 చారారి కారి కూటాటి కనంతిజలి సందికికెందూత్ందారందాసి we have discussed what is the background for this model okay the bubbles two important things because the bubbles when they are going up in the bed they have the clouds and also they have the wake right and second greatest thing is to find out how many solids are there in the wake okay even though sometimes absurd values we will get for the measurements but these are the two important things so that means we have bubbles with very very less number of solids then we have clouds and wakes around the bubble with a different solid concentration around that then we have away from the bubbles the emulsion emulsion is always with at umf with epsilon mf equal to voidage corresponding to minimum fluidization voidage okay so these are the important things now we can imagine how the reaction is taking place right that means the most of the gas is entering in the bubbles and there are some solids in the bubbles okay there are some solids in the bubbles so first is reaction of gas in the presence of because the catalytic reaction unless that catalyst is there it won't react because there are some solid particles in the bubbles so gas will contact them some reaction will take place there so that means some amount of gas is converted in the bubble itself then because of the concentration difference it will go to the clouds and wakes because those are the nearest particles and concentration is more in those definitely in these clouds and wakes then there will be some more reaction so whatever is left after that after the reaction in clouds and wakes then that goes to emulsion so beyond emulsion it cannot go anywhere so under steady state conditions all the gas which has gone to emulsion should react so the the reaction and also transfer of all these things all these steps we will see at the end of the react very easy to imagine I think in very very simple manner I am telling you so now that is what we have to predict now using equations so you have to write equation now what is the reaction in the bubble and what is transferred to the clouds and and whatever is transferred to clouds and wakes there is reaction again transfer to emulsion once it got to emulsion there is no way it can go because there are walls so then it has to react there and the combination of all the steps is the conversion which we will see at the end of the reactor when you measure your concentration of reactant okay so that is the overall picture of this bubbling bed model and what the Richardson not Richardson okay Davidson and Harrison that model what they did was all the solids are there only in the emulsion but they did not identify these clouds and wakes even though they note that they are there okay so all the solids are in the emulsion and gas goes in the bubble there is transfer between bubble and emulsion and then using those two steps only they have developed another model that is called two phase model and this is called three phase model because we have bubble phase clouds and wake phase and emulsion phase right so that we are not actually discussing in fact that will be a special case of three phase model if you remove yeah if you remove clouds and wakes okay yeah it is not so simple the way we tell but I think you know you can imagine that that more easily but more predictions very well accepted one till now is Kuni Levenspiel bubbling bed model that is why we are now discussing because that is a famous model where it can be used for not only for reactions it can be used for mass transfer that means you know for removing gases adsorption you can use adsorbent gas in a fluidizer bed solids okay as adsorbent solids and then gas you send let us say SO2 you want to is a pollutant you want to adsorb that is SO2 on the solid phase no reaction physical adsorption now you fluidize with this gas the effluent gas okay wherever you have that flue gas wherever you have SO2 this SO2 when it is fluidizing this adsorbent right so it will go and stick to the surface because of the adsorption characteristics and even for that same bubbling bed model can be used okay not only that even heat transfer when you want to heat the solids for example you have high temperature in the gas and low temperature in the solids you are heating the solids even bubbling because bubbles you cannot avoid so again how heat is transferred most of the heat is coming in the form of bubbles because you are sending gas to fluidize and our assumption is that most of the gas is going in the form of bubbles that heat is transferred to clouds and wake and after that that heat is transferred to emulsion it is very easy when compared to the reactions so that is a very famous model used for many many cases so that is why let us discuss this one and before that as usual we should have some assumptions and there will be many many serious assumptions like the first one itself is assume all spherical bubbles and most of the time we draw experimentally we also saw that bubble will be something like this okay like hot okay so that is how the bubbles actually when you measure it looks so that is why let us take the assumptions the first one is bubbles are all spherical comma all of the same size d b is another assumption you just see d b d subscript b diameter of the bubble comma and all follow Davidson model full stop thus bed contains bubbles surrounded by thus bed contains bubble surrounded by thin clouds raising through emulsion full stop another continue the same para we ignore the up flow of gas through the cloud because the cloud volume is small compared to that of bubble this is the regime where u b is far greater than u e u b u b is the bubble rise velocity this is the regime where u b far greater than u e u e is emulsion velocity that I think already we have told u b is the bubble rise velocity okay that is one second one is second assumption is inside that there are many assumptions second point is the emulsion stays at minimum fluidization conditions comma thus the relative gas solid velocity stays constant in the emulsion good all these things are given in the paper which are given you last class okay these are given in the formula wise also so next one is next assumption is each bubble drags up a wake of solids behind it full stop this generates a circulation of solids in the bed this generates a circulation of solids in the bed comma up flow behind the bubbles comma and down flow everywhere else in the bed okay this is also good point already I discussed that but I think it should be in your notes also next one is full stop there if the down flow of solids is rapid enough then gas up flow in the emulsion is impeded comma can actually stop comma and even reverse itself that means gas can come down and reverse it okay yeah full stop such down flow of gas has been observed experimentally and recorded comma and occurs when this is very important again u 0 greater than 3 to 11 umf 3 to 11 3 to 11 umf 3 umf 3 umf to 11 umf okay and all our industry industrial reactors will definitely operate in this regime so that is why the justification for bubbling bed model full stop there we ignore any up flow or down flow of gas in the emulsion okay yeah I have given this all this information in the in the sheet hopefully you have not thrown the sheet right okay so if you have that sheet you can see that you can see that in fact figure 7 and 8 that will give all the information about what I have told you till now of course spherical bubbles we are assuming and each of them are having uniform DB yeah you see that you know particularly figure 7 we have divided that entire thing into three regions and left hand side what we have the fluidized bed where all the things are given arrows where we have cloud bubble wake and all that for one bubble and three zones have been identified so then gas is going up in the bubble phase with delta the fraction and clouds and wake is the next region I am talking about figure 7 and figure 8 also of course it is given and then 1 minus delta is the total fraction of you know clouds and wakes plus emulsion together delta is the bubble fraction I will tell you all that a little bit later okay so that is the essential picture which we would like to draw once because I think if it is there on that book itself I do not know whether you will really get it or not so this yeah let us see the simple fluidized bed this is U0 then we have the bubble cloud we have this wake also so then one more bubble somewhere here one more bubble somewhere here the rest are solids yeah so then here I can write please write once this is cloud this is bubble yeah so this will be emulsion and maybe this one alone if I write here okay this is wake excellent good so solids may come down they have to come down anyway not may come down these are the solids coming down and mostly gas is going up this velocity we can write as all the bubbles together as U b and there is emulsion gas also coming down or up this we cannot say how it goes that is why this is plus or minus emulsion gas these are the solids all that very nice good so then we also have one single bubble very large bubble that is the cloud so then now we will have this kind of hunchback just to exaggerate and tell you that is the wake okay so here I have the solids and if I mark approximately this is the you know like we do one bubble we take out and we have written even the last class or last but one or one or two classes we can still connect this one with the first class what we have done okay one particle taking out and writing balance and one bubble taking out one drop taking out the same thing so all this is emulsion this is clear boundary of cloud okay these are the solids okay this is emulsion right so if this one bubble first we have to draw the picture of how the concentration is going and you may have one or two few particles inside the bubble also right so that was experimentally found again so you will have first the transfer between these two this is the transfer arrows okay so this we call it as KBC I hope KBC has a different meaning but here between bubble and cloud okay yeah so then you also have now cloud and emulsion this is emulsion by the way this is cloud this is wake and emulsion cloud yeah of course this is bubble inside itself I can write yeah so this this is KBC and this one is KCE KCE C means cloud E means emulsion cloud to emulsion B means bubble and C means cloud cloud and wake together cloud and wake together we will put together okay so in drawn in a different way once more this is our imagination now for model writing this is emulsion so this is cloud and wake and here we have imagination is that all the gas is trying to go through this but some of it will also go through this okay so this is the one this is bubble phase bubble and we here we have cloud plus wake and here we have emulsion all solids here okay so here also solids but of course here there are very thin solids only and okay yeah so this is what is the transfer we have been telling so this will be KBC this will be KCE so what we are trying to do now is we take a thin yeah okay another beautiful assumption here is that this gas which is going through the bubbles is it in plug flow or mixed flow plug flow okay very obviously plug flow that is one assumption but Kuni Levenspiel they have not done that but in the earlier model Davidson and Harrison model so they have tried all possibilities right so they tried that okay let me also try the n number of tanks in series and also perfect mixing in the bubble phase and dispersion in the bubble phase all kinds of things they tried okay that was the first model so here we take a small balance delta z here this is z this is z plus dz so the thickness will be dz and here okay this is moving with u b bubbles u b right and then we have the transfer to cloud and wake then transfer to yeah emulsion cloud and wake to emulsion right so what we do is we try to write the in terms of words first what is happening how the reaction is taking place then we try to write that in terms of steps then we get the equation okay for example minus r a minus r a u b you have to get right yeah so that is written slightly in because it is a plug flow model where you have already derived this kind of thing so this is delta z okay I think may be order thickness I will write here dz yeah good you see now that you know here whatever gas is going into this it has to come back and then join in the bubbles and then come out okay yeah but overall picture what I see here this is the real one that is the real one then this is what I imagine between any bubble so I am now combining all the bubbles as this fraction and then we are trying to write this exactly same thing what we have discussed in the first class right okay good so now I am not anything okay good so now let us write yeah the balance equation for this that is the mass balance of a yeah and isothermal conditions okay that somewhere you have to write definitely isothermal conditions yeah so correspondingly here I can also just show you when you have z here you have concentration okay this is C A not this is C A at the end because we are now writing for the reaction rate right okay so then this is correspondingly here I have C A and here C A plus D C A correct know that small change like exactly when you are writing plug flow first time when we wrote the balance F A equal to F A plus D F A okay into rate and volume a similar thing it is not given in any book but I thought I will write that once to you so that you will be happy to know that thing what is really happening one balance and here and of course the problem with these things is you know in the earlier book he has given this solids fraction solids available in the bubble phase per unit volume of bubbles and in cloud and wake per unit volume of bubbles and in emulsion also based on per unit volume of bubbles that is what is basis earlier but now we have changed that and then return solids per unit volume of bed and again cloud and wake solids per unit volume of bed that means entire bed so that is changed a little bit but I think if you are having edition 3 all of you will have only edition 3 lounge field book then I think this and what I am going to write will be telling that okay good so that is the one and I think I have to go here so the material balance for m b for a oh okay so before that I have to also write here the rate what we have is minus r a into k c a first order rate no volume expansion you see how many simplified things okay so that we will learn we will not be lost totally so that is the reason so where this is given as I think this I have to be careful that is given as rate in terms of so this is moles per meter cubed solids per second m dot s in the bracket means meter cubed of solids so what will be the units of k k units are meter cubed gas that is you know mole okay and you have meter cubed solid per second those are the units of k okay for heterogeneous system in effect you know because there is no volume expansion it is a first order rate if you dimensionally cancel out meter cubed meter cubed what do you get time inverse okay but I think you know it is meter cubed gas meter cubed solid per time right okay good so that is the one this is also very important for us now what you write is the balance overall rate of a overall okay overall rate of a must be equal to reaction in bubbles plus transfer to cloud and wake okay so what is the next one now next step is transfer to transfer to cloud and wake now equal to a that is all a reaction in clouds and wake there must be reaction now first then transfer so reaction in reaction in cloud and wake plus transfer to transfer to emulsion next one Rajith now you tell me transfer to emulsion must be only reaction in emulsion so now we have to write the corresponding equations for this okay so when I want to write for this one first so again I think I will take yeah this trip okay so now we have to write here yeah you know from this is molar balance what we write there so if I write I think it will be sufficient okay I will first write that okay so entering into this a is u u ca u ca at z what are the units of that it is flux okay so moles of a per meter square bubble because this is flux okay so here we are talking about only the bubbles right so meter square bubbles second that is the time flux and now multiplied by meter square bubbles because finally this balance is in terms of moles per yeah moles per second so what is the fraction of bubbles here it is delta cross-sectional area if you the total cross-sectional area is AC D is the delta fraction of bubbles per unit volume of the bed so this will be delta into AC AC is the cross-sectional area okay so now this will be in fact this will be meter square bubble meter square okay so when I cancel out this plus this you will get only that one so now this is equal to exactly the same thing at z equal to this is very easy yeah dz and I have delta AC all that at delta at z plus dz now plus you have reaction in bubbles plus right here reaction in bubble is minus r a so plus minus r a what are the units moles per meter cube solids per second right so that multiplied by a factor called f b f b f b is meter cubed solids per meter cubed bed okay that is the second term you know that is the reaction term yeah so multiplied by yeah volume of the entire bed because meter cubed of the bed okay so what is volume AC into dz excellent this is meter cubed of the bed yeah but unfortunately I have to go to that side that is still plus plus here what is the next one transport right so the transport is k b c k b c is defined as volume of gas okay meter cubed gas per meter cubed bubbles per second in fact this will be if you see now k b c is nothing but again meter cubed meter cubed if I get cancelled you will get again seconds inverse okay so that is the one and into yeah concentration concentration is C A B minus C A C R that is one important thing yes okay so here yeah please write there this is C A B and here also you can write C A B this is C A C here I have C A C and here I have C A E and with that picture the thing should be very very clear good yeah I think yeah so the the units of this will be moles per meter cubed gas right moles per meter cubed gas that is concentration okay now this multiplied by we have delta delta where delta is volume will come later meter cube meter cubed bubbles by meter cubed bed right okay now this multiplied by A C D Z A C D Z where it is meter cube bed now if you cancel out all that throughout you will get moles per second I think this is clear so the moment you cancel out terms like for example yeah here U C A delta A C this said also you have E C A delta A here you will have C A plus correspondingly to here this is C A plus D C A correct now at that yeah I think that I have to write I say yeah here here sorry but I think if it is understood fine but otherwise this is plus D C A the concentration at okay I think let me write that also this is D C A okay of course another good so correspondingly you cancel out that and also A is throughout common A C A C is throughout common every term has A C you cancel out what you get is okay minus U B delta D C A by D Z and of course if I bring that D Z the other side then what you get here is equal to minus R A F B plus K B C K B C delta C A B minus C A C okay maybe I think I will write first delta and then delta C A okay so that is the equation what you get there yeah but what is minus R A I have here okay let me write delta also here U B D C A by D Z equal to this is K F B K C A first order but C A B correct no in the bubble when I take a small volume I have taken that that no no I have written only here I have written the first one the other two is for you to write okay the other two are very simple no transfer already we have this so that must be equal to reaction that is equivalent term is this reaction term so the third one is transfer term which is exactly same as that one but with C A E and C A C A C and C A E so all that is simple okay so this is the one plus this is first order reaction then I have okay delta K B C C A B C A C this is equation one good so if I write the same thing now that means for the second step for this step that will be delta K B C same equation C A C no C A B minus C A C equal to F C F C K C A C plus delta K C E C E second one so into C A C minus C A E that is equation two good yeah okay what is the next one yeah so yeah exactly delta K C delta K C E C A C C A E equal to reaction what is the reaction F E K C A E this is the one now you will have a pleasure in the examination of so you have to eliminate C A E using this one then it will get in C A C and go to this step C A C you have to substitute there okay and then yeah so then you have C A E you get in terms of C A C then you will get this everything in terms of C A C of course C B will be there and then finally C A C you substitute here in terms of C B finally you will get the equation you know what kind of equation you get very beautiful equation you get so I have also done I have also done you see so many pages all this okay so finally I have done but I do not give you this because you know you have to do on your own then only you will enjoy it okay good so what do you get here is this will be as it is so when you solve equation one two three and eliminate unknown concentrations again class one okay so unknown concentrations you have to eliminate and express everything in terms of known thing then what do you get there after that is this kind of equation where minus delta U B D C A by D Z equal to capital K or overall K O okay K O overall into C A B so this is equation number four where K O okay where K O equal to yeah K O equal to F B small K plus I have to go till that point one this will be delta K C E plus again all this one F C K plus again all this one here one by delta K C E plus one by F E K so this is K very nice equation very nice equation see this is parallel problem this is the first time I think you have come across series parallel even though I give in the small small problems in the examination I think quiz one I give parallel step series and parallel right yeah but I think yeah this is the one and you can see this is the one yeah F B you know the meaning volume of solids divided by volume of bed like that F C is volume of solids in the cloud divided by volume of bed and F E is the again solid volume of solids divided by volume of bed and all other things you know okay right so K B C is the exchange coefficient between K B C is the exchange coefficient between bubble and then cloud this is K C E this is no this is K B C thank you thank you thank you I think I think yeah that is right that is K B C then this is K C okay and now we can easily integrate this oh sorry yeah this is B C B this is only bubble yeah here also I have to write this yeah okay what is the solution for that very simple integrating yeah C A B by C A not equal to 1 minus X A equal to E power minus yeah minus K not Z by because you know integrate between 0 and Z yeah yeah K not Z by U B and that is at any Z okay so if I say that my this Z equal to H F Z equal to 0 correct no oh delta is there sorry delta U B yeah so if I yeah K 0 this Z equal to H F then we will have 1 minus X A equal to E power minus capital K not H F by yeah H F by delta U B so this is the equation and now we have to okay so now we have to find out equations for delta F B K will be given kinetics right so like that F C K C E K B C and F E and also before that what is U B okay so these are the unknowns for us right so if I want to use this equation if I want to use this equation to calculate conversion for a given height or if conversion is given you have to find out the height of the bed okay that height of the bed can be converted into the solids total weight of the solids because height if you know bulk density if you know you can always calculate what is the total weight of the solids so all that so to calculate this I need all those parameters and let us do that now so all my nice diagrams what do I do good of course we have relationships like H F equal to W by rho S A C cross sectional area into 1 minus epsilon F because when you want to calculate that in terms of W right so this is very I do not know whether you get the point or not this is very easy if I write 1 minus epsilon F is the voidage of the bed okay 1 minus epsilon is the solid diffraction okay so that is nothing but you have weight divided by this is the volume of the bed H F into A C and into the density bulk density okay here it comes to density not bulk density because 1 minus epsilon F I have rho S into 1 minus epsilon is the bulk density okay please remember that do not make that mistake because you think that you have done correctly in the exam and you will scold me okay okay good so now to find out the other parameters first we have to start with U B R you need you know because U B I have to find out this is U B U B is not I think here also I have to just explain that is why you know it takes lot of time for me I think I have to take some extra class if you want to finish before Thursday okay so but anyway that we will see yeah U B is the bubble swarm velocity that means all the bubbles how they are moving that is U B okay U B R is one single individual bubble that is raising in the emulsion so that is the starting point for us so I will write here bubbling bed parameters these are all parameters which we should know okay see the beauty here is at the end at the end that means when I substitute everything equations for corresponding F B delta all that you will get an equation in terms of only D B diameter of the bubble epsilon M F U M F and I think H M F because H M F is nothing but your H B height of the packet bed epsilon M F is nothing but your packet bed porosity again and then D B is the only parameter which you have to measure that is why sometimes it is called one parameter model but the main problem Abhinav there is that how do you estimate this D B very accurately that is all entire thing goes to estimation of D B even now we do not have very good methods of estimation of D B in the fluidized bed so if you are interested I told you know I can give you 110 problems in fluidization this is one of the biggest problems what we have okay and the other assumption like uniform solid uniform bubble size is can be extended with delta X because what we do is okay in 1 meter what is the bubble size so another 1 meter because you know bubbles grow that is what what we have shown that is the reality so you make this one as slices of the bed then okay in first 0 to 1 meters I will have let us say 5 centimeters bubbles then next one 10 centimeters bubbles then next 1 meter 50 centimeters bubbles so I can measure that some measurements are there how the bubbles are growing there are some equations empirical equations but even then there are not two representation because when it is particularly 3 D bed 3 D bed we do not know what is happening inside the bed at the centre you cannot see okay at the centre unless you again take you know it goes to technology it goes to sophisticated instrumentation which is nothing but you know sophisticated technology to measure I do not know whether you have seen a movie called total recall long time back excellent Arnold Schwarzenegger you will get S grade now Arnold Schwarzenegger so they have a security system because you know they go from mars to earth and earth to mars like we go to you know our out gate and in gate okay so frequently they travel in that movie because that is highly sophisticated you know science fiction movie so then there is a security system where you know now if you go to airport or if you go to even this Coimbatore bus stand and all that they ask you to remove all your clothes they ask you to remove all your whatever you have in the bags recently they have also stripped you know or Shahrukh Khan yeah so all kinds of things will happen but here the fun thing is you can just move in the corridor the corridors are just you know fitted with x rays every part of you will be seen by the security people every part I think there is no secrecy at all you cannot hide anything okay so then the moment you have gun or something that is extra thing you know easily they can find out so probably similarly we should also develop that kind of instruments where the moment you put something between the bed you put the bed in between and these measurement techniques everything you should be able to see every part of the bed then you can measure very accurately all the bubbles what is happening at each and every point that is left to you for your future PhD okay and then your name will be like Levenspier every one will take Abhinav Abhinav's model Abhinav's model okay yeah so that kind of great name you can get provided you know you do all that you know there are many many things which you can really do in fluidized but even now even now good so the first one is bubble rise velocity bubble rise velocity U B R is 0.711 G D B do you remember this equation this is nothing but you know Tyler equation for bubble dynamics or bubble diameter when you have simple liquid and then gas going in okay that constant is different this constant is adjusted to fluidized bed and that was derived by you know Davidson okay it is adjusted because I told you know he was excellent expert in gas liquid systems bubble columns Davidson okay so that is the one and because it is empirical constant we should mention definitely what are the units please remember always when you have empirical constants okay that is why we are happy to convert them into dimensionless numbers so that you do not have to remember any constant so that this should be only in meters per second so that means this must be in SI units this must be in SI units and of course this you will get automatically that constant 0.77 that is okay my equations gone so this is 567 that is 8 so this will be 9 good so then bubble this is a single bubble bubble rise velocity bubble single okay I can also write here single bubble rise velocity okay then swarm of bubble bubble swarm velocity sometime this bubble swarm velocity U B equal to U 0 minus U M F plus U B R yeah this also must be in meters per second because if you are using this equation this is 10 right so here the assumption is that you have the bubbles just rising okay no okay you have the stationary bed right stationary bed and imagine that you have number of bubbles they are just staying then you suddenly change the velocity from U 0 U M F to U 0 the difference of these two will give you so much amount of gas that is just pushing okay so already U B R having some velocity as a single bubble then the moment you increase that it will simply go like exactly you walk on the this conveyor belt in the airports and all that okay so initially conveyor belt velocity equal to 0 right so but you are walking on that you are walking on that so then suddenly someone came and then switched on conveyor belt then what will be your velocity with respect to solid coordinates I mean I know stationary coordinates okay so that is the one right so that is why this there is a lot of derivations for this one again yeah so this is another thing then we have bubble fraction bubble fraction delta equal to U 0 minus U M F there are lot of equations for that but so many assumptions are there yeah so this is nothing but meter cubed bubbles by meter cubed bed so this equation is 11 and this also can be written as you know if I substitute correspondingly U 0 minus U M F here right U 0 minus U M F if I write then I will have 1 minus U B R by U B confusion simply this U 0 minus U M F U B minus U B R okay good so this is the one and for U B the same equation delta the same equation if I have U B for greater than U M F we can use yeah yeah from this equation we can use simply delta equal to U 0 sorry not U U 0 by U B U 0 by U B this has some but this is for very very large values so when you are going for very very large values in fact you can see now delta U 0 U B you now just see there in that equation delta U B equal to U 0 but lot of assumptions assumptions we cannot do better than that that is all lot of assumptions are there we cannot do anything else over there beyond this okay good so this is another equation for delta so now you have values for two parameters U B you have value delta you have values you know how to calculate but everything is in terms of basic variables like U 0 the amount of gas you are sending even if it is not U 0 it is U 0 and U M F and U B U B is again in terms of U M F right here and U B R is the basic equation okay single rise bubble velocity so all these things good so the next one is let us say K B C okay K B C K B C is of course I will write the equation and then try to write the okay K B C equal to 4.50 U M F by D B plus 5.85 this is diffusivity gas diffusivity D G this is to the power of half this is to the power of 1 by 4 1 by 4 D B to the power of 5 by 4 so that is the equation you have to use to calculate K B C and K B C definition is interchange volumes of gas between B and C or vice versa okay B and C divided by volume of bubbles so in effect again interchange volume volume and volume will get cancelled you will have only seconds inverse dimensionally strict dimensions if you apply okay good so then we have K C E K C E is the simple equation 6.77 epsilon M F again yeah I think you have to write here the D B you know U M F you know right and D to the power of half this D is molecular diffusivity of gas to gas okay yeah and G is the acceleration diffusivity gravity that is no problem yeah so epsilon M F and here also same diffusivity D into U B R divided by D B cubed into D B cubed so equations so this is another form I think I do not have to give equation so this will be 12 this will be the 13 okay so what is K C E definition now volumes of gas exchanging between yeah cloud and emulsion per unit volume of the bubbles per second okay good that is the one K C E then we have this parameter F B it is generally given 0 0 1 2 0 1 volume of solids meter cubed solids by meter cubed bed this is estimate from experiment this is this estimation is only from the experiment experimentally they measured and then they know what is the total volume of the solids in the bubbles then that is expressed per unit weight per unit volume of the bed okay good so then next one is you have F C F C is delta into 1 minus epsilon M F and yeah multiplied by 3 U M F by epsilon M F that is one parameter divided by U B R single rise bubble velocity minus epsilon M F by sorry U M F by epsilon M F again plus alpha yeah so this is equation number 14 you know do you remember alpha that is what is that so that should be given to you that is also experimentally measured value in the problem definitely that will be given to you I have already given I think you have not thrown no last time I have given you some assignment okay in that also it is mentioned if you want to throw it you can come to my room and throw in my testament I can take it and do easy okay good so this is F C then next one is F E oh my god time on yeah so next parameter is F C F E F E again is volume of solids in per unit volume of volume of solids in emulsion divided by volume of the bed right yeah that I think you have to write in somewhere so this equation is very simple in the sense 1 minus epsilon M F into 1 minus delta minus F C minus F B so this is the equation and this one is nothing but 1 minus epsilon F good no so this is also same like for example if I bring F C and F B this side what will happen I have F C F B and F E what are that entire thing total solids that is nothing but 1 minus epsilon F because epsilon F is the wide edge 1 minus epsilon F C F is solid fraction so that is what that balance is simple to understand so this is another equation this is 15 okay good so other equations I have given good I think so this is the one and now you have all the equations and you see for example F C F C is again in terms of only U M F and epsilon M F and delta is in terms of again U M F right U 0 and U M F right so then what else yeah all these F C F E also will come in the same way yeah K B C is again in terms of U M F right all these things are known up to you U M F will be known epsilon M F will be known to you right and of course the properties like diffusivity is acceleration due to gravity all that you know if you convert this entire equation in terms of only D B only D B right so that D B you can vary so that you can also fit the data with your experiment that is what exactly they have done in this figure one this corner is their cutie figure okay so they have mentioned you know different diameters for intermediate large bubbles and all that sorry small particles D P and all that yeah I think none of you have brought the book you know Levenspiel book yeah if you go to this is D P which is representation of what kind of bubbles you get but if you go to Levenspiel figures and then see you will also see in terms of D B written D B 10 centimeters D B 5 centimeters D B 30 centimeters D B means diameter of the bubble so that is why there is a single parameter D B which you have to use finally to calculate conversion in the bubbling fluidized matter that is why Levenspiel model is very very beautiful and very very appreciated Levenspiel at CUNY model okay good so I think anything else equations yeah Savita anything else missing for calculating that first of all you have to calculate K 0 for K 0 you need F B F C F E all that you know deltas you know K B C K C E you know no problems right yeah and U B you know yeah so you can calculate very easily good so yeah like you know you have the delta equation delta equal to U 0 minus V M F and all that but there is a very big derivation for that but we are simply accepting that it is acceptable I have I have that derivation it is yeah this is the one for example you see cut across a cross section imagine that all the gas is in one bubble that is delta all the bubbles are in one bubble imagination then the rest is 1 minus delta right so now through delta delta is the bubble phase Davidson has already calculated through the bubbles how much gas is flowing okay so what he has calculated is delta into U 0 total U 0 is pleated into delta into U B plus 3 U M F that is a delta part right what is the other part other part is 1 minus that is the you know imagine this okay I think maybe if I have a cross section something like this and a bubble like this and the remaining is going down okay so this is delta this is 1 minus delta Davidson has given in this one what is the amount of gas that is going this is nothing but your U B plus 3 U M F yes derived that equation and in the other one you have 1 minus delta in this 1 minus delta how much gas is going 1 minus delta is what first of all delta is the bubble fraction our imagination wise the entire other thing is solids okay so solids are supposed to be at what velocity U M F that is all this is nothing but 1 minus delta into U M F so now you can calculate delta here as U 0 minus U M F if I solve for this U B plus 3 U M F that is the basic equation okay that is the basic equation where if if if U B is approximately equal to 5 U M F yes then you will have delta equal to U 0 minus U M F by U B where 2 M F no no see this is very very approximate right this one oh sorry 2 M F thank you thank you thank you so this is an approximate one where you get this kind of one then if you go for still very very far then your delta is simply U 0 by U B there are many many approximations that is why I want to tell you just one derivation I just would like to tell you you know what kind of things they have done but now you see this one is also is a complete paper where they used Higbee's penetration theory to get KCE equation that is the amount of gas that is going into the emulsion from clouds okay so that like that for each and every one there is a lot of theory behind but finally constants and all that adjusted with empiricism with experiments and then finally this wonderful model has come good I think we will stop here and tomorrow first or we will meet