 నిలుని మత్రంయాసింద్నిందిఆవ్సందాసినిసారెలు పబిసార్సనింది మరెరాక్నటినికొపు నిమరిన్పయక పషని నిందాందిఆరె పపాన్ మరికాటాక this is infinity this goes to zero then only you will have these terms so that can be arranged as t equal to rho b r 3 b c ag k g to x b so this equation is 20 what was the last equation earlier this is 24 this is 24 and we can also try to find out what is the time required for 100 percent conversion for x b equal to 1 that is 100 percent conversion x b equal to 1 we call this as tau as simply rho b r by 3 b c ag k g so this is equation 25 similarly we will have the diffusion control as diffusion control that is 2 as diffusion control which means that we have k g is infinity and k s is tending to infinity that means again these 2 will disappear these 2 terms and the equation now can be simplified as t equal to rho b r square 6 b c ag de 1 minus 3 into 1 minus x b to the power of 2 by 3 that is equation this is equation number 26 again here for x b equal to 1 I will tell you why we have to simplify this and then you can get lot of information from these equations that is why I am just writing so when x b equal to 1 this will be simply rho b r square 6 b c ag de so this is equation number 27 the third possibility is that we have reaction control reaction control where we have k g tending to infinity and de also tending to infinity equation is rho b c ag k s into 1 minus 1 minus x b to the power of 1 by 3 this is equation this is equation number 28 right here again if x b equal to 1 for 100 percent conversion tau is rho b r b c ag k s so this is equation number 29 there is some information here hidden unless you see it carefully now for film control tau is proportional to r for ash control ash diffusion control tau is proportional to r square and again for reaction control tau is proportional to r so this is 30 31 32 okay good so what do we get from this I think this okay all nice equations have been derived but finally we have to use them how do you use for example that information use in the sense that what do you imagine normally I imagine that my non catalytic reaction if it is a non porous particle we imagine we have one particle and then surrounding that we have the gas and it is diffusing through the film and also through ash and reaching the surface and then the reaction is taking place fine based on that that is the phenomena now based on that phenomena we have derived the equations right so after deriving the equations then we have now simplified to find out really what is true what my model is I have to find out that how do you find out we have to find out most of the time with experiments so that is why in any research the experimental part is definitely required for whatever amount of mathematical modeling you do unless otherwise you become it is mainly for simulation people they declare that very proudly that I am simulating many things but all simulations unless it is valid no one is going to believe that right yeah no one is that is why experiment if you do on your own and then trying to find out whether your model as well as experiment both are same or not then you know that is the best otherwise if you are expert in modeling so then you have to model and take some other data and then also you can check in fact that is also very good it is not your experiment but still you should be able to model their experiments if you are able to do that right so one way of doing this is to find out which one is controlling this is one of the simplest test how do I do this I want to find out whether film is controlling or ash is controlling or reaction is controlling the rate that means I do not have three steps I have only one step film control how do I experimentally find out that whether this film control is there really controlling the film okay and then we establish a relationship for different particle yeah so that is the meaning of these equations why you have written all this is that this is the simplest one which I can find out that if I am able to take let us say 1 mm particle 5 mm particle 10 mm particle and 20 mm particle 4 particles I have taken so now I found out the total time required for complete conversion and how to find out is again a big problem I think it is not that easy the way we discuss here okay I just ask you to imagine that I have a coal particle coal particle also can be done on this that also you know similar to the derivations what we have done except ash control may not be there otherwise okay I will take the realistic example not coal the again iron ore reduction right iron ore reduction so then we take Fe3O4 particle or Fe2O3 particle and then send hydrogen around that and we know the reaction is taking place how do you know that it is completely converted because we are waiting till complete conversion this is the time required for complete conversion all these tows capital either tow so for that again you have to imagine a lot what is the mathematical I mean the experimental technique used to find out whether you have really this reaction going on or whether it is changing with time during reaction and at the end if there is no reaction it has to get some you know property which you should be able to measure okay I mean the best thing if it is possible is that you take one centimeter particle and then after five minutes take out cut it open and then see the difference between product as well as the reactant okay so outside you will have the product and inside you have the steel reactant so like that you can see if you are able to see the shrinking core you take the spherical particle so that is one way to find out whether really you have this possibility or not okay people have done that so that is why doing experimentation is also is really thrills me most of the time it is not the solving equation because doing a particular experiment based on on your own thinking so that means you have to think a lot because nowadays the mathematical techniques are most of them are available on MATLAB okay so unless you want to develop a new theory and new way of solving the problem that is fine that is excellent again you will get the real thrill there but most of the time as engineers what we use what we use is packages right so if you just keep that one aside and if I tell you that okay now you take this particular problem and then try to get me the experimental data everything you have to imagine for example what sizes you have to take and where do you measure how do you allow know it is very easy for for academicians to say that you can take a particle send hydrogen it is very easy but when you are actually doing what you do and I do not know whether you heard what is called TGA thermo gravimetric analysis TGA is one of the sophisticated instruments where you can find out thermo gravimetric analysis you know so the changes in the weight during reaction but you should have that change really if you do not have much change in the weight you cannot find out again so in TGA what they do is they just put a small balance and one side of the balance they hang the particle the other side they measure you know balance in the beginning time t equal to 0 then you start the reaction and slowly if there is increase in weight that increase in weight is recorded that depends on what kind of reaction you are taking okay and for example calcium carbonate going to CAO plus CO2 right so you have taken let us say 10 grams of the particle of certain size so during reaction CO2 will come out definitely there will be change in the weight so like that in some reactions there may be weight increase so that you can record and that can be converted as X B that means you know initially how many moles and at any time you know the weight that you can always convert to moles so then that definition of X B will give me what is the conversion so if I am able to plot X B versus here for example X B versus t small t not tau tau is the easy experiment to do right so that means I am now finding out the variation of the conversion and then time so if I plot this equation this is nothing but y equal to mx there is no even c so it has to go through all this is constant because I am keeping my temperature constant I am taking one particle size so r is there then CAG okay at that particular temperature and pressure this concentration of gas I know and of course this entire thing will come as constant this loop right so X B if you plot and then if it goes through origin and then if it is a nice straight line then I know all these things rho B I know rho R I know and this B is stoichiometric coefficient that I know already and CAG I can calculate because I know pressure temperature so I can calculate now I can calculate what is kg you see how beautifully one can do that same thing similarly here or similarly here right but before doing that entire kinetic study is first I have to find out whether which one is the finite controlling step so for that this experiment is very easy right again if you there is a possibility of using TGA that is the best go to this take a particle and then just hang it there and wait till wait till the weight is almost constant that is that indicates the reaction completion so at that point of time you record the timing okay so that is the total conversion and corresponding time tau now if I plot just radius let us say 1 mm particle 10 mm particle 5 mm particle and 20 mm particle the way I said that in the okay this is tau versus R right so here again this equation if this is the controlling one right now now kg is constant because I am putting the same flow for different particles same temperature same pressure right so around this that gas will be going and the reaction is taking place and then I wait till the complete conversion X be equal to 1 so this is the equation we plot this versus this and kg also I need not know kg actually so only R if I know I will just plot then that also goes to goes through the origin yeah and then yeah it should not change first I do not want to draw and then put points I mean you also do the same thing you know so that is why I just want to put the points first and then draw the line okay so that is the line if it goes like this then the slope of this will give me information of course again kg also I can measure there but this again proves that it is film control for me okay but there is a catch this also which one is controlling how do I find out now if this is R square then it is fantastic no problem if I plot okay if I plot this as R square okay or otherwise there are so many ways you know you can plot that on log log sheet log log sheet and then this will be slope will be 2 then that means of course it is R square and of course again when I plot on the normal sheet and then take the slope I can evaluate DE that is fine but between these two we have a problem to identify which one is the correct one how do I find out just can you think how do I differentiate I have now film control and reaction control both are valid if I get you know tau is proportional to R if this is the condition but definitely I have to tell there are two possibilities that is what is the correct possibility what is that what is really do you have any other parameter which you can change which one how do I change mass transfer coefficient yeah so even if I do what will happen I change your concentration as you suggested what do I see the change what changes you expect yeah why mass transfer coefficient also the answer is right but I think you know still a little bit deeper you have to think so that is why doing experiment is not that easy I say so much thinking is required and not only doing experiment anything to do I think so much thinking is required only joining IET you do not require anything and after joining nothing is required okay somehow will pass you somehow you will get marks and somewhere you get because brand name is there you get with the 10 lakhs 20 lakhs salary and all this reward is for not using our brain yeah so this pleasure of doing things you know really I do not know maybe even this Saturday Sunday I went to that Shorathkal for conducting a PhD examination on the way normally always I take some book this time I have taken the grand design Stephen Hawking book one of the latest books the way he explains things so beautiful in fact he changed my mind of you know this teaching I write less equations and then more talking because of that because I would like to tell the entire subject as a story and stories can be understood by even LKG people the way if you if you are able to tell that okay so then such a complicated equations and you will never write in equation complicated theory is so beautifully explained so really I get excited when I am reading that book he was talking about you know string theory M theory of the universe okay all this but the way explained is so beautiful but everything you know you know that you know in his body nothing is working except his brain okay maybe hot steel pumping yeah so that is why he is alive but otherwise you know he is only brain that means he simply sits down and then thinks thinks thinks all those the theories and also his explanations everything is in his mind similarly for us also so that is why I think you know when you are doing experiment there are so many possibilities we know my students they know how the you know what kind of help they are going through not hell helps many okay yeah because simply if you are able to plan very beautifully thinking every point is not that difficult but I do not tell them every point because they have to also go through that mill you know buffet is very easy to eat you know because everything is there you just go put it there and then eat but if you ask you okay you cook and eat then you have to plan many things I say first of all you do not know where rice is available okay and you do not know where vegetables are available and how to cook what temperature should be used how to stir or do not stir so many things are there everything is a question okay so that is why even in teaching also we should not tell everything you know that is why we give assignments we leave something in between steps so that you also will participate in this okay but anyway I think as he said and you need lot of information also to identify which is really controlling here I think that correct answer is that use temperature why we know that reaction is very very sensitive to temperatures because of Arrhenius equation whereas KG is not that temperature sensitive right because why KG is not that temperature sensitive I do not know who invented this word why I think it is really very good yeah tell guess something just tell right or wrong no problem KG why in what way it first of all it changes or won't change it change but in what way it changes here it is exponential the other one directive proportional why university yeah why probability by 2 you see how much thinking is required again okay so that is what I just want to tell you all this try to find out okay you take one equation which KG you know for example we have that our famous equation Sherwood number equal to yeah 2 plus that okay neglect 2 for easy thing and then now you tell in Renard's number you have some properties right you have density you have viscosity okay and in Schmidt number okay you have mu rho both will change with temperature what about D by the way what is D there diffusivity it also change with temperature yeah so you have to individually know that how temperature is varying no viscosity is varying with temperature okay density is varying with temperature diffusivity varying with temperature all that you have to substitute and then get what is the overall change of mass transfer coefficient with temperature okay but if you do all that it will not very much I think it is only 1.2 T to the power of 1.5 depending on different theories okay but it is not definitely exponential so that is why if I change the temperature I may get like this and I may also get may be something like this with the different temperatures so may be this is T1 T2 T3 etc so the change is not that much but on the other hand if it is reaction control this is film control okay if it is reaction control tau versus r so I may get like this this is one temperature like this like this okay yeah now I am doing wrong thing okay so which temperature you know temperature is increasing in this way or in this way which is the direction of temperature increase in that graphs this is likely complicated for you but that one is easy not able to understand question so this is T1 T2 T3 okay whether T1 is greater than T3 or T3 is greater than T1 let me write T1 T2 T3 so which is higher from that graph can you tell T3 is higher why no no I am telling that you know there is no mass transfer here you know this is film control this is reaction control but it is giving clear difference between you know T1 T2 T3 whereas here it is not that much different they are almost close okay so this will tell me that this is reaction control and this is film control but what I am asking you to make you think is that whether T1 is greater or T2 is greater or T3 is greater I mean what direction it has to go yeah why from that graph yeah the slope is okay from that graph can I tell that graph has tau versus r yeah straight forward is it for large temperatures time required is less so which direction I have to show this I have to show this direction okay everything again I tell you I think most of you I think likely we don't have this B Tech guys except Renita because most of you are are going to do some project because in B Tech they have a escape clause that you know if you don't like project escape you don't have to do project they can do two courses and the way they are attending they can attend those courses and then they can get the degree okay yeah not even one day all B Tech have come to the class okay statistics yeah so but you know as research scholars you know all these are very important for us when you are doing experiment because you have to think a lot to do the experiment I am not saying that you should not think at all when you are doing theory theory also required lot of thinking but everything origin is here either theory or experiment right when you are if you are planning experiment at least one week you have to sit in your room close the doors and of course take food also in between and then think what to be done otherwise after one week you will not be there in the room okay alive yeah so I think that is required even when you are trying to do the theory same thing again just shut down and then think and frankly tell me how many of us are doing that if a problem is given how many I think it is a rat race okay that is why I like that movie three idiots okay that fellow wrote a simple word there what is that Rajya Kulashan or something like that ahh yeah the moment he writes that everyone starts you know go to index and then see whether it is given and then pulling the books from others and then trying to find out without thinking whether that kind of word is existing or not that is what is the rat race what we have in education now no time for thinking that is why all of us were lucky you know 50 60 years back happily when you are reading no happy that is all I think you know no motivation if you pass pass if you fail fail that is still okay but I think when you are doing that you know lot of time is there and at that time we do not know we have to think about the problem and all that particularly for us who came from villages but may be the professors sons and daughters and may be educated families they would be thinking that they had sufficient time so that is why they have done wonderful work in fact nowadays all these things is rat race okay so yeah this is one clear way of finding out which is controlling there right because here also tau is proportional to r here also tau is proportional to r the other one is very simple because this if I if ash is controlling then if I plot tau versus r square then I have a straight line but you have to plot r square this is reaction control right and if I plot r square and tau if this is again straight line r square I am plotting now so this this is diffusion control ash diffusion control so this is the one and now once I know this I can even find out what is effective diffusivity or if it is reaction control I will find out what is K s and if it is mass transfer coefficient I can find out what is K g right and this K s I have now 3 slopes okay so then I will have I can find out from those 3 slopes 3 different K s and now if I plot 1 by T versus ln K then I will also get activation energy all that information and if I repeat the same experiment even with diffusivity this is changing even I can find out with different temperatures again diffusivity varies again with 3 to the power of 2 by 3 no 3 by 2 I think again yeah diffusivity with temperature so if I am able to get that variation with different temperatures then I can get information on even how diffusivity is increasing with temperature it is not that blind you know when we say that d may increase with 1 to d to the power of 1.5 because that may be purely for gas gas diffusion but this is effective diffusion where we have other things also coming into picture what are the other things we have 3 different kinds of mass transport in that pores we have Knudsen diffusion we have bulk diffusion and we have configurational diffusion right so all the effect of all these 3 only giving me mass transfer across the that shell whatever shell I take inside the particle all that information we can very nicely get that if we understand these equations how to be used to conduct experiment and then get the overall thing but what do you do if you have 2 controlling or 3 controlling what do you do because always we will go from simple to complicated I mean do you have any thumb rules in your mind I mean all this depends on you know what kind of exposure you have in the subjects I have 3 steps what is the first possibility I can remove that this may not affect most of the time in the in my experimentation one step why answer is right but why again you have to tell pure gas what is less or resistance is less if I do not have pure gas most of the time we do get pure gas in industry but till what point you can go because the boundary layer is almost 0 if you have infinite velocity if you go into infinite velocity particle will not be there this planet will not be there this also will be blown out okay so that may not be the one think some more deeper normally you will forget a small pain if you have a big pain okay so which one diffusion is difficult or mass transfer through the film is difficult mass transfer mass transfer what mass transfer through the film or through the ash layer is difficult because the ash layer diffusion through the force for molecules is difficult when compared to diffusion through the film so that is why general thumb rule is whenever you have a porous particle because after reaction porous particle porous product only you get so most of the time it may be controlling when compared to mass transfer so that is why most of the time we simply ignore film but technically doing it yesterday we had one number biot number okay biot number what is the equation biot number equal to k g r by d e okay biot number should be very large or very small small means what you can neglect or root neglect see you said that internal diffusion is difficult when compared to mass transfer through the film okay so which one should be more there in that equation yes because always the slowest quantity will control is I have been telling all the time all stupid examples also I am giving okay you have a bench car which can travel with almost 250 kilometers per hour and just in the mount road you have the bullock just before that bullock cart so then what is the use of having this that is the fellow who is controlling you and that goes only 5 kilometers per hour if you make it run it may go to 15 kilometers per hour okay so this is the smallest okay and this is very large but when this is controlling this is small you are dividing this by a small number where biot number equal to large so that is why in the beginning itself if you have the parameters like d e and k g you calculate this if it is around 1000 or 2000 5000 happily you can ignore because people also ask you scientifically whether what you are thinking is right or wrong we may say that yes diffusion is small when compared to diffusion through ash layer that rate is small when compared to the film okay these are words only but I think unless you prove that with an equation no one will believe you so that is why if you know these values you can go and then try to find out otherwise you have to do these experiments right how do you do that now because three are controlling we do not know which one is controlling so that is why if I conducted very low temperatures what will happen which will control if I am conducting at very low temperatures reaction controls okay so if I am able to really do that and then finding out you know only this is valid this equation is valid so I have never at least one step right so on the other hand if I have very very fast control I mean high temperature not control very high temperature yeah mass transfer may be controlling but unfortunately here you have two mass transfers right so that is why with the two mass transfers you know you can neglect this in the sense that you know when compared to very high very high temperatures this term can be neglected you have the remaining two so with the remaining two you can try to find out you find out XB at various points now you have this equation okay now again straight lines are better than curves you do not know how they go right so that is why if you are able to use this equation and then arrange this equation such that you will get a straight line you can you can try to do that that again depends on your mathematical skills that is why as engineers you need mathematics and also the experiments even for scientists is always required and generally that combination is very difficult you know good in mathematics and also good in experiments if you are really able to do both I think that is really excellent combination right so like that you have to now think that how do you arrange this part as a straight line and then plot okay time verses XB or it may not be now time verses XB plotting the way you arrange you will arrange as Y equal to MX plus C straight line if you are able to do that intercept should give you one parameter and slope should give you another parameter this is what what we do most of the time maybe you might have not recognized it because you are only worried about whether that problem comes in examination or not most of the time okay after the exam is over delete all files from the mind for next course again you are ready fresh with the hot disk okay that is what is most of the time happening that is why absolutely even if I ask you next semester what is that I have taught I think your hot disk is so clean nothing can be told in that okay good so this is what and also we have an equation for total time required if all three steps are controlling very easy what is that XB equal to 1 all XB equal to 1 and this T becomes tau total time okay so that is just to give you tau T XB equal to 1 will be rho BR 6B CAG 2 by KG R by DE 6 by KS this is equation number 30 okay this is only just for information that is nothing that yeah so one more thing I have to give you is the proof of pseudo steady state yesterday we have assumed that we have pseudo steady state and why is it really valid or not I think I can use this part itself proof of pseudo steady state assumption okay the proof is very simple which I got from professor Anand he told me how to do that is not there in any book when I was teaching in the beginning I asked him sir this proof how do we do it fortunately it seems he has done for something else in US and he told me very simple proof that is why every time whenever I am telling this I still always remember professor Anand very beautiful way he has told me how this can be proved that pseudo steady state assumption is valid for gas solid reactions okay so the equations are same we have already you have the equations which we have written in your book yesterday right so the first equation okay first the second equation is A gas plus B B solid going to R solid R R plus S S gas this equation we know and we have this mole balance DNA by dt equal to minus 1 by B DNA by dt so all that you have done so now I can also write DNA by dt as 4 pi R C square which you have already done dE dC by dr that is the flux okay I mean flux multiplied by yeah that is DNA by dt this is the flux yeah C A C now of course C A C that means I am evaluating it yeah at C also equal to this is minus minus 1 by B and you have this equation 4 pi R C square rho B dr C by dt can you check that actually we have got this equation for DNA by dt right so this can be cancelled and this equation can be arranged as dr C by dt equal to yeah okay let me write this B by rho B this is 1 then I will write dE by diffusivity dr and dr C by dt right and delta C A C this okay this delta okay I think let me write this also delta this I have written delta A C by delta delta R and then dE by delta R what are the units of dE by delta R yeah so that we call it as diffusion velocity this is as V D diffusion velocity okay I have to use may be this part so now that equation I can write minus dr C by dt equal to rearranging that delta C A C B divided by rho B into V D let me write this correct yeah that is fine so now there are two quantities yesterday we have discussed one is the diffusion velocity the other one is rate of shrink of core so these two are present in this equation V D is the diffusion velocity okay I have not written okay V D equal to diffusion velocity and dr C by dt is core velocity oh shrinking core velocity of shrinking core okay dr C by dt that is also velocity yeah so now we have another term here which is relating these two this is delta A C by rho B what is rho B molar density of solid what is delta A C delta C A C C A is concentration what are the units of that again molar concentration so now this ratio for most of the d C A is rho B into of course this is stoichiometric quantity is most of the time for gases that is the one okay yeah if I substitute this there minus dr C by dt will be V D by 1000 this is for gases so what is the meaning there yeah the velocity of the core is 1000 times less than diffusion velocity or in other words diffusion velocity is 1000 times core velocity so that is the reason practically at any point of time when I look my core looks stationary during my analysis okay so but on the other hand and you know the reason why we have to assume this Swami do you remember why we have to assume pseudo steady state what will happen if I do not assume that is a good question always okay you may ask that sir why you should assume that yeah accumulation term will come into picture that is mathematical but physically mathematically I have to take accumulation term because I cannot neglect that but physically what is happening why I can neglect accumulation first of all why I have to assume that I have pseudo steady state I explain that but again I know you will not remember so that is why again I am asking I have to establish that profile inside the ash inside the ash diffusion okay inside the ash if this is the particle this is the one so this that is the film C A S C A S film is there outside okay so this profile this is C A C I need a value the difference between C A S and C A C where I can use this to eliminate my intermediate concentrations but if this C A C if this is slowly continuously moving inside C A C which C A C I take because C A C also is moving continuously inside so that is why in my mind when I am solving the equation that I should imagine that that is stationary at one point of time and that is very well the difference between gas liquid gas solid reactions where the density difference molar density difference between these two is thousand times for particle and the ratio of particle to gas is thousand times or reverse is one by thousand times so that gives me automatically the connection the core receding velocity is thousand times smaller than the velocity of diffusion of gases where that means immediately at any point of time it is already reached I say it is constant almost see it is thousand times by the time it is let us say one mm decreasing so this should have moved almost thousand mm but what is the particle size you are not taking one meter diameter okay so that is why most of these particles are magnitude 2 mm 3 mm 6 mm that is all I think one inch also rare of course for 4 inches they use in blast furnace you know blast furnace okay so that is why here the thousand mm it will move when it is moving one mm so that is why pseudo-strate data assumption is valid otherwise the equation what we have to use is this the equation for complete analysis is dow t that is the accumulation term in fact equal to dow square C A by dow r square plus 2 by r dow C A by dow r ya okay this comes from input and output and this is accumulation reaction is zero because we have inert ash okay so this is what equation we use sometimes people you know this all this is happening in the pores diffusion is only through the pores so that is why some people also write for material balance sake epsilon there epsilon is the porosity okay but for pseudo-strate state for p s s pseudo-strate state epsilon dow C A by dow t equal to zero this is what is the equation which you have solved yesterday right this d e dow square C A by dow r square plus 2 by r dow C A by dow r equal to zero that is what you have solved yesterday to get that equation right good excellent so I think we can close here