 Okay, we will start now, yeah interphase non-isothermal effectiveness factors, okay and we have done something similar to this in the interface effectiveness factors non-isothermal, right, yeah what was the definition there, general definition for effectiveness factor, eta, yeah eta equal to actual rate, yeah by bulk T temperature, okay temperature and concentration, temperature and concentration, right, so this may be R O B, now this R function of temperature and concentration, of course clearly we can also write, okay here, B B at bulk temperature and concentration, good, okay, so as usual we will start single particle and film is there but it has become nice now, it is not affecting anything, okay, yeah this is C B also equal to C S, C S and from here concentration decreases, right, yeah this is T B also equal to T S, T S, yeah so if I take an exothermic reaction this may be something like this, so here T is a function of R and here C is a function of R, this is R capital R, this is small R, this is 0, yeah this is capital R, so if you look at the definition you have to get now actual rate where actual rate is a function of this concentration and this temperature at that point or at this point, for example if I take at this point there is a concentration and also correspondingly there is a temperature, right, so the rate now depends on because this also non exothermal, this is also a function of T B and this one R O B is also a function of T and C at any point inside the particle, so what you have to get information is the rate at every point, so that means if I know the profile we can get the rate at every point, right and also I mean with temperature and concentration, now we have to develop equations how to do that, yeah, so this is a thickness of delta R and here I have small R, this one is R plus delta R, right and now you just imagine because it is mass transfer mass transfer is from outside mass transfer is going in this direction, okay and it is an exothermic reaction so heat is coming in the opposite direction, so coordinate system you have to be careful in the sense that when you are writing conductivity equation, what is conductivity equation, oh yes, what is the equation, q equal to minus k e dT by dr like pixel okay, good, so that equation only we are going to use again here like dE d effective because d effective is not the diffusivity of gas-gas normally we imagine as far as for binary system d effective diffusivity I am talking because now these two gases are diffusing through the pores and pores also will have some kind of influence on the molecules, molecular diffusion like exactly when you are going through narrow gate in your OAT okay you can calculate what is the diffusion coefficient there, when you are going for Saturday's movies okay, so similarly here also when you have small pores and then large amount of gas is sent there A and B so because otherwise A may be going inside B may be forming there and then again coming out binary diffusion so under these conditions that is the effective diffusivity because of these pores and pores are not uniform some may be small, some may be larger, some may be lengthier all that so everything taken into effect as dE d effective and that is what we use in fixula okay j equal to dE minus dE dCA by dr okay similarly here even with temperature we do the same thing we have now effective thermal conductivity why it is not the normal thermal conductivity of the solid normal thermal conductivity of the solid is easy to find out there are no pores nothing but this is a catalyst particle with pores so when the heat is transferred heat will see some voidage then some connection with solid some other voidage some connection with solid so I think again it is random it is not uniform pores or uniform voidage throughout so that is why again thermal conductivity has also has lot of work separately people doing only on thermal conductivity in packed beds thermal conductivity in all other systems again it depends on hydrodynamics and all that okay so that is why KE, DE are very very important parameters in all heterogeneous systems KE is thermal effective thermal conductivity DE is effective diffusivity okay they are not the normal ones which we easily take right which we easily get from the tables and that is why lot of research also has been done on that to give some values good so when I write the mass balance and heat balance I will give you the finally the differential equations okay that means what is entering here for mass 4 pi r square tell me quickly I do not write that I will just tell you for mass first of all 4 pi r square totally rate you know what is the mass mass that is entering rate that means most per time 4 pi r square minus d that is all at r equal to r plus delta r and here at r and similarly when heat is coming out you have to take the corresponding symbols again 4 pi r square KE into dT by dr this is now minus this time because it is going towards the this along the coordinate so again at r and r plus delta r take all these things you know as delta r tending to 0 then you will get the familiar equations which you already know for mass transfer these are the things you would have used many times these equations DE d square C by dr square dC by dr ya minus K C to the power of n if I take nth order reaction equal to 0 and similarly for heat transfer or energy balance I have KE this is the beauty of transport phenomena 2 by r minus I have delta HR into rate of reaction K C to the power of n equal to 0 oh here thank you thank you ya dT by dr very good that is why always you know fresh classes are very important because you will be fresh you will catch mistakes otherwise I think after 2 classes gone going to sleeping mode okay good this is heat of reaction ya so now these are the 2 equations which we have to I mean we have to solve both are second order differential equations then you need 4 boundary conditions what are the boundary conditions here dC by dr equal to 0 at r equal to 0 ya this is 1 ya and C equal to CB which is also equal to CS okay at r equal to R similarly dT by dr equal to 0 at r equal to 0 and T equal to TB at r equal to R good so this is equation 3 this is equation 4 good ya can you solve that equation first concentration equation is it easy to solve for n equal to 1 you can solve okay and say this one second equation from there concentration second C first equation first equation you can solve for C you cannot solve any one of them independently so you need some link in between okay ya correct no because this is here I have that is what I told you know this Arrhenius fellow I hidden somewhere okay if you do not notice that fellow will kill you right so this K is nothing but K0 E power minus E by RT ya so that is there so that is why you cannot solve both of them so you have to solve them simultaneously right so now again observe that two equations can you link them what is the linking factor there Kc to the power of n so if I solve from this equation Kc to the power of n or rate this is nothing but rate okay ya then if I can substitute here then I have a relationship between temperature and concentration okay so that relationship you will get if you write this in terms of this also can be written in the short form no in this form like 1 by R square ya remember 1 by R square short form this is the expansion form D by DR of R square d E by DC by DR okay that form only ya so this form equal to that and if I also substitute this form this will be delta HR all this correct ya I am just substituting that form here and I have this equation this is equal to again 1 by R square D by DT no DR sorry K E DT by DR correct no by R square oh ya ya ya right right right so this is the another equation now okay using these two this equation this is now 56 can you tell me the relationship between concentration and twice you have to integrate that and then you will get the relationship between ya someone did already very simple you do whatever you want to tell me between what is T and C relationship no it is not like management no you do what you are telling is it can be solved sir by taking some constant and all that what I am asking you you solve ya telling to solve is difficult I mean that is only my job to tell that you know you solve I need not even know and you can always take that I mean that assumption you have to do otherwise you cannot solve diffusivity and thermal conductivity are independent of temperature and concentration also okay diffusivity sometimes can depend on those are the questions you have to ask me no sir how can you solve sir unless you tell this okay good I will leave it to you so if you solve that what you get is T minus TS equal to minus delta HR DE by KE into CB minus CS okay I mean there you may have some confusion with science and all that you can always get it when you actually solve okay good you try later you got it right ya you will get it the first differentiation and also you have to use the boundary conditions same boundary conditions and then you will get this equation this gives me wonderful relationship between concentration and temperature so what I do now is you go to this equation KE I have here this KE fellow has this Arrhenius this T so this T I am going to substitute there here right KE has T and that T is substituted from this equation anyway ya CB and CS at any point okay this is evaluated at any point no this is CB equal to CS right okay ya good so that is the one and you solve that those two equations okay so now we have equation 1 2 and 7 with the boundary conditions you have to solve the concentration profile temperature profile and then integrate for the average rate correct no average rate that is the actual rate you will get and then of course you have that as actual rate that divided by bulk rate will give you effectiveness factor we do not have a solution for that so only people have given the results in terms of ya empirical correlations okay so one more step before going to that empirical I mean that graph is this also can be written if I divide by okay sorry ya TB we are using consistently TB where TB equal to TS okay ya so this is the equation now if I can solve also this T minus TB minus 1 equal to minus delta HR DE I also have CB KE TB into 1 minus CB this is another equation 8 where this also can be written as T by TS TB always SMIT equal to 1 plus beta CB this is equation 9 where beta equal to minus delta HR all this DE CB KE TB ya anything missing there KE TB ya correct ya you see like your beta bar here also we have beta good ya and again this nice equation simply relates temperature and concentration we can also find out what is the maximum temperature that is possible here where C equal to 0 okay C equal to 0 then you will have ya when this is 0 you will have ya T by TB equal to 1 plus beta okay so that is what is the maximum delta of course if you bring this this side then you will have the maximum thing I think that I am not doing that okay I just leave it to you your imagination but now I think I can also ask you what is the maximum temperature that is why you have to make a note and then find out okay delta max everything I have to write I say it becomes a LKG all the time okay good so now we will use this information and try to plot because we have to finally say what is the effectiveness factor and how many parameters I have here what are the basic things which we have to plot we have to plot phi verses phi verses eta only right eta verses phi what are the parameters now from this thing what will be the parameters dimensionless you know already one there beta another one because aryan is fellow is there epsilon aryan is a number so these two aryan is a number is one parameter and beta is another parameter okay so now let us plot this for gamma is the no not gamma why epsilon we are calling epsilon epsilon equal to E by RTB okay epsilon equal to E by RTB and of course beta we will try to plot there so I can start here from point one one ten ten okay so this side I may have point one ten hundred approximately do that okay so beta equal to zero means where is beta here beta equal to zero means isothermal okay that means delta h r equal to zero yeah or very very very small approximately zero delta h r okay so that we will plot first that is nothing but isothermal right beta equal to zero which you are already familiar with isothermal so you may get point one right yeah something like this and first simple things beta negative beta negative means again you know endothermic beta negative is endothermic so I may have this may be point minus point two like this right so this is beta equal to zero equal to point two minus minus point two this is okay minus point four minus point six good so now the exciting things exciting things are beta positive that is point two they won't cross anyway so this is this is beta okay let me write again beta equal to point two point four point six okay the top one is slightly awkward I think it has to go till top and then slightly come down okay anyway just analytically the to show the trend good so what can we predict yeah this is this particular thing is for twenty if you draw these curves for ten you will have slightly different okay straightforward things first endothermic reaction always you have less than one endothermic and of course beta equal to zero is isothermal we know already we know how to draw that and as the beta is increasing then you will have even more than one more than ten more than hundred effectiveness factors so what is the meaning that means this rate is even hundred times more than the bulk rate yeah any explanation we know it is possible but any explanation high temperature yeah heat of reaction because of exothermicity the temperature may increase there but you should have a combination of why it is going to hundred and all that right combination of temperature as well as concentration if you don't have concentration again even though if I have temperature there is no use right sufficient concentration that may be happening somewhere down because temperatures may be high but here you have mass transfer resistance five large five large means mass transfer resistance right yeah so that is why you may not have and then that may be falling there good yeah so the things what you have to identify here is there is a maximum for these cars when you go for exothermic and where you have to be really careful because these peaks may destroy your catalyst because thermal centering may come into picture okay or the particle itself may break right all these things the thermal integration okay of the particles may not be there so it may break so all these things may happen so that is why we should have an idea of this temperature as well as the concentrations concentration no problem particularly temperatures we should have an idea and that will come automatically when you are solving this this this to get the actual rates good yeah another striking thing here is that at some five values yeah that is what is the exiting thing there that is what is the beauty there and lot of papers are there on that particular aspect alone how this multiplicity what we call okay are unstable steady states unstable temperatures are stable temperatures multiplicity in temperatures so all these general words are used for this phenomena and then they found that surprisingly it is very narrow region where under those fives only you will get this kind of behavior that means particle will be operating at three temperatures at a particular phi like CSTR we have done those semester okay CSTR under certain conditions we will have three temperatures one will be unstable the remaining it will be stable even here it exactly same thing this is stable temperature what do you mean by stable temperature what we have done here is just nothing but solving heat removal and heat generation it is the balance between heat removal and heat generation okay so whenever you have both are equal then you will have only one point right yeah so under these conditions here I have one point like that and when I am slightly coming in this direction and then suddenly here suddenly it may fall to this or from there suddenly it may fall to this that is not a problem but problem is if I am operating on this at this point if there is any slight change in the temperatures due to many conditions even mass transfer conditions also will change the rate so that rate change rate will change automatically the temperature because rate is directly connected with the exothermicity of the reaction so any kind of instabilities in the parameters will either go from here to here or here to here okay so this point is the reaction snuffed out extension of the reaction almost that means you are coming from hundred to almost one or two here when you are coming in this direction or when you are going from this direction then you have from here to here suddenly jumping to very high temperatures I mean effectiveness factors but that is due to only high temperatures and all this is possible for beta equal to large values just look at beta and then find out is there really something wrong with that beta what is the meaning of beta large there delta hr must be large very good K is low so what is the meaning of that K is low delta hr is large K is low means it is a very bad combination yeah it is always like hot attack and sugar diabetic and hot attack together okay it is a deadly combination so you have large amount of heat in the particle but it is not able to conduct out if it is able to dissipate it out no problem it will quickly come to one of those steady state conditions okay that is the reason why we have and I think no book has given nice explanation except aris book aris goes into very very deep into this concepts as well as mathematics both so he says that this curve is possible only when I have a catalyst with you know the problem is when you are tracking one particular curve okay the parameters are related right you can see the phi also contains K correct no phi contains K and also here here I have K e d e here also I have e all these parameters of course T b will not change e will not change this is not a problem but there is a beta keeping beta alone constant and varying phi is not easy why d e is there in phi you have d e you know in beta also you have d e right so that is why that is why what he says is imagine a situation where catalyst particle with deactivation deactivation simply reduce the temperature without affecting all other things okay I mean rate of reaction it is getting simply deactivated because some active sites are not slowly getting blocked but keeping the particle integrity as it is so diffusivity must be same so under those conditions when you are moving from this to this initially you have low effectiveness factors why because mass transfer is again wake him up I think you are very much interested in subject test that is why I think for you it is like a song okay and the moment you come to the class I am going to my matrix close over okay that can happen with a beautiful song like others you know others did good so when you have at this point at this point here then I have low effectiveness factors almost one or two right that is because of high effectiveness factor sorry high tiller modulus and then you are moving in this direction because anyway when you are moving in this direction towards reaction control regime this is mass transfer regime control regime and this is reaction control regime we are moving to that side so that means the rate of reaction must increase because more and more mass transfer coming right here less mass transfer here when you move this side more mass transfer so slowly you are coming like this and this is increasing and at this point suddenly either it may fall suddenly here okay so that means the reaction which is going on till here suddenly snuffed out I think that is the term he uses or reaction got extincted extinction of reaction and the other side it seems that is not easy to imagine but if you are able to imagine that that means moving this side that means from reaction regime to mass transfer controlling regime okay when you are moving in this direction after some small changes in phi suddenly you may jump from this to this which he calls as ignition point ignition point okay yeah this is very nice because whenever you have see always whenever you have increasing and decreasing trend there is no thrill there is no kick if you plot x versus y maybe it is increasing and in other case it is decreasing always increasing decreasing no kick but kicks will come whenever you have it reaches maximum and falls okay or when it reaches minimum and again increases suddenly so then there is some phenomena which is changing its gear somewhere due to some reasons and then you have to find out those reasons so that is why as research scholars you should be looking for those kicks instead of asking for smooth variations like x is increasing y is increasing okay x should increase and y should increase the way you never expected that is why we have a journal called A C H E journal okay then you have of course journal of fluid mechanics and all that if you have normal way x versus y when you plot when it is going smoothly up they will never publish the paper okay so they publish happily those papers when you have a graph this is y this is x so it goes like this goes like this goes like this goes like this very good A C H E paper because you do not know what is happening here okay so this goes suddenly sometime what happens it comes back again goes forward and again goes down goes up so this kind of complicated phenomena if you are able to explain reasonably so these are the complications so that is why you are getting this kind of behavior immediately acceptance letter welcome your paper has been accepted because no one could understood okay so that is why I think we are publishing leave it good so that is the kind of kicks you have to really see far when you are doing your PhD okay so that is why observe do not try to have only smooth curves and straight lines smooth curves and straight lines nothing will happen okay no kicks will be there good so this is about non isothermal effectiveness factor and what you have learnt here is out of course there are many things there is a criteria where this particular zone can be avoided there are equations available okay depending on beta and epsilon if you know beta and epsilon because many books use the gamma gamma as minus e by r t so if you know beta and epsilon and calculating using these two there are some criteria like vice preter criteria right where they have found that to avoid this what should be the value of beta and epsilon or combination of beta and epsilon okay nice information so that you do not have to operate here operate somewhere here which is more stable and better okay highest you are getting so temperature I mean the effectiveness factors are very high catalyst particle is very very active so you operate under those conditions and your reactor volume will be automatically very very small reactor volume because rate of reaction is very high always in the design v by f not equal to dx by minus r a the down in the denominator so that is what is the chemical engineers dream how do I increase my r a at the bottom so that I will get maximum minimum volume for given conversion or maximum conversion for a given volume okay so that is the criteria excellent good so the next one which I would like to do is the combination of internal and yeah interface and interphase combination of internal and external effectiveness factors okay so that we will do good I think you have I enjoyed always but you know I do not know whether you enjoyed or not okay yeah I mean now you are enjoying anyway okay you are always in your wonder world what is that who is that girl named Alice in Wonderland yeah okay so that is why I think you always enjoy Alice enjoys her wonder world in Germany you know German professors they will have this board one part and another part another board so you know when professor completes first time of course till that point so when he is writing here his student will come and then wipe it out this side okay yeah and when again he comes here and then goes that side he fills up so like that and their classes are one and a half hours classes most of the classes okay so German professor German professor really fantastic okay so this is inter and so we have now combinations now we are talking about inter and interface isothermal effectiveness factor and we also have combination of inter and interface non-isothermal effectiveness factor that we are not touching right now otherwise you know earlier we used to have one course separately on catalytic reactions only on catalytic reactions so that is why we used to do all that there I used to do when we had a stream in mtech where you know chemical reaction engineering stream I think that is 87 88 90 that time some 10 years or so it was there at that time we had five courses in reaction engineering that in non-catalytic reaction also was a separate course reactor theory was common for everyone and there was also a course called chemical reactor design for process plants only design reactor design very thorough but problem was you know operating problems we did not have sufficient students not reaction in a reaction engineering transport phenomena all transport phenomena is another another stream so these two all and process control these two always used to have there are other things like enrollment engineering biochemical engineering polymer engineering so always those three were empty that means only one or two so then we thought at one point of time let us merge all mtech so that they won't learn anything so we combined master of all jacks of none so I think that's what what we did so inter and intra phase isothermal effectiveness factor ya so ya first one I think as usual we will start with the diagram these diagrams are very good I said ya good I have become expert in drawing particles on the on the board and this film also you never see this kind of film anywhere in reality okay so this is the one in fact this is a very very good habit of drawing the profiles for whatever problem you take okay so this is r equal to 0 capital R small r equal to 0 this is r good so in the film now we have to show the profiles this is C B C S and C good ya and this side if I have exothermic reaction T B T S and you may have like this this is T S not linear the shape also exactly I don't know ya this is this is T S isothermal ya I am just plotting the whole thing there okay ya you are right here you can always assume that T B equal to T S equal to T right this is the whole thing these are the profiles what we are taking now that is right you are correct isothermal only we are talking right now good ya so when you are talking about isothermal ya this is a general profile it is not that I am now solving this problem using these profiles only these things good okay so we already have an equation for I think I will write here so we have R O B equal to ya under steady state conditions this mass transfer through the film must be equal to diffusion at the surface right that is what how you have done for interphase effectiveness factor so then you have K G A K G A C B minus C S here I have Eta bar K and not Eta bar Eta this is logical for me right so this Eta I know already that is internal interphase effectiveness factor so this is the intrinsic rate that multiplied by this will give me the rate on the surface okay now film is coming into picture now that film is because of this gradient C B minus C S right so what is the procedure eliminating C S and finding out what is R O B this is R S where the rate is based on the all the surface area of course right ya external surface area only but this Eta will take care of those gradient inside this one I do not know whether you have understood this or not okay Eta when I multiply okay if Eta equal to 1 then it will be straight there is no effect of interphase mass transfer no resistance so like that so because when Eta equal to 0.5 for example you may have this kind of profile so that will take care of that so then we have this is equation 1 ya solving this you will get these two C S equal to or C S by C B equal to I have 1 by 1 plus Eta K by K G A please check this this is step 2 quickly right or it is common sense correct good so then I have R O B equal to C B divided by 1 by K G A plus 1 by Eta K ya so this equation 3 we will try to arrange this in terms of an observable okay ya how do we do that this is R O B L square that means I am dividing this by L square and dividing by D E C B okay or otherwise of course I can bring that C B this side okay this side L square by D E and C B also okay let me and the same thing we also do here if we substitute that then I will have L square D E 1 by K G A just it is a manipulation that is all nothing new no equation is new and only mathematical jugglery here and there to get some beautiful information so 1 by Eta K 1 by Eta K correct ya so now this equation can also be written in terms of R O B that is an observable this is an observable D E C B which is which can be written as K L square by D E again manipulation there K by K G A plus 1 by Eta okay and this I know already K L square by D E what is that 5 square okay so that is the one now this particular part I have to also just manipulate D E C B this will be 5 square divided by I have here again K L square by D E here alone I am doing that again K L square by D E and I can also arrange that K G A L square by D E plus 1 by Eta just again I have divided by L square by D E L square by D E but remember oh no here remember K G A what is K G A A ya so what is that ya specific surface area for unit volume yesterday we have seen V by S X is what ya so that is also a characteristic dimension for us so that is why that L and this L can be cancelled so now I have K L by D E K G K G A L by D E what is that number Sherwood number will come only when you have single phase okay ya this is Biot number Biot number Biot number will come when you have more than one phase okay good ya so that is why we can write this this one again 5 square by 5 square by Biot number plus 1 by Eta okay good now Eta let us say for slab what is Eta for slab 10 inch 5 by 5 okay you can substitute that and then try to simplify that what do you get for slab for slab Eta equal to 10 inch 5 by 5 did you get this R O B L square by D E C B equal to 5 10 inch 5 divided by 5 10 inch 5 by Biot number plus 1 it is numbers gone total ya so this is 4 5 6 7 could you get that R into the substituting 10 inch 5 by 5 1 by Eta so 5 will go up and then you get something cancelled and you will end up with that equation okay good that is nice everything is nice okay so this is do you remember this is Eta R O B L square D E by C B only 24 hours gone I say Eta 5 square only yesterday only we have discussed about that is Eta 5 square so Eta 5 square equal to this observable equal to this okay and when you plot this this is what what you get okay this is Eta 5 square which is nothing but your observable this is Eta this is again isothermal so maximum is 1 this side you will have 0.100 somewhere here 10 somewhere here 1 1 10 so when I plot now Eta is the parameter not Eta what is that Biot number Biot number is the parameter so then you will have that is fine so this is Biot number equal to Biot number mass you know heat also has Biot number heat Biot number also is there so that is why we put this one as for mass so this is equal to 1000 this is equal to 100 this is equal to 10 this equal to excellent yeah take send it back sufficient send back yeah you have either you draw that quickly drawing is very good because you will have one more experience you know by drawing it that is why I did not give this first and how do you use this Eta 5 square is observable our idea is to find out the overall effectiveness factor okay I measure the rate I know the dimension I know DE correlations and C B so now I will go here read and then get the corresponding overall effectiveness factor okay so for that if both are controlling so this is the one what we have to learn and this is very simple now because I do not have to explain that much and yeah one more thing what we have to observe is that you know 100 and 1000 almost there is not much any difference so what do you conclude mass transfer yeah yeah right yeah controlling when it is all together reaction control mass transfer is not controlling so that is why when you design a particle you have this information on mass transfer coefficient and all that diffusivity you calculate Biot number once you calculate Biot number if it is 1500 like that then you say external mass transfer is not coming into picture at all so then you concentrate on internal interface effectiveness factors you know how nicely one can do K g is known I have correlations DE is known I have correlations but you have to find out at what temperature pressure and all conditions and then calculate what is Biot number if Biot number is generally 500 and 1000 safely ignore external mass transfer okay good I think when Nassim already was telling this is Sherwood number Sherwood number will not give you that kind of information this is only within one phase Sherwood number so here Biot number will give me between two phases okay diffusivity is inside and K g is within the particle I mean in the film yeah surrounding the particle the film so it combines those two film and then interphase so that is the reason why Biot number is more useful for heterogeneous systems and Sherwood number at all that is normally for single fluids that is you have all the correlations okay and of course there is no way you can also call this one as a Sherwood number 2 but you should know what is that diffusivity coefficient and what is the mass transfer coefficient is Sherwood number what does that give you Nassim already Sherwood number what is the information you will get from there K g divided by what is that D which diffusivity mass diffusivity it is within the same phase binary diffusivity so that will give you a ratio of convective transport by all that within the phase whereas here between the phases so that is the difference between this and that okay good yeah and the moment you cross the door anyway you will not remember anything okay but anyway examination is coming