 in the last class we have discussed the design equation the general design equation for gas solid non catalytic reactors if you are focusing on solid phase because solid phase we take it as a macro fluid so the general design expression design equation for macro fluid is 1 minus x bar B equal to 0 to tau 1 minus x B single particle ET dt this is the general expression so now depending on your reactor whether it is following mixed flow or whether it is following plug flow this ET we have to substitute and this 1 minus x B has to be written in terms of in terms terms of time and then we have to integrate so for ideal contacting we have mixed flow and plug flow first let us take plug flow for ideal contacting design expression for plug flow design equation for plug flow will be 1 minus x bar B 0 to tau this 1 minus x B single particle we will write SP single particle now onwards into ET dt is direct delta function T T minus T bar P dt where ET equal to delta T minus T bar P that is for plug flow this is equation number 1 if I start giving numbers this is equation number 3 so now in this equation we have 2 terms 1 minus x B for single particle and then the exit as distribution function equation because it is plug flow we know clearly so we have written that now depending on what is controlling whether film is controlling or ash diffusion is controlling or reaction is controlling corresponding equations we have to substitute here and then we have to integrate that and we know integration with direct delta function I think you had some examples which I have given you sometime back in reactor theory so with that help how the direct delta function can be used for integration and all that so with that help you can get the equations so now this is mathematics but logically what do we expect if I have plug flow I have single size particle may be all are uniformly 1 centimeter particles right and I have plug flow right so this equation will it give the design expression as single particle equations like for example single particle equation you know for film control what is the equation for single particle you should remember that yeah T by tau equal to x B now I will substitute that instead of T by tau there that is x B equal to T by tau there T by tau equal to x B so 1 minus x B we have to put here so 1 minus T by tau so what do you get yeah so for all plug flow the single particle expressions directly will be used for the design but only difference is what is the difference between that single particle equation and plug flow equation what is the difference Renitha not able to follow at all you are there in the last class or yeah I am talking about single particle single particle film control you have an equation 1 minus x B equal to 1 minus T by tau okay single particle T by tau so when I use the same thing some equation here and then integrate this equation what do I get you said that the same equation must come but same same okay it will be T bar equal to tau see again this equation gives me the relationship between x bar and yeah 1 minus T P bar by tau so that T is simply replaced by T bar okay that is all so that is why for all the for example reaction control you can take but only slightly difficult thing is even though it is same I mean integration wise it is same thing only you will get and as diffusion you have two terms and then you have to separate x B separately right x B alone because you have T by tau equal to 1 minus 3 into 1 minus x B to the power of 2 B 3 and the slightly complicated equation is there but instead of the even then we have to substitute for single particle and then integrate but the lesson what you learnt is here now is for as far as plug flow is concerned the same equations for single particle equations are also useful for the design of plug flow reactor is it logical or not logical it is logical you have a single particle and you have the relation between x B and T so this particle when he is kept inside the plug flow reactor depending on its time which we prescribe okay may be 10 minutes 20 minutes like that the same relationship is valid but the difference is that T is replaced by T bar P where T bar P is T bar P is yeah average average residence time which is also defined as T bar P equal to W by F W is hold up kgs and this is mass flow rate kg per hour or kg per second whatever that is what is T bar P for tau bar P also I mean sorry for T bar M also same equation W by F okay good so now this is one that means all single particle equations are simply valid for valid for plug flow okay good so now that will come later I think that is okay you can make a note of that somewhere okay all single particle equations can be directly used to calculate T bar P if X bar is given and X bar if T bar P is yeah what did I say T bar P if T bar P is given X bar you calculate or X bar is given T bar P you calculate okay good so now for mixed flow for ideal contacting again if I have for for mixed flow for mixed flow of particles right you know what is ET for mixed flow of particles good so I think yeah okay let me say that for mixed flow of particles you have again 1 minus X bar B equal to 0 to tau 1 minus X B single particle S P ET DT now ET is here is E power minus T by T bar M by T bar M DT this is the equation which you have to use good okay I think when you separate again we will take mixed flow and plug flow these are the general expressions and we do not have any other reactor if you are not considering a non-ideal reactor if you are considering non-ideal reactor then ET has equations for different non-ideal parameters right if I have a moving bed where the bed is the particles are slow like blast furnace the particles are slowly coming down but they are not coming down so smoothly some particle may be staying a little bit up above one then some particle may slide more so on the whole you may have slightly a diffusive region that is what what we call yeah dispersion okay I think I have told you last time my example in Tirupati Q right so that is what is the dispersion axial dispersion so the people will be pushing you forward and the people front also pushing you backward so you will be moving yeah back and forth but that movement region may be maximum one meter okay or may be two meters right so like that only restricted region you just move so similarly here when the particles are simply sliding down some particles may go faster some particles may all are uniform size particles we are talking here okay yeah so then even if they are not uniform particles the same dispersion should be happen very limited region some particle may go like this and go up go up come down come down go up like that it may happen that is what is the diffusive region for that we already have an equation for ET that equation you have to substitute here again for plug flow means yeah here this is ideal plug flow if I have dispersive plug flow then there is also called dispersive plug flow generally okay DPF people call okay so there also you have to substitute the corresponding ET equation and if I have a fluidized bed with dead space and by pass we have an equation right so that equation can be used here for finding out the relationship between conversion and t bar m tau is fixed tau you get from kinetics right tau you get from kinetics this is this comes automatically when you are putting the integration limits and tau is known apriary to you which tau is that for example it may be tau for 3 controls or 2 controls or each individual step control that information you have that is why we said first we should know kinetics and then contacting and of course before that you should know what is input right so all that 3 will give you the output output again so many times I am repeating the output again can be in the form of given conversion what is the residence time what is the volume okay residence time is related with volume this is kg and you know flow rate this W if you know the bulk density then you can also calculate what is the volume of the reactor so at the end finally it is the volume of the reactor but you should know conversion or otherwise if you already have a reactor where volume is already known to you then you can calculate conversion that is all all the time these are the 2 design parameters which you should know either this if it is conversion is known volume if volume is known conversion you have to find out good yeah so this is the equation which we have to use and now we will take one by one like for example first let us take plug flow for details for plug flow if you have the each individual step controlling yeah and there are assumptions also here I have plug flow but you are nice student so normally you do not ask any doubts know yeah all the assumptions what we made for kinetics also should be used here like for example spherical particle okay and to start with constant size okay and also there is another assumption where you have lots of reactant gas around the particles okay that means the entire bed even though the moving bed is completely moving from top to the bottom okay this gas composition is so much inside that there is no depletion of gas for the reaction to take place you able to follow me yeah I mean imagine that you have a batch system you put all the particles there and then you are feeding with CA0 and CA0 concentration is not changing outside concentration I am talking but once the particle starts going through the film concentration drop may be there controlling or not controlling then afterwards you will have again concentration drop in the pores again you have to take whether it is controlling or not controlling but in reality generally they may be there and then reaction may be smaller slower or faster so all these steps are there but in the environment of the reactor you will not have any depletion of that means uniform gas composition throughout the reactor otherwise what will happen imagination just imagination abhula otherwise what will happen yeah your CAG is not CAG throughout okay just imagine that I have a very lengthy reactor and then the concentration somewhere at the bottom you have less okay or if I am sending counter current it will be more of course okay counter current it will be more and then slowly when the gas is going up you will have less concentration of gas and when you are coming down that means the particles will see first the low concentrated gas at the top correct no your counter current so particles are coming from the top so the particles will first see the gas with low concentration and the particles at the bottom they will see more concentration of yeah of gas so that is why I have to take that variation also into account that design is slightly complicated mathematically but conceptually it is fine and you know one thumb rule as in industry what they do is okay I will take the arithmetic average let us say here five moles and on the top may be three moles okay decreasing counter current so then I will take so this is I said how much four plus so seven five plus three eight so four is average so your CAG is four okay so of course all kinds of thumb rules are used in industry because their idea is not to understand and then not to solve the problem you know mathematically but they are understanding their aim is to run the plant somehow so that is why even before chemical engineering was there excellent chemical engineering factories for that okay but only thing what we are doing now is that okay when you are operating that are you able to understand what is going on in the plant otherwise even without understanding also we will produce many things okay simple example is are you not producing every day curd and are you not eating curd every day particularly now I mean South Indians North Indians may not use every day okay yeah then do you know how it is actually produced do you know the scientific basis for curd production there is a lot of biochemical engineering there lot of biochemical engineering okay so that is why temperature should be maintained if it is very hot the you know you put the earlier previous days you may not know but you ask your mother so she will put the earlier days curd or buttermilk and she will not pour for one liter of milk she will not pour 10 liters of buttermilk very small that is all she simply touches that is all one drop and after six seven hours beautiful and again in the winter it may take more time in summer it may take less time if it is too hot too much summer you may not get any curd because microorganisms die because they also need air condition like us I mean correct temperature like us okay so that is why that is why I think you know all these things are possible but I think without learning anything about curd we are happily eating so like that I think without knowing anything people have been producing the chemicals but only when you have the knowledge you know how to handle it that is why I give another example always that you know a mechanic driving a car and people like us driving a car is totally different his confidence is totally different I mean most of us we do not know except steering where it is and where is the brake and where is the other you know clutch and accelerator that is why sometimes even we get confused for clutch for brake brake for clutch and also accelerator these two is okay at least but if you confused with the brake and accelerator instead of stopping your accelerating so gone accidents but mechanic will not do that because experience knowledge experience knowledge so you will happily stop I think even without touching the steering also he will run the car so that is the confidence so after doing this chemical engineering degree when you go to industry you will be in that driver seat if something happens you should be able to tell that okay this distillation column is not well because of this reason okay our reactor temperature is falling so I think conversion will be less so this is not correct I again increase the temperature so that is the kind of thing okay so that is the reason why we are trying to discuss all these things which one the average in use side right this average is very easy you know it is automatic average why we are going for average is to make mathematically simple so that everybody can understand if you understand the physical phenomena you know where to simplify okay yeah if you know understand but without understanding also people are able to simplify things because that is experience through experience may be 5 moles and 3 moles or 4 moles there is not much difference for them as far as conversion is concerned why because always you have factor of safety in industry okay so instead of getting 90 percent exactly even if you get 85 percent it is okay for that is the reason why but nowadays that is not correct in chemical industry okay still bulk industry is no problem but very that what you have been telling you know that a pharmaceuticals and there is a name for that specialty chemicals specialty chemicals you cannot 90 percent means if you get 90 percent it will be economically viable otherwise if you get only 85 percent then you have to again purify that to 90 percent then it may that may be another big process for purification itself okay so the cost will further increase so that is why nowadays chemical industry also is is going deeper and deeper into what is called the product design product design we did not have till now it is only mechanical engineers we had product design all the time okay but I think chemical engineering now also started thinking not now I think even 10 15 years back they started thinking what is the product design so that means if you want a particular type of soap okay so I will ask may be by market survey I will collect the information okay now 10 million people wanted only this kind of particular soap you know like nowadays you see the advertisements you know okay I think Google I don't know you are get advertisement for soaps or soaps so now what is that called soap is not soap it is only milk or something you know what is that that white dough yes correct yeah dough soap yeah dough soap yeah I think you know it is not soap it is milk and then on bucket will be pouring milk there okay so then if it is really milk then where do we go and buy we have milk in the house you can use that you can happily use that so those are very those people are very very quality conscious oh my god my daughter also I mean purchases so many things I think you know in her bathroom so many not 1 2 3 4 5 6 7 10 and I don't know what are the 10 I will only look for shampoo because I have to wash sometimes you know so I have to see 10 times you know all these bodies which is shampoo so this is shampoo with the color removal shampoo with more color and shampoo with hair shampoo without hair all kinds of things are there I think you know this really funny but they are making a lot of money I say yeah she purchases this much and then say it is 500 rupees yeah I think those people they have to be very very quality conscious so the process parameters should be very thoroughly controlled so that is why that averaging all that will not happen there but bulk chemicals like you know producing millions of tons of sulphuric acid okay plus or minus 10% does not matter for us so like that even sulphuric acid or hydrochloric acid nitric acid all bulk chemicals so that is why this you know the product design again is a very engineering very interesting course in chemical engineering I wanted to introduce but I don't have time now I think this hustler there is a book product design chemical product design it is a wonderful book and I think someone who knows all chemical engineering only must teach that it is not that specialized now we are told you know specializations so for car left back backside tire only I know or wheel I know I don't know anything else that kind of people cannot teach that you should have the entire picture of chemical engineering and then teach product design it is simple problems again so what he has discussed you know for example ammonia how do you make a quality ammonia whatever you require so if I if I want 99.99% ammonia what do you do okay so bulk industries we never bothered about the quality it is only quantity but now it is quality and also quantity because on this planet people are just coming now I think now it is 1 billion when did you see yesterday may be by this time the class is over may be another 1 million 2 millions also would have been added to the planet okay but death is not that much because of again chemical engineering okay because we have so many pharmaceuticals and so many medicines so if he is about to die also you give him an injection keep him another one day two days like that you know so that is why if death and birth if both are equal it is steady state nothing will happen to the planet all of us would have not been here if that is true okay good let us take now plug flow and then write those equations okay the assumption why all these started because of the one assumption that uniform composition of gas throughout the reactor okay otherwise when I have okay let me draw that because I think most of the time I am drawing in the air yeah if this is solid coming solid going out this is gas if there is change in gas concentration initially I may not here I may have here CA naught somewhere this may be CAF final gas concentration okay so solids are of course here have X B equal to 0 here I have some X B so when the particles are slowly moving assuming that we have plug flow as I told you already this particle will see a concentration of CAF almost CAF whereas this concentrate this particle will see a concentration of CA naught okay so but this particle has already been converted to certain value that depends on your design what is the flow rate and how high use your reactor that depends on that so that is why at every point I have my CAG changing in your equations so in every equation for single particle you have CAG that CAG is continuously changing with so that has to be integrated throughout that also has to be taken into the equation then integrate throughout from this point to this point if it is X B equal to 0 X B equal to some X B one or whatever okay right so that is the reason why we are not taking right now that complicated phenomena it is only just you know conceptually only you can understand but the rest is mathematics okay so for simplicity we are now saying that throughout I have CA naught or CAG so that every particle is exposed to the same environment so then it is only I have to concentrate on solids I do not have to worry about gas that is correct okay solids so that is why what we are doing now okay for plug flow you have again the assumptions I am telling plug flow constant size spherical particles only we are talking okay constant size and uniform CAG throughout throughout the reaction yeah and here what we are discussing now is sinking core model is valid in all these models sinking core model okay so under these conditions I have given you those equations yeah for film you please tell me for film is film controlling that is what is the meaning so 1 minus X B equal to 1 minus T bar by T bar P correct no yeah and for reaction control what you will have for reaction control you have to remember that exam is coming next week okay reaction then we will write diffusion because diffusion is slightly complicated 1 minus X B equal to 1 minus T bar by tau whole qubit is there now you see the equation whole qubit right yeah and for the other one you cannot write so easily you can only write in terms of T bar by tau equal to T bar P by tau equal to 1 minus X B and all that so you have to solve that okay so for diffusion I will just write here now T bar P by T bar P by tau equal to someone can tell that 1 minus 3 2 by 3 plus 2 into 1 minus X B so this is the equation yeah in fact now if you have a reactor which is already operating that means you know T bar P okay then you have to calculate X B means it is cubic equation for it okay but the other one is easy if you want to find out T bar P for 90% conversion very easy okay substitute here 90% conversion and then you can calculate what is T bar P so this assumption we have plug flow of solids constant size uniform composition and uniform size particles also all particle constant size doesn't mean that uniform particles I have only one size particle constant size means I think that is understood okay good so this is about plug flow for single particles that means constant size particles yeah yeah good I think I thought of I ask you when I was about to write but I think I now forgotten but anyway yeah you tell me I think does that matter for plug flow even if I put X bar P or X B both are same why will particle will spend exactly same time that is why even with this average or single particle it is same but if you are still feeling comfortable you can write that okay nothing wrong even if you don't write also but that you cannot write for mixed flow because every particle will have different conversion there what you see is real average okay good so this is one aspect now yeah if two steps are controlling then you have to write that T by tau equal to all that thing you know in terms of X B one minus X B yeah if three controlling same but solve for X B yeah one minus X B and then substitute here analytically it may not be possible so that is why you have to go sometimes graphical integration or you know numerical integration or so okay good I think the procedure is same but easy for us to remember these equations so that is why we are writing this good okay so now other thing is next thing what to tell is for plug flow I have constant size particle only particle is not changing with reaction the overall size but I have mixture of particles not single size what do you do under those conditions so I have plug flow of solids mixture of particles mixture of particles uniform gas uniform gas composition mixture of particles but constant size again you know even if the mixture of particles if I take 1 mm particles 1 mm 5 mm 10 mm three size are there but 1 mm will remain as 1 mm till reaction is completed 10 mm 10 mm 5 mm 5 mm I think that constant size is maintained but now I have instead of one size one size 1 mm 1 mm 1 mm 1 mm I have 1 mm 5 mm 10 mm mixture of particles okay so what do you do under these conditions we have to simply average the contribution of each particle fraction okay I am sending continuously let us say 100 kgs per hour out of this 100 kgs per hour may be 30 kgs are 1 mm particles another 30 may be 5 mm particles and another 40% is you know 10 mm particles okay so now I know the fraction 30% 30% 40% okay so now we have to average weighted average of this particular contributions okay from each size so that is what what we do okay good under those conditions what we can write the equation is mean value of I will write the this is more realistic condition so mean value mean value for for the fraction of b unconverted unconverted this is the one equal to fraction of reactant fraction of reactant b unconverted in a particle size of ri the ri may be 1 mm or I may be 5 mm 10 mm this is the one okay so I have another term here fraction of feed which is in fraction of feed which is of size ri and finally you have to all sizes summation that okay so numbers I have forgotten so this is 4 5 6 7 okay so this can be 8 so if I write this sigma of ri equal to 0 to r m where f is the total flow rate of course sometimes this will be more than 1 sometimes it will come so that is why that limitation must be given there so this is equation number 8 I hope you understood now this x bar of ri is this is the conversion which you got for 1 mm particle okay or 5 mm particle or 10 mm particle and this one is f of ri is the flow rate of 1 mm particles I told you we have 100 kgs per hour so 30% of that means 30 kgs per hour of 1 mm particles 30 kgs per hour of 5 mm particles 40 kgs per hour of 10 mm particles so that is what the meaning okay so that is the equation that means you do not have to use any new equation you have to only just take the weighted average of all these contributions right so otherwise the same equations are useful if it is reaction control or if it is film control or if it is diffusion control same equations are useful good okay so to more realistically manipulate this equation we can write this lower limit as 1 minus x bar p sigma of rm that is maximum size and actually I do not have to take the contributions from this particular size or less than that size into 1 minus x b of ri okay I do not know whether you understood this or not this is 9 this is 10 so what is the meaning of that t bar tau you know is a plug flow reactor then imagine that you know we have our beautiful plug flow reactor is conveyor belt right so you put on this conveyor belt 1 mm particles 5 mm particles 10 mm particles okay good and for 1 mm particles tau is 3 minutes okay and t bar p is 5 minutes not able to follow because for each small the particle individual particle you have tau tau is the total time required for that particle to completely get converted so 1 mm particle just as an example is getting converted in 3 minutes and 5 mm particles may be getting in converted in 6 minutes and 10 mm particles are getting converted in may be 12 minutes okay 3 6 12 minutes okay just giving some values but your residence time that you provided for plug flow in plug flow is 3 minutes sorry 5 minutes okay then what will happen to the contribution of 1 mm particles completely converted so what will happen 1 mm is x b of that that is what is the meaning of that so take only those particles which are above your t bar p that is all okay so you can take that into account and then only try to design that is the meaning of this particular equation good so I think the examples we will do it later but the next one is the mixed flow reactor mixed flow reactor also exactly the same thing we do what is that what are the equations for 1 minus yeah here okay reyneeta this x bar b is okay yeah in fact if I put this x bar b I have to put there double yeah so that is why we remove x bar b here that is your convention okay if you are comfortable with x bar b where still I will average sir no problem because every particle is giving okay conversion 1990 1990 1990 90 90 90 90 90 90 divide by 1000 is again 90 only okay not 90 divided by 1000 corresponding ones yeah so then if you still feel that no no no I will just put again only that bar it is okay when you put bar here you have to put double bar here that average again you are taking here okay so that is why we will remove here okay for single particle but when you to talk about mixed flow reactor definitely you will have one bar here and definitely you will have double bar there two, okay. So mixed flow also what we do is mixed flow is not this simple mixed flow is very very complicated again what is conceptually happening I have the mixed flow like fluidized bed and continuously mixed flow we have, okay that is the distributor this is the gas we have the solid particles to start with uniform first then we have the bubbles and all that here bubble breaking small bubble, okay. So solids continuously coming gas continuously going, okay yeah so when particles are continuously coming and continuously going some particles may come in a very short time because of mixing we know that very well so even at a time 0 plus you will have some particles coming out so in that there is no concentration almost and those particles which are staying over time 100 percent conversion all that average only I see at the outlet as conversion. So that is why there because of the residence time distribution you have to take that inter count and derive the equation that we will do in the next class.