 So, these two are the equations which we have derived in the last class. And you can see the meaning of this equation. This is six, this is seven. Again you can see this meaning. This is the accumulation time. Normally we do not take that into account. Accumulation, because all statistications. So if you do not have a statistic reaction then you have to take this term. And then this is the axial dispersion term. This is axial dispersion coefficient correspondingly that change in concentration with length and this is the radial dispersion term and radial dispersion term will have this because of the cylindrical coordinates, okay. So then this is the actual convection term flow, actual flow and this is the rate of reaction term, right. So in ideal plug flow what we do is we block this, block this steady state and then you will have only these two convection and reaction, okay. So that is why if you say flat velocity profile this u is same across any cross section but in fact it may not be true but still we are taking but the moment you imagine that we have some kind of axial mixing then and radial uniformity is not is assumed radial uniformity and axial mixing it goes like this, okay. So in that case this is block and only this is there of course if you take this also this term will not be there steady state if you take. So but most of the time in the last reactor theory class I have proved that most interesting thing will be only radial dispersion but not axial dispersion, why? Under what conditions I told that? Do you remember? Again this is another quality of our all of us as students. We remember only the current things not last semester what has happened previously what has happened. I think what we remember is only alphabets which we studied in the schools that is all. Afterwards nothing we do not remember at all. None of you remember? RTD, during RTD time we have done that if you have L by DP certain ratio you can safely neglect axial dispersion, okay. Do you remember those numbers? L by DP for 5 percent away from plug flow if you want to be 5 percent away and it is I think 54 or so, okay and if you have if you want to have only 1 percent deviation from plug flow then it is 5 times of that I think 190 or something it comes. So that means L by DP must be 190 that means if you take 1 centimeter particle the length should be minimum so 1.9 meters. So this criteria is easily followed in industry. Most of the time in industry the height of the packet bed will be 4 meters 5 meters around 20 feet, okay yeah and not 1 meter 2 meters. So that is the reason why and diameter of the particle will not increase more than half an inch either 10 mm or 12 mm. So that is the reason many times you do not have to consider this term, okay but most of the time you have to consider this term again why most of the time the moment you want to go to the, best thing is to say that no no I do not care about any non-ideality this is ideal world so though I will follow only ideal world so then I take only plug flow means fine no problem nothing is there only these two steady state reaction, okay. Think I say think think think extend your memory a little bit of thinking yeah Prabhu, Prabhu I will ask I will come to you. The question is clear or not clear at all? I know this is the way that is I say matrix move you are here but literally you are not here you are somewhere else your mind somewhere else, okay what I said was most of the time we can neglect this you know why we have we can neglect this? All these smart question I mean smart answers please leave I say smart answers will be good for MBA, okay but not in engineering I tell you definitely not in engineering all the smartness I think you know management you can use but in engineering you should be solid you should know or do not know, okay in science also, right because I said so what is that I said so I said so many things but are you following all those things, okay there again you do not follow, okay but write down value you will follow because you want to escape temporarily the question so that is why I said you said so that is why it is right as if you are believing me every second every minute I have just explained the length of the reactor if it is more than 190 packet bed reactor 190 means if size of the particle is 1 centimeter particle size, right so the minimum length for ignoring this term should be 1.9 meters so that means L by DP must be let me write 190 L by DP, right yeah so if I have 1 centimeter particle the length minimum should be 1.9 meters this we can prove through residence time distribution studies where we have discussed about axial dispersion model, okay there I proved that so this condition of 1.9 meters and normally we do not use beyond half an inch particles half an inch is tolame so 10 centimeter 10 millimeters also very frequently used in industry so that is not varying much but whereas the length of the reactor will be 3 meters 4 meters 5 meters 6 meters 20 feet also 20 feet is almost around 6 meters, right yeah so that is the reason why axial dispersion will not affect us the conversion is not affected by axial dispersion most of the time in the industrial reactors whereas next I asked the question this term this is the radial dispersion term that cannot be neglected most of the time can you think why first of all why that term came radial dispersion term why it has come there first of all what is that we are assuming there radial non-uniformity, okay good that is why that term is there now just imagine in the actual operation under what conditions we get radial non-uniformity when the diameter is comparable to the size of the particle it will be uniform if it is small diameter it will be very uniform flow, okay if you put only in 1 inch diameter only you know maybe 1 centimeter particle flow will be uniform but only thing is why it is will not be uniform because somewhere you have more why it is somewhere you have large why it is and all that but on the whole in the cross section the flow will be uniform, right but when you have large cross section and not that it is not only large cross section that is one factor most important factor is most of these reactions either exothermic or endothermic you have to either remove heat or add heat when you are removing heat it creates temperature difference from the center to wall because wall is the lowest temperature when you are removing or center will may be the highest temperature because you know reaction is taking place across the cross section and packet bed is a lousy conductor yeah packet bed is a really lousy conductor because you have voidage plus solids and the fluid in between all kinds of things are there so that is why it is not a good conductor if it is not a good conductor naturally you will have gradients, okay that is the main reason if it is endothermic reaction you supply heat from outside and wall will be at high temperature and center will be low temperatures and automatically whenever you have the temperature gradient immediately concentration gradient also comes into picture it is called which effect sorry it effect in mass transfer, okay yeah the other one is dufor effect or something is there which is reverse, okay yeah so that is the reason why most of the time this term is very very important for us logic is simple most of the time these reactions are either endothermic or endothermic either you have to remove heat or add heat so when you are adding heat or removing heat definitely there will be concentration or temperature gradients across the cross section this temperature gradient automatically creates concentration gradients so that is why this term and this term are much more important in normal packet beds when compared to these two axial dispersion terms, okay then the other thing is okay fine so this is the convection term this is the ray term this is again convective heat transfer term and this is the heat removal through the wall, okay and here you have again heat generation term, okay good so these are the equations and these are the differential equation so I have to write definitely the boundary conditions some of these boundary conditions already you know this is C A equal to C A naught T equal to T naught at Z equal to 0 for all R and also have those C A actually all these things are bulk up please remember because it is a pseudo homogeneous model, okay pseudo homogeneous model so pseudo homogeneous model means we are talking about a packet bed where we never feel for the packing that is present there we treat as if only fluid is going through the packet bed so that is why it is bulk concentrations, right somewhere please write down otherwise you may be confusing all these and temperature also is bulk because I did not specify in the beginning the reason is that we already talking we are only talking about pseudo homogeneous models pseudo homogeneous means there is no presence of even though there is a presence of solid we do not feel that there is a solid, right as if it is empty cross section where only bulk flow is taking place that is the reason, yeah so dou C A dou R equal to 0 dou T by dou R, okay also equal to 0 these are all at R equal to 0 for all Z, yeah so another condition is I think I will write here dou C A by dou R equal to 0 at R equal to capital R for all Z I will ask the meaning of this then we have another boundary condition U C A not minus D dou C A by dou Z at Z 0 plus plus U C A 0 plus this is another boundary condition at Z equal to 0 for all R and we also have another without forgetting so if I give the numbers what was the last number only 6 and 7 so this is 8 9 10 11 12 good, yeah so again some physical meaning Kavya is this correct what is the meaning of this equation 8 where are we talking about this boundary Z equal to 0 Z equal to L now it will be very clear this is R 0 capital R, okay this entire thing is packing things go inside they come out, yeah it is very easy now know, yeah so Z equal to 0 for all R what is the meaning of for all R, yeah but it has to be specified clearly what will happen if I do not specify, yeah so that means what when that is only at Z equal to 0 the material is entering by plug flow at the entry level, okay so that is why along the radius there are no non uniformities it is going as if the concentration velocity you know temperature everything is uniform across the cross section when it is entering once it is entering then you have these problems, okay so that is why we have to specify that if you do not specify that then we will have a problem of what is the temperature here because these two equations will give you information at each and every point throughout the reactor at any R at any Z temperature and concentration if you are able to really solve these two equations without dropping out any terms, okay you will get all the information about the packet bed but normally we do not do it because as engineers we would like to simplify and tie our same so that is the reason why we will only drop out some terms important terms and important terms throw them away so that is the reason why we do it otherwise if you are able to do this as a pure mathematician you get all the information solution of this may take place I mean you may get only in 10 years but still you will get beautifully all the information so that is why all these points are very important this one this condition so I mean you are telling something what is this condition symmetrical but where where are we talking at the center because of the symmetric diffusion the gradient equal to 0 so beyond that there is no transport okay so that is what is there for temperature and concentration and this is at R equal to 0 for all Z what is the meaning yeah at any point draw any line there along the z line so this is valid what is this one dosa by dow R equal to 0 at R equal to capital very good no mass transfer through through the wall and which is valid for all Z at any point because there should not be any holes in the reactor okay yeah but this is what is the condition I was telling if it is a membrane reactor this condition will be different you will have some flux going through the wall okay that is again diffusion flux fix you can take that is the one which is going at R equal to capital R along all Z some flux going out for membrane reactors the same equations exactly can be used for membrane reactors also okay good then this one yeah flux balance at the bottom so what is the meaning of this term what is the meaning of this term what is the meaning of this term convection I think I will ask these people these people have been exposed in last time do you remember Rajit Karthik what is the meaning of this at Z equal to Z minus at Z equal to 0 minus just before entering here you have this equation and now you have this and this what is the meaning of that because you know just at the place of entering we are assuming that we have everything entering by plug flow flight velocity profile indirectly coming in the other boundary condition also okay so after the moment it enters because we have axial dispersion term here then this will split into diffusive flux and convective flux I told you know example if I put lot of perfume here and then stay in the room without any like this no still air no there is no movement of air so then it will be only by diffusion which is this by concentration difference if someone puts me my behind any fan then you have the convection so the moment it comes into the reactor because of the dispersion in the reactor you have some by dispersion some by convection and what is the name of that boundary condition blankets boundary condition blankets in fact intuitive boundary condition so excitingly I told you last semester that one because later this is by intuitive boundary conditions later all the people try to prove him wrong and all the you know they could not prove him wrong he was right these are the best boundary conditions for a packet bed right yeah and of course the other boundary condition at the outlet was dca by dz equal to at z equal to l 0 that is also another one which is not required here but still I think I can write okay dca by dz equal to 0 at z equal to l again for all for all okay that is another one this is 13 and equation 11 and 13 are called dank quartz boundary conditions dank quartz intuitive boundary conditions okay you are not feeling well why I can't say for all l why I can't say why why you cannot say tell me the reason what is your doubt why do you feel that that is not true discuss I say no problem whatever you have discuss if you ask Prabhu Prabhu will give a very smart answer Prabhu what spot answer you have for that dca by dz equal to 0 he says at z equal to l but since I can't say for all l I will give you the clue I also told you sometime back you also should have definitely learnt that there are what are called open open vessels and closed to closed vessels anytime remember Reneeta have you know long time I looked at you tell me what are closed to closed vessels at open open vessels not boundary conditions I am just asking vessels how do you define a closed to closed vessel in chemical reaction engineering not English wise English wise close to close to means close to one closed is enough why two close respect to macro flow you know it is a reactor so flow it has to enter it has to come out so both sides you will have to have flow otherwise after sometime it will break with a loud noise because one side you have flow other is not coming out what will happen okay yeah pressure will increase so much than dome open open vessel flat velocity profile not there no but what is the meaning of open open vessel molecule and you can't come back so there is no back mixing at the entrance and there is no mixing what do you call that the bad mixing like there is no mixing what do you call that entrance once a molecule enters reactor it can't come back yeah but what do you call that there is a name no where is back mixing equal to 0 plug flow so definition of closed to closed vessel is the material in any vessel if it if the material is entering by only plug flow leaving by only plug flow then it is called closed to closed vessel okay examples are packet beds in fact how packet beds normally designed is I am going to LKG all the time I say see this is one lousy way of designing packet bed reactor but which is the closed to closed vessel example because yeah here I have small cross section here also small cross section when compared to this right so definitive velocity will be high we know that when we have very high velocity you are moving towards plug flow right so this is the reason why we will say here that I have at all r because r is only here okay at all r I have the plug flow that is why at all r dca by dz equal to 0 at the outlet okay and I gave the explanation again I do not want to give the explanation that takes I think minimum for me 15 minutes to explain that why dankwatt said this is okay easily understandable even dankwatt could explain this one very easily but in this place why dca by dz must be equal to 0 he has beautifully physical intuition he has had and then he explained that in terms of dca by dz equal to 0 is the correct boundary condition at the outlet you have to read that 1953 first paper on residence time distributions we also need some kind of history I say you should know who are your grandfathers in chemical engineering okay some of us are already grandfathers because we are about to retire okay we may not be that great chemical engineers but still professionally grandfathers but great grandfathers also you have to remember dankwatt I think that is a beautiful explanation what he has given 1953 paper I am not I have that paper but I do not want to send to you because if I send it it will be e pollution unnecessarily because I know I will send it to you it will be on the computer you will never download you will never read why unnecessarily wasting computer space so that is why otherwise you know if there is interest I could have sent many many papers I have some wonderful papers one or two dankwatt papers not dankwatt denby papers wonderful papers where without maximum mathematics with minimum mathematics how beautifully he has told yesterday I gave that condition know dynamic program condition where 0 to you know in and out dough of dough dough by dough or of 1 by minus r a into dx a equal to 0 intuitively he told that that must be the condition for optimal design of any reactor I mean either plug flow reactor or packet bed reactor later people proved that dynamic program through programming you get the same condition that is what is the great intuition for engineers that is required scientist can have time to do things and then finally you will say that yes is intuition and also mathematics both are matched but engineers without mathematics also they should have intuition which later can be proved by mathematics and if you have that kind of intuition then you have your excellent engineer okay because every time if you start proving you will not have you will not have time to do things you know as engineer engineer has to run the world okay I told you know I have my famous thing engineers run the world doctors save the world yeah and yeah what the people like Brahma Vishnu Maheshwar you know we have three gods you know very famous gods Brahma Vishnu Maheshwar yeah so what is the job of Brahma create what is the job of Vishnu protect what is the job of Vishnu I mean Maheshwar destroy so like that we also have this you know as engineers we run the world we produce the way we produce things also and then run the world so Brahma and Vishnu in one place engineer okay really same half half 50% 50% otherwise you choose you know if you are only theoretical engineer not producing anything you are only half of the body of those two yeah and anyway there are so many things which I can tell you but I think I have to I thought I will complete packet bytes today I am not able to even start packet bytes today okay anyway so all these equations yeah this is the reason why you know we can say that if I have this kind of flow right so then I can say that I have at all r the dispersion is 0 flat velocity profile then I can say that condition is right that is only valid for closed closed condition okay yeah then we can have the open open where the material enters by dispersion and also leaves by dispersion so that is an example of tubular reactor normal tubular reactor where we have this is the tubular reactor there is no small tube here and another small tube there for things to come out good so this is the tubular reactor where material enters by that means same dispersion whatever is happening inside here if I have the dispersion same thing also is happening here same thing is also happening here so this is called open open vessel open open vessel and this is called closed closed vessel okay good I think this class has become a joke class and I want to finish these packet bytes and then start fluidized okay good so now we have the terms normally what we have to use okay this equation 6 and 7 we can choose that means if axial dispersion is not important this may not be there this may not be there and if radial dispersion is important this will be there and this also will be there because both are connected okay but under one condition both are not connected this you may have radial dispersion and you may not have okay for adiabatic reactor I think I instead of asking you that answer for adiabatic reactor what will happen which terms are important in this adiabatic reactor means there is no heat input no that means you are not taking anything from the walls for adiabatic reactor which terms are important equation 6 and 7 important or not important that is not there adiabatic means totally this is not there okay so out of the rest of the terms also some will not be useful some will be useful because there are so many conditions here in this those two equations you can imagine for example if it is adiabatic reactor and let us say adiabatic very small diameter what are the terms which are important I think this you should be able to think adiabatic system but diameter is small this one that means actual dispersion term is there radial dispersion will not be there why because diameter is small you can expect uniform things okay so then this one will not be there this is the main convection okay then this is the term okay but now relance number is more than 40,000 in that reactor yeah this also will not be there steady state this also will not be there okay so depending on each condition is a flow condition other a heat transfer condition whether you are heat removing a unit adding or adiabatic so depending on that you have to choose the corresponding terms okay so like that we have to simplify them and I think I have to give the conservative equations where normally people use okay so tomorrow I will give I have a table which I have taken from H H Li there is a book heterogeneous chemical reactor design heterogeneous reactor design H H Li okay good book so that is why I know my notes always will be from various books but I am mentioning Carberry I will take something and Fremont and Bischoff I take something Li I take something this information is good for us but I think what is the uses I am disappointed most of the time because after the examination is over or even before the examination is over if you know that this is not coming in the examination you never bother to remember Prabhu sad no what is the ultimate idea is ultimate idea is when you are a chemical engineer there are some things which you should know okay at the end at the end of chemical engineering but now I think you do not know chemical engineering spelling also you will forget very quickly I do not know whether chemical engineering spelling how many people write because most of the time we write camping no that is all I think everyone thinks that that is the expansion of chemical engineering that is why it is very very sad otherwise there are so many nice things okay and I was also telling in the department you know like see B Tech now I think B Tech have anyway it will come to M Tech also it is not very far B Tech you have a choice now you know not to do the project and go for only courses yeah I discussed in the department I think in chemical engineering we are so lucky we have so many flow charts so many problems and if you give that problem as a project student will learn so many things in that see one project can be solving these two equations I said where I get fluid mechanics here velocities and I get heat transfer there I get reactions there I get transport phenomena there because dispersion and all that that is correct you know that is traditional way of doing chemical engineering project throughout the world but I think in IIT last 25 30 years I have been fighting this so give them that kind of projects where they learn everything but no no no these people are very very intelligent so we should not give them that project we have to give the research project research project anyway they can't do so finally what is the solution so no project so really that is why so many things they will learn if they solve the problem I mean this kind of equations in the in their project same thing after five years I tell you M Tech students are not interested in the project that is the logic why B Tech students are Senate also accepted unfortunately Senate also thought that when they are not interested why should we give them project let them take courses so that is why you also don't do the project now M Techs okay so after five years everyone will realize that M Techs also not doing then coursework another four courses you don't learn anything anyway so that is the kind of thing what is happening so so many interesting things are here okay maybe I think I will tomorrow I will I think you can now run