 yesterday we had been discussing about design of noncatalytic gas solid reactors and we again went to my diagram that is kinetics contacting input output and all that so then we have also discussed what kind of contacting patterns you will get depending on the type of reactor you are taking that means if it is a blast furnace rotary kiln fluidized bed moving bed all various reactors what you have for noncatalytic reactions వచికంలు మవిన మారానినిని నామ్నంసంక ఆల్లానినే కాకుర్లినినినీ వనోరి పెరికికుటంపంది నెరినిర్తిదినంసి ఉపర్లాచిసి నంమదిం. to summarize I will give the equations which can be or which how to be used for the design. So the kinetic expressions what we have are kinetic equations for gas solid non catalytic I will write again NC non catalytic reactions. So you know if it is only film control I think anyway for 3 control you have the big equation and that has to be easy at the end but individually if I write if I have film control what you have is equation tau sorry t equal to rho b r 3 b k g c a g x b this is the one which I ask you to design. So tau is rho b r 3 b k g c a g we know this equation comes when x b equal to 1 and t by tau is x b this is very nice simplest equation t by tau good. So now if I have diffusion control ash diffusion control number 2 you should remember this otherwise in the examination you will have the problem okay. So t alone individual steps rho b r square 6 b c a g d e so this is only slightly complicated 3 1 minus x b to the power of 2 by 3 2 into 1 minus x b this is the equation this equation is 4 and tau is rho b r square 6 b c a g d e so this is equation number 5 then t by tau equal to 1 3 x b to the power of 2 by 3 2 into 1 minus x b so this is equation this is equation number 6 there is a meaning why I am writing all this okay then you have 3 reaction control so here t equal to rho b r b c a g k s k s into 1 1 minus x b to the power of 1 by 3 this is 7 tau equal to rho b r b c a g k s this is equation number 8 now t by tau for reaction control is 1 1 minus x b to the power of 1 by 3 this is equation number 9 and all this is for constant size particle right okay constant size particle of course here you can also list out there just leave a space and then you can list out all 3 controlling that equation also you can just leave it there I am not writing that and now if you have changing size if I say that this one as a b changing size particle that means during the reaction size of the particle continuously changes and here we have for film control film control you have t equal to rho b r b k g not c a g into 1 minus 1 minus x b to the power of n plus 1 by 3 divided by n plus 1 this is equation number 10 so tau is rho b r b k g not c a g n plus 1 this is equation number 11 then t by tau equal to 1 minus 1 minus x b to the power of n plus 1 by 3 correct now this is 12 I hope you know the value of n for small particles large particles n varies from half to one okay depending on the other size small particle and large particle and next one is reaction control for reaction control t by tau is given by equation 9 t by tau okay for reaction control t by tau is equation 9 do not have to repeat again so these are the equations you should remember for the design because this is the kinetic models right so which will give you the relationship between x b and t by tau or t because tau also you know good yeah so once I have this then the next one is contacting right contacting and yesterday we have discussed about contacting saying that whether you have blast furnace the flow through that for both the phase or plug flow if you have rotary kill again both phases move in plug flow and if you have fluidized bed then solids may move in mixed flow and gas may move slightly difficult to find out and when we are doing fluidization fluidized bed reactors then you will know exactly what kind of pattern you will have that I will tell you later but to start with if you want to assume you can assume when you are near minimum fluidization velocity it will be almost plug flow right I mean there are people who can assume even the mixture flow for gas also but that must be at very high velocities of gas you know in fluidization there is the minimum velocity that is required for fluidizing the particles to keeping the particles in suspension right and there are very high gas velocities where almost terminal velocities are reached so if you reach terminal velocity it is no more a fluidized bed it is a transport bed you know all the particles will go out okay so that limit and all those things we will discuss that later and if you have a moving bed moving bed means you take a cylindrical tube may be one foot diameter and height may be 20 feet 30 feet 40 feet and you may send for example your iron ore slowly from the top right and from the bottom you may send your hydrogen or CO so then this will react gas will react with the solid and you should have sufficient conditions for the reaction to take place then what kind of contacting pattern you can take for plug flow sorry for moving bed both are plug flow okay because the moving bed you are allowing such that you fill up the entire column as a packet bed and then just remove the you know some small hole you know with a control of the wall so that fixes the actual flow rate flow rate from the top and bottom then the entire bed will be slowly moving down that is why we call it as moving bed reactor so that is why any reactor you take we can always break into our basic contacting patterns whether mixed flow or plug flow so once we do that then we should also think one more thing we should also take one more thing into account this I have mentioned in the beginning as usual I think you may not remember I told you that our most important phase is solids in non catalytic reactions in many cases okay almost 99% of the time we are more worried about what is the conversion in the solids rather than what is the conversion in the gas gas phase so that is the reason why solids are our important phase and once you have solids as important phase then we have to treat the solids as what is called macro fluid I don't know whether you have okay I will write here macro fluid okay and that is why the knowledge in reactor theory is also important here if you have forgotten what is macro fluid and how do you get an equation for macro fluid all that then it may be slightly difficult for you but I will just try to give you the brief connection there okay good and I have also given you the formula sheet for reactor design you know sometime back in the class earlier so there also for macro fluid what is the equation and how did you derive that that is very important here so conceptually what you are thinking is that in one of the problems with RTD is that it cannot solve or it cannot give complete information to calculate conversions and RTD can give the information only for first order reactions why because first order reactions conversion depends on the time and RTD gives you that kind of information so direct combination there so RTD gives me the time what is the definition of RTD you know the time taken by you know the fraction of material which is coming between time t and t plus delta t so if I look at my reactor and if I want to find out what is the fraction of material that is coming between 10th minute and 11th minute right so that fraction is ET DT where ET is the exit rate distribution function and DT is the time interval that is the fraction so now if I can add all those fractions then I will know that the total amount of material how much time it took then I can take the average of that then I will take the mean residence time or average residence time all that you can get from RTD but if I take other than first orders in second order okay there is one more parameter which comes into picture that is called mixing whether the mixing has taken place early or mixing has taken place late and one of the beautiful examples that was given for this to demonstrate is this I have a plug flow reactor then followed by mixed flow reactor and just reverse the same thing yeah these are the two models right so this is ideal plug flow ideal plug flow ideal mixed flow ideal mixed flow right so I have this system two systems and if I conduct a first order reaction whether here or here I get the same conversion right and if I conduct the second order reaction this will give me more conversion and this will give me less conversion so the idea here is that here in the second order reaction we have to keep the concentration as high as possible because we have kCA square as the rate so if CA is large then you will have more and more yeah more rate of reaction and then more conversions correspondingly okay so that is what is happening that means you should not allow mixing for higher order reactions greater than one okay you should not allow in the beginning itself so that means you should delay the mixing as much as possible delay the mixing here it is delayed first I have plug flow where by definition there is no mixing we are talking about action mixing okay good so whereas here if I conduct the same second order reaction it gives me low conversion because here at the first instant itself you have the concentration decreased because of perfect mixing the idea here is when fresh concentration you know CA naught is just put inside then this will mix with already there is a continuous flow continuously reaction is taking place so this fresh CA naught is mixed with already converted stream within the reactor where the concentration falls right product has a different concentration in the outlet or here it is same concentration so that is why it now gives low conversion here and orders other than less than one this will give you more conversion and this will give you less conversion it can be proved and you can derive on your own it is not great E equal to MC square type concepts here it is very simple so that is why you can derive your on your own and then try to find out so that is why then now the question came here that okay you are telling about this is late mixing and this is early mixing okay this is late mixing and this is early mixing can you say how late it is or how early it is because early and late are only you know relative terms I think they are loose terms general English only you know we will use that but you have to be very precise in engineering to say that late means how many days late how many hours late so that is why we cannot get that kind of information in these two systems right so that is why we imagine a fluid where if I have broken up all my fluid which is continuously going inside in terms of packets where these packets are not allowed to coalesce or allowed to break I hope you know what is meaning coalescence and breaking of course you know okay yeah so we should not allow that means once the packet enters as a packet it comes just like that outside okay this we call it as segregated fluid okay or we can also call this one as macro fluid macro fluid and the other extreme is the micro fluid where we have individual molecules freely moving throughout the reactor so that means any molecule if there is a chance for reaction any molecule can react with any molecule because there is perfect mixing in this perfect mixing is possible in micro fluid you should not get confused here but even this micro fluid there are two restrictions the status of the fluid itself fluid can easily move in the form of individual molecules that means there is a possibility for the molecules to interact with any molecule in micro fluid right but that is one restriction the other restriction is that suppose I put that in a plug flow reactor these individual molecules micro fluid now I put in a plug flow reactor but by definition plug flow reactor is not mixed reactor you are not allowing I mean the molecules are capable of moving but your system will not allow the molecules to move that is why even micro fluid behaves as macro fluid there okay so in a plug flow whether I have micro fluid whether I have yeah macro fluid or macro fluid it does not matter for me because the treatment is same what we see is every particle is entering and exactly leaving at the outlet okay with the same time whether it is packets or whether they are individual molecules so the residence time is fixed for the packets that is macro fluid or for the individual molecules and the reaction is taking place when it is slowly moving and that is why we say that we have infinite concentration across the cross section okay I mean not infinite concentration infinite mixing so that means infinite mixing idea is here at any cross section if I look my concentration is same but that concentration is varying with axial direction okay so now good so as far as these individual molecules are concerned we have understood now okay if these individual molecules are we are able to put in mixture flow reactor they are capable of easily mixing so actually it will get diluted so that is why you get also less conversions now how do you put this same information with macro fluid I will now take these packets of molecules and then pack here and okay continuously I feed and then they come out now even in the packets the concentration is uniform because if I look all the packets will be exactly moving like this okay exactly moving like that so then I can find out all the concentration here is same concentration here is same concentration here is same okay within that line so again is easy for me to imagine for plug flow but if I put this same packets here in the mixed flow reactor then by definition the residence time distribution that means molecules coming from the start to end it is 0 to infinity that means some molecules may stay almost 0 time and some molecules may stay infinite time that is molecules wise even if you take packets some packets may stay 0 time some packets may stay infinite time and each packet now you can imagine as a small small batch reactors they are not allowed to mix with any other packet right so that is why the reaction that is going on is only limited to that packet that packet alone right so now the concentration or conversion in that packets depends on how much time it has spent inside the reactor so some may spend only 5 minutes some may spend 50 minutes some may spend only 5 seconds so in 5 seconds packet the conversion is almost 0 I mean let us say very very small or 5 minutes you will have some corresponding conversion or 50 minutes almost the conversion would have been completed okay conversion should have been 1 so that is the reason why the average of all that that is the one which you see outside and I can also tell you like a story you know like a story only I am now trying to tell you so now we can tell that if I have macro fluid okay and if I have second order reaction second order reaction macro fluid mixed flow reactor remember 3 conditions mixed flow reactor second order reaction macro fluid and the other case is again the okay mixed flow reactor now not micro fluid micro fluid not micro fluid micro fluid and second order reaction which one will give you more conversion okay you are telling something why yeah so what is happening correct answer is right yeah excellent excellent concentration is retained by that packet as high as possible so that is why you get more conversion when compared to micro fluid right okay I think this is the general information I think this has to you have to remember all this is it is a wonderful concept and remember this which will make everything clear okay so that means if you put first PFR and then MFR and reverse it and you will see that here you have more conversions for second order here you will have less conversions for second order and then the reason you have to question why why that is happening that is happening because that is late mixing this is early mixing for second order reactions so then everything falls because I cannot define even now we do not have complete picture in reaction engineering how to get that information we are asking that means you know one extremely you are discussing that means you are allowing only packets in one case and all individual molecules in other case but in reality you may have some packets and some molecules individually then under those conditions how do you do that oh many people tried wonderfully and there are very nice models and all that but still we are not perfect in trying to get the conversion in a chemical reactor okay now what is the connection between this this what I told and also now the design in non catalytic reactors right the I told you that our important phase in most of the time non catalytic reactions is solids would you take solids as micro fluid or macro fluid yeah if I take iron ore particle of let us say you know may be one centimeter then inside that we have so many yeah molecules of Fe3O4 right so that one is like that you can imagine all the particles are as macro fluid right so once you have that macro fluid now how do you find out the conversion for this macro fluid even here if I have this I told you that imagine that only we have mixed flow reactor the other one also we can imagine but this will give you slightly more information then continuously I am putting the packets and continuously they are coming out under steady state conditions but when I am concentrating at time t equal to 0 let us say 100 particles 100 macro fluid packets I just put it there then out of that one may come out right immediately then afterwards may be 5 minutes 10 minutes all that you may come out now what you are trying to do at the end you have to average the concentrations of all these packets within that batch what you focused so like that continuously anytime you imagine that is what is happening continuously I am feeding the packets continuously they are coming out but depending on their own residence time you will have the conversions and all that average conversion is what I see at the outlet okay so imagination here is that all those packets have been broken into molecules and then try to find out what is the conversion there so then that is what what we see but actually what is happening is each individual packet will have a different conversion you are averaging that all that so that means what is that mechanism what is happening there I have let us say between may be 5 minutes to 50 minute to 60 minute how many packets have come what is the conversion in that okay how do I get the information between information on 50 minute and 60 minute how do I know how many packets are coming do you have information or you do not have information question is clear or I am just asking I want to find out in any reactor anyway here you think mixture flow we are talking so what is the what number of packets are coming between 50 minute and 60 minute do you have that information RTD instead of packets instead of fraction I said packets that is all so RTD will give you between any two times what is the fraction of material that is coming out and now the kinetics if you have first order or second order that kinetics will tell you now what is the conversion in those packets right so between 50 minute and 60 minute I have 100 packets or the fraction may be 10 percent of the material okay so within that what is the conversion and that conversion is dictated by kinetics so that is why kinetics and contacting this RTD is nothing but contacting right and if I take ideal mixture flow reactor I have a specific RTD right because mixture flow I am telling so when I have mixture flow specific you know reactor then I note definitely my RTD the reverse is not possible that is also a wonderful point which we have discussed at last time so that means given a reactor you note definitely what is the RTD okay here if I have if I give you I have a mixture flow reactor then you know the equation is yeah ET equal to E power minus T by T bar by T bar that is the equation what you have right but if I give you this equation and then ask you what is the contacting pattern you can generate infinity infinite number of possibilities are there is again wonderful point I think you have to remember okay we have discussed that in the last semester so you understood the question know I think it is very nice given a reactor you know definitely what is RTD but given a RTD function RTD equation you can never say what kind of reactor you have but only one possibility one condition is that if you have plug flow direct delta function no other system can give you direct delta function direct delta function is the RTD for plug flow no other system can give you that direct delta function except plug flow but whereas with mixture flow there are many many possibilities there is nothing new or totally weird thing what we are discussing because it is already been discussed by Levenspiel for long time in his book okay all other books also right but Levenspiel gives that you know beautifully so that is why I think you have to read once more all this information okay so that is why when you go for actual writing the equations then we need these kinetics because this is what is gives me what is the conversion that is going on in each and every packet this one all these equations depending on film control depending on ash diffusion control or reaction control or depending on three controlling that is also right that will that will give me in those packets what is the conversion and my ETDT will give me what kind of reactor we have okay so based on that only we have derived that equation for segregated fluid okay design expression for macro fluid okay here we have the macro fluid for solids yeah we have 1 minus X bar B equal to 0 to infinity 1 minus X B of single particle single particle into ETDT so this is equation 13 so this is the equation I have written this in slightly different form then what I have given in the that formula sheet okay so actually what is this one here yeah CA bar by CA not and this is CA bar by CA by yeah CA by CA not batch okay here it is single particle now you can see yeah if I have this ETDT for plug flow reactor so then this becomes direct delta function if I have mixed flow ideal reactor so then again you have here e by yeah e power minus T by T bar by T bar so those things you have to just substitute there and it is not only ideal reactors any non-ideal reactor also this is true yesterday Satyam was asking me for in blast furnace where if you don't have ideal plug flow what do you do if I don't have ideal plug flow what is the nearest non-ideality that comes for plug flow axial dispersion we have an equation for ETDT or ET for axial dispersion that is what you substitute here if I don't have ideal mixed flow reactor then we have a mixed flow reactor with dead space because mixing is not perfect and that happens many times in fluidized bed also because some part of the bed is not that vigorously mixing that vigorously fluidizing and some other part in the bed because in industry you will have 3 meters 4 meter diameter fluidized beds okay 4 meters I think you know 3 4th of this room that the diameters what you have so some part it may be very active some part may not be active because of non-uniform distribution of gas so that is why you have dead spaces there so you also have a model for ET if I have dead space and bypass okay so that is why you have all that information and I also gave you last semester that you have so many models also possible combination of models for ETDT you know for residence time distribution models I also gave a handout there at the time so that is why all that information can be used in this equation right okay so normally what happens here is that when I put 0 to infinity as the upper limit is it really realistic or we have to change that value physically what do you mean by infinite time what will be X B for infinite time yeah so that entire integration disappears so then what is the logical upper boundary excellent tau tau is the logical so this equation is 1 minus X bar B this you write first then this is 0 to tau see beyond tau you do not have X B equal to 1 so that function disappears okay yeah so that is why you have to put only this maximum is tau this is single particle ETDT this is the actual equation yeah and of course now if I have this ET for ideal plug flow or ideal mixture flow now I can substitute and then get the values okay this is the general expression for yeah I think I will actually write maybe tomorrow all these things the actual equations we write for plug flow as well as mixture flow reactor and what I would like to tell is here this X B is not an easy function except for this correct no this X B is simply 1 minus I can add 1 minus 1 minus X B okay so this 1 by see this equation yeah I think I do not know whether you have realized that or not ETDT is in terms of time and X B also have to convert in terms of time so that is the relationship what I have been telling you know you need the conversion time relationship for kinetic model from kinetic models so that is why this equation gives me that relationship 1 minus X B now we express it in terms of t time tau is constant anyway that those things I know how to calculate so that equation you have to substitute here if it is film control here I have to substitute 1 minus t by tau here and then here I know if it is a packet bed or I mean plug flow or if it is ideal mixture flow then I can integrate that equation integration is not easy I think it is normally complicated equations only you get okay now you see diffusion as diffusion control yeah so this is the equation which I have to solve first I have to get from this equation 4 1 minus X B that 1 minus X B should be expressed in terms of t or t by tau okay this equation equation 6 this is easy instead of writing all that things so t by tau so this X B 1 minus X B has to be solved and then substitute there and integrate so you know this is a cubic equation now and then it is not that easy and even here the other one here yeah this is easier than this okay so that is why I have written all these equations in this fashion so that you will have immediate relationship between this design expression and then that this is the final design expression what do you get from this if I give you time may be you know this yeah we have in terms of time here right if time is let us say 20 minutes 30 minutes 40 minutes what is the conversion okay or of course if you substitute after the integration these limits that will be in terms of t by t bar t bar by tau tau is the tau is associated with individual particles please remember that tau is associated with individual particle t bar is associated with reactor t bar is associated with reactor yeah I think may be you did not catch my point here if ET if I write as e power minus t by t bar by t bar this is let me say clearly m right so this t will disappear because it is a definite integral okay this t small t then you will have in terms of t bar m and tau okay so that is what so you will have the equation only in terms of t bar and tau tau we know anyway okay that is no problem now given x bar be calculate t bar t bar is associated with again I am telling you the average residence time of the particles how do you calculate volume by volumetric flow rate right but in this case it is not volume by volumetric flow rate it is hold up by w is hold up kg is by f what we call kg per time so t bar I know now okay so we have to find out this t bar if x bar is known if t bar is given to you I will say that okay I have so much you know they hold up inside the reactor and I know what is f the flow rate that is actually your input f is your input here okay so then I can calculate t bar using that t bar I can now calculate what will be the x bar given is the same thing same design if you know the volume calculate conversion if you know conversion calculate volume you cannot do anything if you do not know both if you do not know either this or that okay at least one you should assume one you should know okay so I think we will stop