 so till the last class we have discussed about the reaction rates we started with that 7 steps we started with adsorption desorption surface reaction that is the pure reaction rate without any mass transfer and heat transfer coming into picture then we have taken interfacial gradients and interfacial gradients and then finally now we have a rate expression so if i go back to my original diagram which i have been drawing all the time that is ya kinetics all that we have come there so here we have input output kinetics contacting here we have chemical physical batch and again we have p f m f all ideal contacting then the performance equation output as a function of input contacting kinetics sorry kinetics and contacting ya this equation also we have seen what we have done was this one thoroughly entirely for the heterogeneous catalytic reaction so here we have kinetics for design finally you need the input which all of us know how to find out kinetics which will give you minus r a and then contacting what kind of reactor you are going to take and fortunately for us in a continuous system we have only two most of the processes are only continuous and then that is why these two reactors one of them you should choose and contacting will give me type of reactor kinetics will give me the rate and that rate only thoroughly we have done using chemical and physical aspects chemical aspects is the intrinsic rate what you have derived for using langmer initial input kinetics okay that is minus r a course you know all the time we will say it is only first order reaction but it can be any order or there is no order at all for most of the reactions so this entire thing will give me minus r okay so now if i take for example a packet bed reactor where it is a plug flow reactor then the equation for designing packet bed is this in terms of w where w equal to weight of the catalyst 0 to x a dx a by minus r a okay this entire expression is due to the contacting pattern right and in this expression i have minus r a kinetics and in this expression i have input the f a not okay good so this is what now we are trying to go to we are trying to do now next few classes that means we would like to have we have to design using this information the packet bed reactors or fixed bed reactors right packet bed reactor or fixed bed reactor i do not know which one you like you like packet bed reactor or fixed bed reactor which has got more meaning packed a why catalytic reactor ya why do you say packet bed packet bed is better than fixed bed just like that is there any meaning for packet bed and fixed bed ya packet bed can mean i think more correctly if you say fixed bed is the correct one because bed is fixed bed is fixed at least between two two plates distributors otherwise it can fluidize and the entire bed also packet bed also can move right entire packet bed can go up that is what what we call moving bed and it is called entire bed so fixed bed is the correct word even though many times we say packet bed packet bed packet bed okay good so now for the design of this you can go to any level whatever you want okay i will just describe one i think now nowadays one of the new subjects are the future of chemical engineering people say i also believe that is multi scale chemical engineering multi scale right so multi scale means you have to start from the smallest may be from nano or less than nano to to the almost to the planet level and planet level so that means almost chemical engineers are associated with molecular level to planet level so i can give you the example for example taking catalytic reactor packet bed reactor itself what is the starting point for us starting point for us the chemical reaction which occurs at molecular level if it is a catalytic reaction then you will have a catalyst that catalyst you have to prepare how do you prepare catalyst again various techniques we have discussed may be crystallization one of the techniques and if you take one specific example like zeolite zeolites can be made beautifully through chemical reactions whatever structure you want you can get it depending on what kind of reaction you are going to conduct that means you can control the structure of force using zeolite catalyst for many other catalyst still is not possible to have that kind of structure okay so then if i have a particular structure known to me and also my molecular size my molecule size known to me these two so i will take a zeolite catalyst with some force zeolite crystals for example it is not even pellet zeolite crystal i have to make first these crystals have the holes the pores so that pore naturally should be slightly larger than the pores larger than the catalyst sorry the molecular size okay now what happens if you are theoretically if you want to theoretically find out how many molecules are getting diffused you have to go to quantum mechanics level because these molecules are driven by thermal forces at molecular level and because of this thermal forces the molecules will vibrate translate and rotate right so for example you have to find out the speed of these molecules how they are going and entering through the pores if you want to calculate the rotational speed okay then you have to use this Schrodinger equation to find out or even just before that if you temporarily assume that there is no catalyst there then you have the kinetic theory of gases and at room temperature you know that what Schrodinger also has given this value i think you can calculate also that 200 meters per second at room temperature the molecules can move according to kinetic theory of gases they are moving in this room also in that same speed what only thing is we are not able to see okay but when you are having more and more molecules that means dense gases then what happens this free flight you see at what level we are discussing the free flight of individual molecules are restricted because there are many obstructions because other molecules are there it is like i do not know whether you have experienced this i have also not gone long like 20 years back i have gone now it must be worse there is a street in Chennai called Ranganathan street Ranganathan street or Ranganathan street have you gone to you would have got okay that is what is highly dense packed packing okay you cannot move you have to make really effort to go through that packing so when you have that kind of so many molecules very close the free flight of this individual molecules will simply become the phenomena of diffusion otherwise that is why kinetic theory of gases is for more number i mean less number of molecules per unit volume dilute gases for example if you go to very very dense gases then the free flight of the movement of the individual molecules will become diffusion that means it has to fight to diffuse through the other molecules okay that means diffusivity coefficient will automatically come there and Schrodinger equation will automatically come there because you should find out the orientation of the molecules that means rotation and all that so that the molecules should properly orient to go into the pores otherwise the molecules particularly when you are talking about our catalytic cracking then you have the long chain molecules long chain molecules when they are moving in this direction they may not enter chain is in this direction so they have to rotate in this direction and then try to enter through the pores that is why in jewelites most of the time i have told you you know three types of diffusion right what are those nudes and diffusion bulk diffusion and configuration diffusion and the specialty of configuration diffusion is that the molecule is always in touch with the surface of the catalyst and whereas the bulk diffusion it may not be even seeing the catalyst surface that means the pore wall whereas nudes and diffusion nudes and diffusion what is the size of the pore for nudes and diffusion no we have talked this about in mean free path ya if mean if the pore diameter is almost around five times the mean free path of the molecules in kinetic theory of gases then you will have the nudes and diffusion but our idea should be to go for configuration diffusion why because each and every molecule should easily see the wall because the reaction is happening on the wall so when the reaction when the molecule goes and gets absorbed there is surface coordination between the molecule and then the surface then because of the energies involved bond energies some of the bonds will be loosened okay they are broken then the molecules will form other molecules will form you see all this is at molecular level and we never bother to know all this 50 years back because what is that i want there i have a catalyst send the gases analyze the gas find out how much is converted but now because of the efficiency you know to improve the efficiency this level also people are thinking that is one scale nano scale right sorry i think even less than nano scale around nano scale only all the phenomena what i have described so now you have the pore and through the Schrodinger's equation and also kinetic theory of gases you can try to find out now how the molecules oriented and then trying to diffuse into the pore so now diffusion automatically comes there because is the configurational or nudeson or bulk because most of the time we would like to be configurational diffusion because surface area should be always in touch with the molecules or vice versa right so after that then these catalysts crystals geolite crystals you make them as because crystals as it is i cannot use in the reactor you have to make them as pellets okay so now these pellets may be of 3 mm diameter or 4 mm diameter 5 mm diameter right so when they are making this one as 4 5 mm diameter then the diffusivity will come automatically into picture so that means you know Thiele modulus right so that means what is the total surface area now i know how to calculate at individual molecular molecules level and then pores of individual crystals now all those crystals came together as a particle right now i have to say that yes now what is the diffusion that is going through this and there is a beautiful definition which i have i think did not occur to me at that time to tell you the bigger the particle you will have more time for diffusion right i mean it takes more time for diffusion and that is why there are 2 times involved diffusion time and reaction time which one should be large and which one should be small you have 2 times diffusion time for the molecules to diffuse through the particles and reaction time for the reaction to happen which one should be smaller yeah if you are able to make diffusion time much much smaller than the reaction time that means all the surface area is already used i mean it is being used by the molecules so now that is the thing the in fact the ratio between the ratio between diffusion time and reaction time is nothing but square of Thiele modulus i will just explain diffusion time i think i will write here diffusion time tau d is r square by d where d is the diffusivity r square is the radius of the crystal okay when it is diffusing through okay or the particle right okay and tau r reaction time for first order it is very easy to remember how do you find out tau r for first order reaction time i am not talking about integral C t and all that what are the units of k time inverse so that is an indication of first order reactions this is the time this is seconds here also i have seconds because this is meter square per second meter square there so the ratio between these two tell me is it Thiele modulus tau d by tau r x yeah phi square phi square that is a beautiful definition again this did not occur to me when i was discussing earlier okay so this is wonderful that means you should always try to have you know always effectiveness factor one means the phi should be as small as possible okay good that is nice so that is the level what you are talking and then now you take the particles and let us say phi m m particles so is it important whether we have phi m m particles or 10 m m particles or 20 m m particles is it really important to know what is the size why diffusion time because that means there is an some optimal particle size that is as far as the reaction is concerned there is another thing also again particle size that comes when i am now talking about the reactor because we have the molecules crystals particles now you come to these particles are put in the reactor level in the packed bed reactor so when you are putting in the packed bed reactor what is the use of particle size i mean will it will you use that in a calculating any other parameters for reactor size pressure drop Ergun's equation so if it is too small pressure drop will be very large okay if it is too large then diffusion time is large so somewhere you have that optimum that is coming at reactor level right so once you have the reactor designed properly this reactor is kept in the plant level correct no you do not have only reactor alone even though that is my dream that you know only reactor dream dream reactor okay dream flow chart where you have only reactor nothing else that can happen only when you have reaction at room temperature 100 percent pure reactants 100 percent conversion atmospheric pressure all the reaction is happening then whatever you have is pure reactants coming into the reactor and then pure products coming out straight away yeah storage tank to supermarket straight away you are sending the material to the supermarket then they can open the tap and sell and sell to you wonderful wonderful if that is really possible but currently that is not possible so you have the reactor along with the reactor you have so many other equipment that is plant and if you see the molecular level is nano level okay and then particle level is micro level okay and then the size of reactor and then plant is at macro level plant also you are thinking no then definitely we know that chemical engineers are the main people to pollute the atmosphere in many cases okay so that means the waste gases from the plant going to the atmosphere now you have to model the atmosphere to understand how the dispersion of pollutants you see no molecule to plant level but fortunately all the all the gases are only not able to leave the planet but it is only within the atmosphere of the planet if that is also throwing away material to outside to some other planets then it will become universe level so those are the scales chemical engineers are working appreciate at least now and lifelong you can have your entire research based on only molecules and Schrodinger equation that is your research area that is all and some people can concentrate only on the tillo modulus okay tillo modulus means you know again interface gradient interface gradient and all that some people can spend all their life only to develop Langmuir Hinshelwood kinetics that is all throughout their life so similarly some people can also concentrate only on the Erguns equation it seems Erguns failed in the PhD examination when he first told about that equation and that is what the story someone was telling me some UK professor you know that Ergun has been failed because there is no theory in that it is only empirical correlation correct no Erguns equation is only empirical relation where they have two terms one is viscous term and the other one so it seems after doing that work and then he submitted that one as a PhD thesis examiner came through that thesis out no theory only empirical so no PhD for you and that is the equation what we have been using all the time I think on this planet wherever there is a packet bed same story with Monard's equation also that was also first not accepted by the journal I do not remember that journal it was rejected by the journal Monard's equation not Monard's Michael S. Menden equation Michael S. Menden equation because that mechanism and all that many people did not know at that time so that equation also was that paper was rejected but afterwards it was accepted by some other journal and then you see everywhere without bio without Michael S. Menden equation and Monard's equation both are similar no except one is for enzymes the other one is micro algorithms both are same so I think now everyone talks about in fact entire bio chemical engineering only two equations correct no what is that yeah so only thing is you add some kind of deactivation terms what is that we are called deactivation or inhibitions inhibition terms and all that for the reaction to take place so these are the scales we are talking so when you talk about packet bed catalytic reactor design what level we are talking we are talking about only reactor level but before that and after that there are many many things okay so that is why that story I thought I will tell you so for packet bed reactor design alone again if you take okay I will let me concentrate only on that scale reactor scale how many ways you can design okay taking the entire system as simply plug flow reactor ideal plug flow nothing else or otherwise you can take this one as axial dispersion correct no because that is the nearest non-ideality okay and you can also take radial direction then it will become two dimensional model because variation along the length of the column variation along the yeah diameter of the column which is true many times because particularly when you have and we are talking about heat and mass transfer together it is not no more isothermal okay we are talking about energy balance and heat balance because temperature profiles as well as concentration concentration profiles both are important for us so radial direction and axial direction all this what we say is for homogenous pseudo homogenous models what do you mean by pseudo homogenous model I do not know whether you heard of this or not pseudo homogenous model for packet bed reactor yeah what is the meaning actually yeah what do you call that is pseudo what is pseudo that is one dimensional what you are saying is if I say one dimensional I am not bothering I simply do not care what is happening in the cross sectional area okay yeah what is the meaning of that right right equations are similar to homogenous models but what is the meaning that is why you are calling pseudo homogenous in a packet bed what is the imagination when you say pseudo homogenous what happen now the situation is close to human homogenous situation since the single particle packing and not moving and the liquid is uniformly distributed so yeah very close then then then what do you call it is pseudo homogenous it is what is pseudo homogenous it is not actually but under what conditions we are saying that you know this can be pseudo homogenous yeah you know I know yes I want you to think that is all it is not that I am trying to because before I give the answer yeah so that means what is that you are trying to because there is a catalyst yeah very close but I think still simply we do not feel the presence of solids take the as I said bulk properties alone when you take the bulk properties alone and then try to find out the actual conversion along the length if it is only one dimensional model that means simple plug flow okay when you have that situation we say we have pseudo homogenous which is very famous in industry because we do not have to bother about what is happening in the particles right how do you take care of that you already have that because particle whatever happening inside if you are not able to find out exactly right multiply by eta okay effectiveness factor so you are making converting that into anyway bulk so that is lot of amount I mean large amount of data available on pseudo homogenous models that is one section so then you have you can also use the heterogeneous system in heterogeneous models in heterogeneous models you always treat gas as gas or fluid as fluid and solid as solid now you can imagine solid is one part all liquid is another part or fluid is another part now you have the concentration gradients in the liquid okay and also in the solid if you again assume that at any cross section you have perfect mixing of the liquid then you have horizontal some concentration here and then when they touch the solid then you have the gradient inside the solid temperature also same thing okay that is in one direction may be you can do this only in axial direction where the temperature and concentration gradients are existing within the particle so now what you have to do you have to first solve the concentration and temperature gradients within the particle then you come to the surface and if there is interfacial again again solve for surface and then through the film to the bulk and now you come to the bulk right the equation in term based on bulk so you have to first solve the particles for temperature and concentration you know what are the equations you have to be used question number 3 number 3 or 4 or question number 4 same equations so you will find out now T and TS depending on the boundary conditions then from TS to TB same thing C to CS and CS to CB now once you have CB that is where the plug flow is happening in the axial direction or you may have the generally we do not take axial mixing and all the there because the mathematics will be very very very complicated so normally we assume that we have plug flow but consider the temperature and concentration gradients along the radial direction sorry along the axial direction and also for two dimensional radial direction but two dimensional heterogeneous models are very very rare there are few only few academicians have tried to solve those equations so that means what when you are solving this interfacial and intrafacial gradients you assume that at any cross-section if you take one particle and then able to write the equations throughout it is same that is what is the assumption so along the radius so in the two dimensional model is that it is not uniform if I simply solve only one particle and then assume that throughout it is same it is not correct because there is also concentration and temperature gradients along the radial direction not only in this direction okay that means if I have the temperature in the radial direction each particle will have a different kind of you know there is a temperature and concentration gradient there otherwise one line and over that you have axial mixing or you may have simply plug flow that is all what we have so these are the possibilities what we have and I will also write that for your notes otherwise I am sure you will not remember anything okay absolutely you do not remember because what is your interest it is not coming in the examination because normally I do not give this kind of theory okay that is what your expectation but I can also give anytime okay so that is why let us first write these models and then we have to go to the corresponding equations yeah classification of packed bed I am using packed bed only reactor models yeah so let me here we have pseudo pseudo homogeneous models where we take t b equal to t s c b equal to c s then we have heterogeneous models good if let me take 1d first one-dimensional the simplest one is ideal okay basic basic is plug flow because for packed beds we know why we are taking for packed beds plug flow right yeah that is the one and okay let me finish this first then we can go to heterogeneous systems then in the same thing I can also have basic plus axial mixing it is still one-dimensional correct no you have plug flow over that you have axial mixing but the all the changes only in the yeah along the length of the reactor so this is the one then we have 2d basic plug flow basic plus basic is plug flow plus radial mixing basic plus radial mixing right basic will tell you variations along the length and radial mixing okay so when you come here for heterogeneous okay let me also have this boundary somewhere there yeah here you have basic plus interfacial gradients interfacial gradients and in the next again one-dimensional this can be basic plus intra and interfacial okay inter and inter and intra facial gradients so here basic plus radial mixing also taken into account so these are the general things what we do okay and we are always that means we are always trying to tell that in a cylindrical most of the time we use cylindrical part cylindrical reactors okay packed beds why why do not we use square cross-section or rectangular cross-section the distribution is not proper distribution in the corners and all that you will not have proper flow that is the reason and because we have that symmetry so that is why we will take only two-dimensional but we will not take this direction because cylindrical also we will have r theta no yeah that one we will not take because of the symmetrical diffusion okay good so this is the one and based on this now I am not going to say anything about this except may be if there is time I will write one or two equations how do you write those equations that is all and those equations are already known to you because for separated plug flow you have to write and separately for particles you have to write and combining them is your T s boundary conditions at boundary what happens okay all these equations are known to you but only thing is you have to bring them together may be as an example I will show you one set of equations for solving but anyway for this course you do not have to really worry because they are very very complicated models but course is not writing for the examinations course is a preparation for your future work okay that is why most of the time because you are not going for chemical engineering you are not caring the chemical engineering subjects okay so that is what unfortunate things that are happening you know in this country where we do something and then finally end up doing something else totally something else unconnected okay yeah so that is the reason why but anyway as teacher I cannot assume that and then keep quiet because I do not have to teach you at all if I assume that anyway you are not interested in chemical engineering so that means if I have that attitude then you do not have to I do not have to come to class you do not have to come to class okay so that is why I cannot assume that so more amount of time we will try to spend on pseudo homogeneous models and I will give you one or two set of equations for you unfortunately in future if you become a chemical engineer dealing with at least you will remember that okay I think I know this somewhere you may not know who taught you you may not know which subject but still memory you know sometimes it will come with neural connections when suddenly something happens so lot of chaotic theory also there in the brain suddenly some connections will come okay so one of those connections may tell you that oh maybe I think I saw this kind of equation somewhere so that is the reason why we have to do this and a little bit of this okay and before that I have some other passion for my design without writing equations how do I design the reactor okay this is graphical design you need some equations but not solving differential equation here differential equations automatically come because packed bed is a distributed parameter system okay so that means the changes are occurring along the length okay whereas if the same thing if I want to write design equations for mixed flow reactor then you will not have differential equations but you will have algebraic equations that is the main difference between distributed parameter system and lumped parameter system okay so this I think you should know each one of even if you not becoming chemical engineer also you have to know this system because that will automatically differentiate the type of mathematical equations what you are trying to use okay right good so and with minimum amount of mathematics how do we actually design this is the graphical design which I have done something in the last semester reactor theory when you are talking okay so our reactions we will say that we have reversible reversible exothermic and reversible endothermic here does not matter reversible endo exo both are here endo and exo okay good yeah and for reversible reaction I have an equation maybe reversible irreversible irreversible reactions a general order if I take I will have minus r a equal to k c a to the power of n as usual I am going to take n equal to 1 so this is minus r a equal to for n equal to 1 okay this becomes k 0 e power minus e by r t c a which is nothing but c a not into 1 minus x a that means I am also assuming epsilon a equal to 0 this again only to simplify the equation procedure is same exactly same even if you put epsilon also does not matter even if you put second order also does not matter I think it is the same procedure procedure is same so now I have there are three things for me I have minus r a minus r a is one parameter t is another parameter x a is another parameter I have t x and minus r a so now I can make plots using any one of them on y and x and y the other one the other becomes a parameter okay these two are variables good so yeah the other things are constant because k 0 I know frequency factor e r is k e r is the gas constant and c a not also I know initially so when I write this equation in terms of okay I may write here x a I am solving equation 2 this will be 1 minus minus r a e power minus e by r t plus r t by k not c a not so this is equation number 3 okay now I can plot this particular graph yeah as yeah the convenient way as I told you any two parameters can be chosen here third parameter will be automatically there okay so but most convenient for design is x a versus t and rate will be the parameter so it is a irreversible reaction so maximum conversion will be x equal to 1 x equal to 0 okay t we cannot say anything then if I assume a particular minus r a let us say point not not one point not one that values we cannot say but you have to start from lowest to the highest and you have to start for specific reactions some specific values these only by talent error you should get otherwise there is no universal law there but anyway minus r a equal to 0 means what equilibrium okay but here there is no equilibrium because it is irreversible reaction right it is not reversible reaction so that is the the other extreme right so but minus r a very very large value I can start so very small value I can start I go to go to large so then you will get curves like this all of them tending to 1 all of them tending to 1 this is for example minus r a equal to 1 10 100 log scale I am going 1000 10000 okay okay they may touch here also this is one set okay so I will stop here say here run okay we will stop