otherwise it is not possible right so that is why the next thing what I do is combination of reactors reactors and just before going to that some people have still some uncomfortable feeling with the difference between tau and t bar okay because we have now taken individual reactors and then thoroughly seen but we have three equations for batch it is t by C A not equal to 0 to X A d X A by minus r A and for P F V by F A not which is unfortunately written t by tau by C A not correct no can be written equal to 0 to X A d X A by minus r A and the other one is M F is V by F A not also equal to tau by C A not X A by minus r A these things you have to remember right yeah and unfortunately the first reactor what you have learnt batch reactor is having this format t by C A not okay so that is the reason why we would like to also express this tau for the other two reactors that is the only reason otherwise you do not really require for design see in the design what you do I say you will either find out conversion if volume is given so volume is I know here okay or otherwise find conversion sorry find volume if I know conversion that is all what you do so that is why why tau is required but you think we should have some psychological feeling that we also know tau and also sometimes it is easy to write in terms of tau rather than V by F A not right yeah V by F A not that is the reason why what we have good and I will ask you this question what is the natural time that automatically comes in the design is it tau or it is t bar we are telling t bar is the mean residence time where it is defined as the molecules spending you know sometime on the average t bar no mean residence time on the average what is the time spent by the molecules inside the reactor okay that is t bar the time spent by the molecules on the average inside the reactor and space time we know I think space time is you know it is the time taken to process one volume of one reactor volume and we also have another complicated thing space velocity the reverse of it unnecessarily we are blaming I mean we are loading our CD okay our hard disk so that is why I think you know all those things are not required I mean as far as I am concerned but still we need that because sometimes you can write very simple equations in terms of k tau like del you know del del is the simplest form of writing complicated equation similarly sometimes k tau will be a parameter where you can nicely put it and then k tau also has a number called damkohler number again I think to respect damkohler we can also learn what is k tau but my question is what is that naturally coming one naturally coming time in the design he says t bar I am not saying he is right or wrong abdul right or wrong but I mean what is your opinion what is that means naturally it has to come you do not have to do anything else naturally it would not come naturally it has come means it has come in the equations naturally without your any problem without any extra effort that is why lounspiel nicely uses a word natural performance measure time for example if you take what is that natural performance measure it is tau bar is it t bar or tau you know the difference you know tau is defined as volume by volumetric flow rate where the volumetric flow rate is mentioned at the entry correct no at the entry whereas t bar is the time on the average actual time on the average that is spent by the molecules inside the system which is the natural measure t bar tau yeah Rajeshree in between you said something tau why do you say tau is the natural measure actually tau is the natural measure of you know performance for any reactor why because it is naturally coming in the material balance equation because what I did this v by f a not I know what is the relation between you know volume by volumetric flow rate c not f not and all that that is why simply I substitute I have not done anything extra there but whereas t bar to find out is not that easy and unfortunately this natural measure will not give me a clear picture yeah but why do you want to check because it is not coming in the design why do you want to check you do very well in the examinations the moment you know that this chapter I am not focusing oh it will not come do not touch okay but why do you want to know t bar where you are not using anywhere where are you using t bar unless in the examination if I ask you find t bar where do you use where do you use tell me because what is the what is the aim of this entire reaction engineering to design the reactor okay to find out the volume of the reactor where is t bar coming you tell me even non idealities where is t bar coming there for the design I am talking for the actual design there also what you do is that you try to measure the amount of non idealities for example if I have dead space okay I have 1 meter cube to volume I would like to find out out of that 1 meter cube that means 1000 liters out of 1000 liters how many liters are in dead space not very actively participating that is all what I want why do I need t bar one way of finding out that is t bar that is okay so that is different so that is why as far as we you know design is concerned particularly this ideal reactors 99% of the time you go for only ideal reactors non ideal reactors we do not go the reason is that particularly mixing flow is very easy to manage you get almost perfect mixing so I have the simplest equation right and plug flow normally for gas phase that is why gas phase reactions we use the flow rates are normally very fast reaction timings are very fast so that is why you will get almost flat velocity profile you are sending at very high Reynolds number through the reactor right that also can be managed and maximum thing what you can add there is axial mixing there also you do not go for dead space and all that so that is why all the time we are in the we are only in the ideal world are very close to ideal world so that is the reason why the t bar and tau where you have to I am telling all this because I am not against t bar or tau what I am telling is when I am naturally trying to find out what is the volume or conversion from the design expressions where I have written material balance and all that t bar is not coming at all it is only the tau which is coming if you want to use tau actually there is also not required why V by F not directly I can get okay good but in spite of that I may ask you a problem in the examination or sometimes you write a you know K tau K tau K tau tau is very easy okay to find out because volume by volumetric flow rate but you have to specify what is the point at which this volume is defined you know you also heard of what is called liquid hourly space velocity LHSV liquid hourly space velocity space velocity of course is inverse time okay so that is not the point but why we use that is again engineering simplification right so you have actually vapor phase reaction good but before making this vapor we have only in the liquid form the liquid then you vaporize then it becomes vapor and then enters into packed bed or may be plug flow reactor or may be CSTR okay depending on the temperatures so at that time when you are doing it for me easiest measure is particularly in industry is liquid right so you now send some 1 liter or may be 10 liters or 100 liters which is corresponding to that flow rate which is just entering the reactor okay but actual basis is this equation where at the entry at the entry but you are not considering that what you do is okay how many liters I will now flow through the rotameter liquid because it is available in the liquid form then afterwards it goes to the heat exchanger from heat exchanger it goes to the reactor now I have calculated what is that vapor velocity that has to enter at that point or F A 0 that is V by F A 0 that comes there but instead of remembering that one I will now remember that is equivalent to how many liters of liquid that is what it is called if you are able to convert this time in terms of liquid velocity then that is called you know liquid hourly space velocity now it has become a old-fashioned no one is using that similarly gas hourly space velocity so that means somewhere else you measure see normally at room temperature it is very easy to measure all your rotameters and all that at room temperature why maintenance is simple it is at room temperature I do not have to worry about you know suddenly putting my hand and then burning jumping because at room temperature if I am able to flow make the gas flow this I will remember but actual entry is at may be 300 degree centigrade 400 degree centigrade what will happen to gas gas will expand I will calculate that is my basis for the reactantism that is the basis not this but for for me to remember to operate this is the easiest one that is why industry people will try to remember those things much easier I do not know whether you noticed or not I mean when I first came to Madras I noticed I mean I am a lousy observer that is my problem because I think once I observe 99 percent of the time I am right in my observation I am not wrong you know looking at people and then trying to find out and I also like each and every small small detail small things so when I came here first you know there are on the mount road and you know our transport is bus only and you know in the bus bus stop people will stand one person will come out there you will say like this okay and all other people will be waiting for the bus to stop so then afterwards I think I observed some four five times and afterwards I just asked someone why only is doing like this and then all other people are standing there it seems that is the signal to driver that he is from PTC the person who is doing this is PTC so even if it is not stop because you know all buses may not stop in that particular stop all the buses so that is why if he says that he will at least go forward and then stop this guy will go run and then climb there that is the signal because that is the easiest one for them similarly in industry also we have some thumb rules liquid actual reaction is vapor phase but he knows that okay by doing this liquid for him so that is why I think there are some I think in even railways also they will say that okay 47 down has gone you will say example 47 down I mean it is only numerical value for us 47 down but he knows that 47 down may be coimbatore express okay that means he is actual name he is trying to convert into the simplest language which they can understand much faster that is what exactly the meaning of this LH SV and also I think GH SV this is question mark I am not exactly remembering LHSV so that is why it is only convenient because suddenly there are many books nowadays in reaction engineering many books have come last 10 years so that is why some books also industry people have written not akramishins so that is why they may be using these terms still so if you are able to see that should not get confused and V by FA not only is the correct one okay where this FA not is the actual one which is entering mass will not change okay so only the velocities will change depending on you have volume large or small depending on what temperatures you are sending and if it is liquid reaction also taking place in liquid phase it is a constant density system again whether you measure here or there or anywhere it must be same volume also mass is same okay good so this is what and tau you know how do you find out volume by volumetric flow rate but that volumetric flow rate normally is V by V not okay where tau equal to I write here where tau equal to V by V not where V not is normally expressed only at the entrance good so how do you find out T bar T bar is the actual flow I mean this is very easy for imagining for a packed bed mean residence time I think I have to cover many things mean residence time T bar this one is called space time tau okay space time tau so tau is very easy to define V by V not and the actual definition is you know words if you want to tell that that is time required for processing one volume of the reactor okay space velocity is inverse of that where you write per unit time how many volumes you can do that okay that is the space time good that is nice so now mean residence time is normally T bar okay so how do you actually find out this so if I have a plug flow reactor which is easy to discuss plug flow is easy to discuss here so then I have here maybe I think it is entering V not it is coming at V F and I may have C A not X A not equal to 0 this is C A F X A F all that okay good now it is a wall variable volume case where one mole is going for 4 moles and it is gas phase gas phase okay by the way if it is liquid phase that I think let me also tell that if it is liquid phase T bar equal to tau or constant density system I find out people say that rho constant density constant okay good so this is very clearly known now we are talking about only variable volume normally gas phase reactions okay so if I look at the gas when it is entering that is the phenomena that is happening and here also I have now tau equal to volume by volumetric flow rate because V not is entering here right so I have this definition already but in the actual case and you know this tau is not talking anything about what is happening inside it is not talking as far as time is concerned nothing is you know is told when we say volume by volumetric flow rate equal to tau now so when I look at this reaction A okay one mole has entered let us say as an imagination okay so then it is moving from this point to this point let us say I have 90 percent conversion of that one mole one mole per liter right so that means here I will have only point one moles per liter as far as concentration is concerned then how it is moving I have here when it comes the moment it enters one mole is going to four moles let us say there is some conversion okay let us say there is I think you know ten moles easy for remembering ten moles just entered and here when I look conversion is ten percent that means concentration is nine okay good so during this time what is happening to the volume because conversion is okay moles entering and moles at that point leaving okay divided by number of moles but during the reaction this one mole of A is producing four moles of R it is a constant rigid pipe so when volume of the gas increasing what should happen to the gas molecules now yeah it has to accelerate the velocity will increase right so when it is increasing with the velocity then it moves faster now our volumetric flow rate V not is it same here no it will change so it will change to somewhere E V V1 V2 like that finally VF corresponding to 90% conversion whatever right so now what is the I mean how do you define T bar T bar is the volume of the reactor if I take a small volume here volume element so the volume of this divided by volumetric flow rate is the time it is the time so when that volume is increasing then it will much it moves much faster because this is not constant because one mole is going to four moles this will expand a little bit when it is expanding a little bit cross sectional area is constant so then what you should happen length should be more that means that will appear in terms of velocity faster right so that is why the basic definition if I take small dt bar is d V by V okay now this is not uniformly linearly expanding why that depends on the kinetics if it is second order it is different if it is first order it is different if it is third order if it is different in between any order it will be different so that is why T bar depends on the kinetics whereas here tau depends on only volume of the reactor divided by entry volumetric flow rate entry volumetric flow rate that is all we are not taking the volumetric flow rate inside whereas I am taking this volumetric flow rate inside this is changing throughout and this change I know I have to find out so I will now try to find out integral of this and here the volumetric flow rate can also be imagined as same for isothermal and constant pressure system V equal to V not epsilon A okay for ideal gases here also we are assuming volumetric flow rate also per unit time okay how it is changing so that is why I have to substitute this V here and what is this V d V change in the volume do you have an expression for change in the volume here recall your memory the first derivation of plug flow what is the basic differential equation for plug flow F A not d X A equal to actually this d V you see F A not d X A equal to okay you see how much time you are taking to tell this it is the basic design differential equation for plug flow and we have only one plug flow nothing else okay that means no practice absolutely no practice okay we do not worry about whether plug flow is increasing volume decreasing volume all that you do not that is a problem so now using this equation this equation this equation calculate T bar I am not telling that I am I do not tell you that I gave the clue in fact in no other class I gave even this definition always I will welcome them with a surprise test all zeros very easy for me to correct okay all zeros I mean I cannot even one right yeah so then I will explain in the class why you got zeros this is how you have to do but now I gave you these clues now you do it but if you are only worried about all other things except C R E again it is easy for me all zeros to put you have to derive now T bar in terms of this and then see the difference between what is tau by C A not and that T bar good very nice that is the definition of T bar and you can now calculate for first order reaction you know what is T bar what is tau for second order reaction what is T bar what is tau for zero order reaction also you can calculate what is T bar what is tau right so all these things you have to make some way some corner in the nodes and then that you know this fellow is dangerous fellow he may give this one in the examination all that you have to write and then you have to prepare for that not one day before now itself okay good excellent so this is again now T bar tau is it clear remember this figure this volume when it is entering it is different you know it expands a little bit then expands then expands then expands that expansion depends on whether you have first order reaction second order reaction and whether number of moles 1 going to 4 or 1 going to 2 or 1 going to 100 okay 100 may not be there but I think large values so all that will come into picture because use that the moment you say someone says T bar your mind should go to that volume that volume is no more constant it is moving different speeds at different locations average of all that speed is called T bar mean residence time that is the thing that is the funda right average of all that because it is not uniform if it is linear throughout then I think you know we know how to easily find out it is not linear some case it may be linear some case it need not be linear okay the time changes depending on the volume expands and depending on the volume expansion time of these molecules change inside the reactor okay good so T bar and tau also now it is clear for you so next one is yeah the reactor design expressions also I have given now good so now using these equations how do you calculate actual design expressions integral expressions these three you have example if I have batch reactor as the first order reaction simple k into ca so what is the design expression for me in terms of integral equation this is the one which you have to integrate but minus r is a general one so finally when you are actually calculating this time that time depends on this kinetics which is first order second order reversible first order reversible second order the reversible second order one what you have done in the yeah in the in that assignment you know to find out the volume of them yeah so but you know if the integration is difficult then what do you go for numerical if numerical also may not be giving you exact answer what is the best one graphical area under the curve but you should know how to count area under the curve if you do not know then it is a problem so that is why I can also give you just one or two equations for example batch reactor is okay we have done for first order second order and all that okay and variable volume also we have done these are all batch so that is why now I will give you some equations for plug flow and mixed flow variable volume because constant density is very easy right but all the time what you have to do is you have to only use these two expressions v by f a not dx a by minus r a substitute here for minus r a k into ca do not try to integrate automatically without converting into x a okay because there may be people who do not know that x a ca different okay simply they may use it right so that is why you have to convert this into the otherwise you also have other possibility of using this equation in terms of concentrations but that is a special case special case like what Swami you can convert one from the other that is not a difficulty that is not the one when do you use ca when do you use x a constant density system because many people may have doubt why this fellow always writing in terms of x a and in earlier course beta we also wrote in terms of ca okay you may have that doubt so the answer for that is that x a is a general expression whether volume change or no volume change that can be used but ca if I write it is restricted only for constant density systems that is the reason why people use this kind of x a equations repeat why do I use x a why people use x a all the time very good that is a specific case so now that is the one so that is why we use only these equations and I will give you integral design expressions or design equations for ideal reactors okay batch reactor already given you may not know I say you may not know I say so that is why because you will search because when I ask you what is the basic differential equation equation for plug flow no one could say no they went again you know flip flap sound I have heard so going to the old papers and then only trying to tell that that equation also you cannot forget f a not d x a equal to minus r a into d v what is for mixing for reactor minus r a into f a not into x a equal to minus r a into v is it difficult to really remember that so easy even for batch also you can write that differential equation okay good so then you have plug flow reactor first order 0th 0th order I do not write first order first order reaction so now there are so many cases here a going to 4r gas phase because you know I am a very tricky fellow my question papers will be very good okay I will give for r r going to I mean sorry a going to 4r but I will write somewhere it is liquid phase reaction okay but immediately you feel that a going to 4r volume change so let us now take epsilon and all that you will get 0 mark okay that is why be careful read properly right so a going to 4r is not automatically gas phase reaction right it can be also liquid phase reaction that is why I have to give if I do not give you scold me like I am scolding you can also scold me no problem okay good so this is gas phase reaction what is the equation what you get so design expressions we are talking v by f a not equal to integral 0 to x a d x a minus r a so this is okay this is first order reaction I told no minus r a equal to k into C a okay here also I have to write it is elementary reaction otherwise you do not know what to write right if elementary only it is first order if I say non elementary then you give me the rate equations or we have to take you have to ask me good so this is equal to d x a by k C a again which is equal to 0 to x a d x a by k I can bring out yeah this is C a not C a not that also I can bring out but let me write first time epsilon a x a this is what you have to now integrate right yeah and again I have not told this one is pure gas or impure gas right if I say that it is only 50 50 percent you know units and all that then epsilon will change otherwise you have to take pure gas means so all these statements must be given there so the equation what you get for this for easy one I am taking k tau tau equal to volume by volume flow rate so this design expression okay k v by v not so in terms of f a not what is that come or f a not equal to v not into C a not so when I write that also v not if I write f a not by C a not okay so this is also equal to please check me k C a not by f a not correct is it right so many subroutines inside okay so this is the equation yeah this is the reason why I was telling sometimes tau we use because simple to write instead of this equation this equation is simple okay yeah now this is equal to this you have to derive minus yeah 1 plus epsilon a ln 1 minus x a okay minus epsilon a x a anyone remember this during their B Tech class now I have to go to 1 2 3 all this is 4 this is 5 so I just gave this example because this is slightly more complicated to write so that is why k tau is easiest one right that is the reason if it is second order second is first order second order this is simple a going to r k C a square or maybe a going to 4r also you can take so there is volume change maybe this is 4r right so you know there are so many cases this can be a going to 4r or the other one may be a plus b this k C square a plus b going to products again okay some change again 4r right but C a not equal to C b not I will get again same equation only you see so many conditions that is why reaction engineering is hell because one small thing changes entire equation changes okay so otherwise the moment I change this is C a not is not equal to C b not then you have again complicated equation that means it is not 1 mole 1 mole initially it is 1 mole and 3 moles C b not is 3 and C a not equal to 1 then again you have the problem so that is why in these 2 cases it is same equation that means C a not equal to C b not so many conditions so the equation what you get here is C a not k tau equal to 2 epsilon a 1 plus epsilon a ln 1 minus x a plus epsilon a epsilon a square x a plus 1 plus epsilon a square x a by 1 minus x a this is the equation just to give you you know some kind of taste for you and this is what you have to derive in the examination if I ask you second order with volume change okay you can simply this is general expression you can simplify when epsilon equal to 0 tell me when epsilon equal to 0 what do you get from this equation good that is what is for constant density system what you remember which is also same for which reactor batch reactor batch reactor also you get so all constant density are this also make a note somewhere all constant density systems batch reactor and plug flow reactor same equations rho constant PFR equations are identical to batch batch reactor okay so I think in a law and spell book many integral expressions are given you have to derive them don't blame me at the time okay so unless you derive on your own definitely you won't understand all that right and the mixed flow reactor is very nice guy mixed flow reactor why it is very nice guy there is no integral there okay so if it is first order reaction let us say k c a square sorry first order reaction minus r a equal to k c a may be I also have a going to 4 r gas phase reaction what is the equation v by f a not equal to x a by minus r a where minus r a is written as x a by k c a square sorry k c a which is also equal to x a by k c a not 1 minus x a by 1 plus x a right so like that for second order also now I will have c a square then I will have now here c a not square all this whole square so like that it is very easy because it is not a integral expression it is simple algebraic expression and if I ask you what is the value what is the value for 90 percent conversion epsilon you will know because I have to give other pure or impure and epsilon you will know x a 90 percent c a not definitely you should know in the beginning you are sending then k from kinetics right and you can directly calculate f a not is known to you and v can be calculated that is what is that performance equation v for given x 90 percent epsilon equal to 3 in this case and c a not equal to some 10 moles per later then k equal to 1 all that you substitute and get v right yeah so f a not is the input and kinetics k and c a you know this first order will give me this k into c a is the from kinetics and this entire equation is from contact right very good so this is how we have to develop the equations and this is very easy for us particularly mixed flow reactor and for mixed flow reactor there is another way of writing the material balance here you got you know f a entering f a not entering f a leaving and what is reaction under steady state conditions what we have done but sometimes we also write for mixed flow this is also let me tell you this is very convenient way of writing yeah here I have v most of the time it is liquid phase reaction okay c a not f a not v c a f a this is volume v right so one way of writing the material balance is what is the entering input equal to output plus reaction plus accumulation this fellow is zero we do not like accumulation steady state people okay good so input is what is this actually writing moles per moles per time so this I can write as v c a not what are the v units push a and what are the units of this moles per meter cube so meter cube to meter cube you will get cancelled you will get moles per second okay so then this is equal to v into c a what is minus r a I mean a reaction k c first order reaction is a first order reaction but in general you tell me for normal any order v into r a plus v into minus r a okay this also gives you a simplest form c a minus c a not minus c a is equal to all that you can this is another way of writing for liquid phase reactions when you write these in term you know without writing in terms of conversions algebra will be a little bit simpler algebra will be simpler