 So, we will discuss or we continue our discussion on non ideal reactor models and we are looking at a multi parameter model or a two parameter model. So, in the last lecture we just started our discussion on this of course, like depending on geometry will have different kind of model is not a it is not like like you will have just one equation which will have two parameters and that is applicable to any reactor system. So, I will just give you one example right now and then we will see how it varies from reactor to reactor and we will have the two parameter model applicable to different reactor system, but then the equations would change accordingly. So, what we are looking at is the formulation for a two parameter model fine. So, we are looking at a normal CSTR and now looking at a geometry of the CSTR. So, it is like this and you have a nozzle here and you have a nozzle here and there is a stirrer and you have a feed coming in and there is the overflow that is happening and then there is a product that is coming out. Now, you will have nice mixing inside because of intense agitation right, but there is a possibility that you will have certain pockets in the reactor which are away from the stirrer or impeller and they may act like stagnant pockets or dead zones or relatively dead zones or relatively dead and there is another possibility that you will have a feed coming in and part of it will go and mix here or get inside a core, but then there is a possibility that part of it will bypass and go along with the outgoing stream. So, apart from dead zones you will have another non-ideality coming in picture which is called as bypass. So, there are three possibilities that you have a bypass then you have dead zones and of course, you will have the main core or well mixed zone right and all I know is this that well mixed zone is something similar to a CSTR whereas, these two things I do not know how to incorporate them so for at least how to incorporate them in the actual reactor model, but then if I know that there is something called as bypass there is a possibility of bypass there is a possibility of dead zones in that case I should incorporate them and one parameter for this and one parameter for this you will have two parameters as simple as that. So, it becomes a two parameter model and this is applicable to a stirred reactor that I am talking about if you have some other system there also you may have multi parameters not necessarily two, but bypass and dead zones are normally observed in many reactors many real reactors and there is always some one parameter associated with one of these effects. So, in general a reactor or reactor model that considers bypass and dead zones can be considered as a two reactor sorry two parameter model for a mixed reactor for a mixed reactor or back mixed reactor or stirred reactor I am not calling it as CSTR now because CSTR is ideal reactor that we always say this is stirred reactor stirred tank in which you may see bypass and dead pockets. I am going to look at some other situations as well where you will have will come across two reactor two parameter models or multi parameter models this is one example. Now, we will try and elaborate this further try and give a mathematical treatment to it and try and predict the conversion if possible. Now, first of all how do we know that there is a possibility of bypass how do you know that there is a possibility of dead zone who will tell you again the tracer experiment. See that is the importance of tracer experiment is such a useful technique that tells you the health of the reactor health in the sense what kind of flow patterns are there inside what is happening inside a stomach. So, I may treat this particular reactor this is my real reactor equivalent to something like this see bypass means it is not spending time in the reactor at all. So, it is just going out sorry it is going out. So, bypass is directly going out that means not spending time in the reactor whereas, some part is remaining here dead zones which is not seeing turbulence at all. So, I can look at this particular reactor as a set or network like this. So, there is an inlay that is coming in part of it is going to a CSTR sorry a CSTR part of it is bypassing and how do I represent the dead zone there is a dead zone here. So, some volume is dead nothing is going inside nothing is coming out of it is dead. So, from outside I will look at a big reactor, but part of it is dead means the actual active volume is much less or relatively less. So, V D is something that I need to subtract from the total volume. So, V D plus this I may call this as V S. So, V S plus V D is V. So, V is equal to V S plus V D right. So, now this is my inlet concentration is C A 0 I am giving now a mathematical treatment to this particular system flow rate V 0 part of it is going here V S and part is going here bypass V B and just calling it as V B. Now, what is coming out or what is there inside is C A S right this is C A 0 same C A 0 nothing happening. Whereas, because the reaction something is happening here C A S is different from C A 0 right V D I do not have to worry much about only volume wise I have problem there. So, V 0 here the total volumetric flow rate I am assuming it to be constant say liquid phase reaction which is nothing but V B plus V S right and this is C A the resultant concentration. So, ultimately I am going to look at C A. So, this is nothing but C A here this is C A right and this is C A 0. So, C A 0 coming out is C A here again coming out is C A this is C A 0 what happens inside is I have just formulated a network. So, this is how I am going to look at this particular reactor this reactor for me is this as far as the modeling is concerned. Now, it becomes very easy for me to write down the equations for this because every part of this particular network I am well aware of how to write equation for it how to give mathematical treatment to it. So, let us do that let me draw it again because I will I will always need to refer it again and again while writing equations sorry. So, I will spend some time in this C A C A 0 right V B V S C A S C A 0 V 0 is equal to V B plus V S all right. Now, so coming in in is equal to out. So, let me write whatever is coming in and I may take a balance at this particular point at this point at this point whatever coming in is C A 0 into V B V B into C A 0 plus C A S into V S is equal to plus C A S into V S is equal to C A into V B plus V S or nothing but V 0 right. So, let us go ahead C A is equal to that means C A is equal to V B C A 0 V B C A 0 plus C A S V S divided by V 0 that is outlet concentration. I am interested in outlet concentration. So, let us go ahead C A is equal to that means C A is equal to V B C A 0 V B C A 0 plus C A S V S divided by V 0 that is outlet concentration. I am interested in outlet concentration so in this I know the inlet concentration I know the flow rate what I do not know is what is V B or what is V S once I know V B I can get V S because V S is nothing but V 0 minus V B I do not know what is C A S right these two are known and these three are to be found out. Once I get these I can get a value of C A and that is nothing but my conversion as simple as that. So, fine let us go ahead let us assume let me retain the same figure and then keep writing equations. So, same figure now that balance this one C A I just will write expression for this now before. So, let me as let me say that alpha is equal to V S by V 0 what is V S? V S is the volume of this reactor and what is V is a total volume that is alpha is a fraction of volume which is active sorry fraction of volume which is a total volume that is active that is nothing but alpha right then let me assume beta is equal to V B by V 0 V B by V 0. So, your total flow rate V 0 so out of which V B goes as bypass so this is bypass ratio bypass ratio beta beta V B by V 0. In that case my this expression look at this expression C A let me write it here C A is equal to beta into C A 0 plus 1 minus beta into C A S plus 1 minus beta into C A S plus look at this expression I am using beta there I just divide numerator and denominator by V B and solve fine or no need to do that C V B by V 0 is equal to beta and V S is nothing but 1 minus beta. So, I have express C A in terms of C A 0 and C A S in terms of and again with the parameter beta. Now, what about C A S now where does how will you get a value beta and all will see later by the way like can you guess where will it come from it will come from a tracer experiments how much bypass is occurred tracer experiment will tell me right. So, beta values to be determined through tracer experiments independently. Now, in this expression if beta is known what remains is only C A S. Now, how do we calculate C A S? C A S is your simple reactor here you see it is a reactor and C A S is the outlet concentration of this reactant this reactor is well known to me no dead volume no bypassing it is the area which is fully active as far as mixing is concerned is well mixed stirred tank reactor nothing but a C S T R. So, outlet concentration of the C S T R can be very well found out by writing material balance here. So, C A S I just write a balance for this I just write a balance for this I write a balance for this C A S to be found out. So, V S into C A 0 inlet plus sorry minus V S into C A S outlet minus reacted K C A S into V S is equal to 0 is a normal C S T R balance is a C S T R balance. So, C A S now can be written in terms of alpha and beta. So, this expression this equation and this equation I do not need this equation for this. So, let me say the C A S is equal to C A 0 into 1 minus beta into V 0 divided by 1 minus beta into V 0 plus plus plus plus plus plus plus plus plus plus plus plus alpha into V into small k by the way why alpha is there because I am multiplying it by capital V which is a total volume because from outside I am just going to look at a total volume I do not know how much is the active volume. So, total volume into alpha total volume into alpha. So, V S for V S I have substituted and got alpha into V. So, this expression you can derive it from this I get this is very simple because you know the expression for V S. V S is 1 minus beta into V 0 1 minus beta into V 0 1 minus beta into V 0. So, this V S is going to come out as 1 minus beta into V 0. So, you can derive it on your. So, this is the expression that I have got for C A S and that is what I want in this equation. So, substituting for C A S here from here from here I substitute for C A S here what is this? This is out late concentration overall out late concentration overall concentration that I get it out late as a net effect of these two strings bypass and this. So, I have incorporated bypass I have incorporated bypass I have incorporated dead volume as well through alpha. Fine, now I get the expression by substituting for C A S if I do that then C A by C A 0 is equal to 1 minus x you know that no C A by C A 0. See, I had expression for C A I am just dividing by C A 0 that is nothing but 1 minus x I want conversion ultimately. And if I substitute for C A S I will get the expression you can derive it on your own see what is more important is a methodology and not a final expression that you get of course you need to get a right expression, but then methodology is important more important rather. What do I see on the right hand side? Beta and alpha these are the two parameters these are the two parameters this takes care of the dead volume this takes care of bypass case rate constant first order reaction I have considered here and tau is a residence type there is no total volume do not forget that nothing to do with V S total volume that I I know of in the sense I just go into measurement in the in the actual reactor I measure the diameter I measure the length and from that I calculate a volume blindly I do not see what is the active volume I just see why this is a liquid level height and this is a diameter pi d square by 4 into L. So, this is what I am going to calculate so that is so tau will be based on that volume volume divided by volumetric flow rate. So, this is the expression for conversion is a two parameter model now the question is how do I get this these two that is the only question once I know how to get these values I have a two parameter model for a CSTR not CSTR what is third tank which is behaving in a non ideal way with bypass and dead volume I can predict the conversion for the given volume given flow rate. So, the next question is how to get alpha and beta of course I am of course the answer is tracer experiments E curve or C by C T C T versus time curve. So, for the same system can I write unsteady state balance look at this this particular reactor system I will say network can I write unsteady state balance for this that is simple because this is there is no hold of here there is no hold of here. So, I can write this and there is no interaction with this part of the reactor. So, I have to write a unsteady state balance for this part only right. So, what is that unsteady state balance let us write that unsteady state balance for the CSTR for the CSTR for the CSTR it is V s C T 0 now this tracer experiment C T 0 minus V s C T s coming out of CSTR is equal to can I write reaction though I am calling this CSTR is no reaction this third tank tracer experiment. So, D N T s divided by D T D by D T of N T s accumulation term which is nothing, but V s into D C T s by D T unsteady state balance for tracer no reaction no reaction. And what are the conditions for the positive step input at time less than 0 C T is equal to 0 and I have to write C T is equal to at time greater than 0 or equal to 0 C T is equal to C T 0 C T is equal to C T 0. Tracer equation I know the conditions for the positive step inputs are this now what I need to do because see again what I am going to see at exit H sorry at exit concentration is this. So, I need to combine these two now these and this. So, I need to take balance at this particular point now what is that balance we already seen that. So, balance at this point for the tracer is C T is equal to V B into C T 0 plus C T s into V s divided by V 0 sorry at this point I hope it is clear I am just writing balance for the C T coming out here C T is here and it is a combination of these two. So, they have two terms here and we know V s is equal to alpha into V V B is equal to beta into V 0 plus C T and tau is equal to total volume divided by V 0. Now, just a matter of integrating this equation integrating this equation and substituting for substituting for V B and V s. So, what we get is the final expression in terms of alpha beta and tau that is rest of the incidence time. So, the final expression after doing all this is C T s divided by C T 0 is equal to 1 minus exponential minus 1 minus beta alpha T by tau. So, this is the expression for the tracer experiment or tracer concentration at exit C T s. Oh sorry not at exit sorry sorry this is this is at at the outlet of C S T R at the outlet of C S T R here sorry here because I have the equation for C T s the unsteady state balance has been written for C T s. So, if I solve this I get expression for C T s, but that is not enough because what I am going to see through my tracer experiment is the concentration see sorry here this is my C T I am going to see this concentration. So, this is no mean I am not going to measure this I am going to measure this right and for that I have the expression I have the expression for C T what is that expression we already seen this C T right this expression. So, now I know the relationship between C T s and C T I have already determined C T s this. So, I need to just get C T. So, just substitute this here and get C T that is it. So, if I do that I get an expression for C T which is nothing but C T divided by C T 0 is equal to 1 minus 1 minus beta exponential minus 1 beta divided by alpha T by tau s. So, this is the outlet concentration this is the inlet concentration for the tracer beta alpha and tau with respect to time. So, I have got a relationship out C T changes with respect to time if I have these parameters out of which tau is known because total volume and volumetric flow rate are known to me all right. So, in this expression if I know beta and alpha I get C T or the other way round if I know C T as function of time I get a values of beta and alpha I get a values of beta and alpha. So, I can probably for calculation purpose I need to I will just rearrange this equation and I will put it in this particular form C T 0 C T 0 minus C T is equal to l n 1 by 1 minus beta plus 1 minus beta divided by alpha into T by tau the same as this you can you can spend some time doing this. But why I have done this because now I have a linear plot of what time versus C T 0 divided by C T 0 minus C T. So, of course, if I if I do log log. So, if I if I plot l n C T 0 minus sorry C T 0 divided by C T 0 minus C T 0 divided by C T 0 minus C T l n versus time what I am going to see is a straight line you see is a is a time y is equal to m x plus c y is equal to this m x plus c right. So, I am going to see a straight line for which the slope is what is the slope slope is 1 minus beta divided by alpha into tau and you know the intercept it is l n 1 minus beta right. So, if I do experiments in the laboratory and plot this versus this I get a straight line the slope gives me this and the intercept gives me this. So, with these two expression I can determine beta and alpha right. So, if I determine beta and alpha at my job is over because I know the conversion in terms of beta and alpha and tau because I already derived an equation for the conversion before what is that equation this is the equation right beta alpha and tau k of course is to be known and now I have told you the procedure to get alpha and beta right through experiments through tracer experiments. So, once I know the values of alpha and beta I can get a value of conversion two parameter model stirred tank with bypass and pockets I have incorporated both these effects through two different parameters called alpha and beta is one example you may come across some other non ideal reactor or a real reactor where something else is happening right. So, what is that? So, let us consider another example just to appreciate that it is not that difficult to consider the effect of flow patterns or non-ideality what is more important what is different here we have used the same reaction engineering principles that we have learned earlier. Only difference is the tracer experiment is able to give me the idea about a flow pattern so that I can formulate a reactor model non ideal reactor model with different parameters in it right and this model is able to give me the right conversion fine. Now, this is another example where you are talking about a reactor a reactor where you have a nozzle and there is a stirrer it is a long reactor relatively long reactor and is a pipe that is coming in sorry that is for the field. See what happens now kind of flow pattern I do not know like it all depends on kind of impeller that I am using so it is quite possible that you have in the upper part you have some circulation happening the lower part also you have some circulation happening sorry. Now, what is the difference in this reactor and normal reactor difference is that you can see that is no flow path or rather there are very few flow paths which are taking streams from here to here there is always some exchange between these two, but you can see that there are distinct zones formed there is like upper part where there is a mixing happening lower part also there is some mixing happening there will be some exchange there will be some exchange this is not stagnant there will be some exchange, but then this concentration may not be same as this concentration because of the extent of mixing that is happening and the exchange that is happening between these two. The exchange is such that the concentration is not same the concentration is not same in these two regions. So, can we treat this as a normal CSTR or ideal CSTR we cannot because ideal CSTR means that the concentration is uniform everywhere constant everywhere right. So, how can I formulate a model for this particular reactor now first of all whether this happens or not who will tell me again a tracer experiment. So, sometimes the tracer experiments tell you about a flow patterns. So, the see now if you are exporting this field then the moment you see the tracer experiment results C T versus T or E T curve you know what what is happening inside a reactor. So, it is like doctor examining a patient based on a symptoms he would know what exactly is happening similarly based on this tracer experiment you know what is happening inside. So, try to understand importance of tracer experiments E curve right fine. So, this can be looked upon as two CSTRs exchanging matter and a feed is going to this CSTR and the product is coming out from here as simple as that you see there is a CSTR here feed is going in product coming out and this is exchanging mass with the upper CSTR. If this nozzle was here I would have put a outlet here as simple as that this is this is a network now this is the network. Now, what are the parameters for this network this is v 0 this is v 1 and v 1 and this has to be v 1 and v 1 otherwise there will be accumulation. So, whatever going in would come out whatever going in would come out and of course, this has to be v 0 this would be C A 1 and this would be C A 2 they cannot be same if they are same then they know point in making two different zones why they are two different zones because the concentrations are different right. So, this is my network model now you can identify the parameters what are the parameters first parameter will be v 1 divided by v 0 this is one ratio and the other parameter would be the volume of one of this CSTRs divided by the total volume. So, v 1 divided by capital v this is another parameter. So, this can be beta this can be alpha the meaning of alpha and beta are different here and they are not same as what we either it was that there it was bypass ratio, but in order. So, I can use some other symbol is not an issue but then it is very clear like say v 1 by v 0 this flow rate is one parameter and this is another parameter what am I going to do now I am going to write equation for both of them right both this CSTR combine them solve them together now it is it is like exchange. So, I have to solve this equation simultaneously see I have to solve this equation simultaneously. So, mathematical exercise will become slightly difficult because your model is a relatively complex model, but not nothing to worry much about it. So, solving the model system what you see is like see v 1 is equal to beta into v 0 capital v 1 is alpha into v capital which means v 2 is equal to capital v 2 is equal to 1 minus alpha into v tau is equal to v by small v 0 this is capital v this is total volume total volume. So, if you can derive equation write the equations of both the CSTR solve them together what I would get is C A 1 is equal to what is C A 1 look at this C A 1 and that is what I want that is coming out here it will have impact of what is happening here as well for this reactor see is going in and coming out going in and coming out from this reactor. So, this is in what is coming out from this reactor is inlet for this. So, this has two inlets here. So, this will be determined not only by this, but by this as well fine. So, C A 1 is equal to C A 0 divided by 1 plus beta plus alpha tau k first order reaction beta square divided by beta plus 1 minus alpha tau k this is what it is I not writing equation solving them in front of you, but then you can very well appreciate I just write those equations for both CSTRs and then solve them simultaneously I get this expression and this is nothing, but of course this is nothing, but C A 0 into 1 minus x sorry 1 minus x is conversion. So, see the similarity between this problem and earlier problem though geometry was different methodology is quite similar I am just formulating a model coming up with parameters getting an expression for the exit concentration which is related to conversion and. So, I am I know conversion how it depends on the parameters alpha beta right tau of course it has to depend on tau residence time and the rate constant and inlet concentration of course this will get cancelled C A 0 C A 0 for the first order reaction alright. Now, again the same procedure how to get alpha and beta how to get alpha and beta same question and same answer pressure experiment I have a system where I have a CSTR which is exchanging mass with another CSTR it is not a dead volume do not forget it this exchange happening earlier dead volume there is only one line that I had drawn now there is a arrow which is going here and there is an arrow going here right. If I write unsteady state balance for this non reactive conditions because the tracer that is going in the tracer that is that I am that is going in. So, if I write unsteady state balance for both the reactors and convert it to beta alpha tau and k I get expression like this tau alpha you can derive it on your own right now I am just writing it directly to show you or explain or rather demonstrate how similar the entire exercise is compared to the earlier one. So, this is for this reactor 1 for 2 there will be another equation tau 1 minus alpha look at this 1 minus alpha d C 2 by d t is equal to beta C t 1 minus beta C t 2 every terms tells you something. So, you can imagine very well imagine how this equation is come nothing great about it is coming in going out accumulation similar thing happening here. So, but then now what is happening here is like you have two equations they coupled they coupled because C t 2 is appearing here and C t 1 is appearing here. So, I cannot independently solve them. So, I have to solve them together I have to solve them together this is a set of O d is to be solved together you can use numerical technique or one can do it analytically also right and get a values of alpha and beta. Now, the in I have to solve this earlier case it was quite easy for us to get a value of alpha and beta because it will linear plot we plotted some l n C t 0 divided by C t minus something versus time and then that plot slope and intercept gave us some values of alpha and beta. Here it is going to be a bit difficult because I have O d is I can even get some equation after solving this after solving this I may get some equation and that equation analytical expression for C t 1 because C t 1 is something that is coming out and I need to compare. So, this plot of C t 1 versus time I may see something like this I am just giving you some picture just guess something that C s t r. So, it is going to go like this I will get value at 0 also very close to t is equal to 0 right. So, I am going to get something like this let us assume that now this is by experiment and by solving these equations by solving these equations I will get some behavior. Now, these equations if I want to solve these equations I need to have values of beta and alpha. So, the procedure may be you assume some value of alpha and beta you may get a behavior like this which is much different from this. So, there is some error there is some error right and this error we need to minimize you minimize it in such a way or other assume the values of alpha and beta in such a way that this error is minimized right. I think we have seen this before also you have to have use a least square optimization technique. So, it is a numerical technique it is not so easy to determine the values of alpha and beta in this particular case because I do not get a straight forward linear relationship I cannot rearrange those equations. So, that way this particular problem is slightly complicated compared to the earlier one, but methodology procedure is quite similar. So, go on changing the values of alpha and beta systematically there are techniques available such that this error is minimized or least square error is minimized that means c t 1 predicted minus c t 1 experimental square divided of course, you can normalize it c t 1 experimental square if this this is minimized. So, this is you have to minimize this for the values of alpha and beta. So, this becomes a optimization problem. So, get a values of alpha and beta in such a way that this is minimized this function is minimized and you can do it using software or rather they can be a solver they can be an optimizer a least square method which uses least square method and you know why square and why it is called a least square method and all that square is because like I do not want to consider the sign of the error. Otherwise two errors may compensate each other and they may give you the function to be they may say that function is 0, but that may not be a situation you are much away from the actual real values. So, that is why square. So, it make sure that you have a positive values all right determine alpha and beta. So, like this there can be different situations that you may come across a simple scenario. Now, there are many possibilities how do I identify what kind of model I may I should go for from the e curve itself. So, let us let us consider a situation I have a tubular reactor, but not a simple tubular reactor where I just have one inlet there are two inlets at the reaction of A plus B giving C. Now, C sorry B is coming this way as jets all along the periphery. So, I have I am showing two jets, but they can be multiple jets along the circumference and is one reactant coming in this A and all the B is going this way. There is a reason why because I want these these two mix get well mixed here both these reactant as and when they come because if there are some zones where only B remains or only A remains then there is a problem because it may go undergo side reaction. So, in this zone I want them to mix thoroughly and later on I do not have any problem let them go together or in a plug flow manner once they are mixed well. Now, this is one reactor this is one reactor how do I model this reactor it is very simple is not it. If I before that if I just do a tracer experiment on this that means in a non reactive conditions suppose I inject a tracer at the inlet what will I see at the outlet there is a material flowing like this I am just injecting tracer now there is no reaction I am just sending some inert in this flow pattern is similar when the reaction is taking place. What is what what will I see here I have injecting a tracer a pulse what do I see here if it was a simple plug flow tubular reactor I will see a pulse at time tau right, but now because there is some mixing happening I am going to see something different. So, let us assume that this part of the reactor is well mixed. So, this behaves like a CSTR and then it behaves like this particular part behaves like a plug flow reactor. So, what I am going to see is initially at this point suppose I take measurements I will see a CSTR type behavior right. So, this is a CSTR. So, this is a CSTR this is a CSTR and here it is PFR. So, after this it is going to behave like a PFR. So, this this particular response is going to come out at this point exactly at as after the time interval equivalent to the residence time spent here. So, the behavior is going to be like this. So, if I see an E curve or C curve whatever like this then I may say it is a combination of PFR and CSTR in series of course, you have to remember one thing whether it is first the PFR or CSTR because in both the cases you are going to get a same E curve. Then you will have to think about what is a real situation. Now, I know that I am sending z CSTR. So, definitely first part is CSTR. So, without knowing this just based on this I cannot make a reactor model. There are many possibilities of combination of these reactors that can give rise to same E curve. So, one E curve is likely to give you different reactor models, but knowing more about the real reactor you can shortlist them and come up with the final model. So, for this particular case my network is this a simple network is this a CSTR followed by PFR. Once I know this I can definitely find out the conversion what is the parameter for this model? There is only one parameter here I know the total volume how much of the total volume is covered by or other is taken up by CSTR and how much of it is PFR that is the only parameter now. It is a one parameter model, but if for some reason I see some dead zone here that means area under the curve for E curve is not one. Traces still comes out like it keep coming out long time in area under the curve calculated based on the total volume is not matching one that means there is some dead volume. So, that means there is some dead volume sorry I can say that there is a dead volume here not here sorry this is a real reactor. So, this is a dead volume here that is how I look at it. So, then the dead volume becomes another parameter alpha something that we used before. So, that is why that is how we look at different types of reactors and at the same time from the E curve or C curve I can very well get some idea about what kind of flow patterns you may have inside a reactor otherwise. So, there are many possibilities just one of them you have a tubular reactor and some streams are lagging behind some streams are going ahead from. So, it gets divided for some reasons this is going very fast and this is going slow. What is the model for this? The model is that you have a feed that gets divided into PFRs right is again one parameter model how much goes here how much goes here oh sorry it is a two parameter model how much goes here how much goes here and the volumes of this. So, again is a two parameter model there is a possibility that you have something like this there is a mixing happening here and this is going straight. So, you this is equivalent to sorry this is equivalent to PFR and CSTR in parallel right again two parameters how much goes here and what is the ratio here the volumes right. So, many such scenarios you can have even more complicated that you may have some 3 4 CSTRs in series then there is one parallel PFR. So, many things happening depending on what kind of situation you are in normally for a chemical plant our reactors are well shaped reactors, but if you are talking about some natural reactors reaction taking place in space or reaction taking place underground in that case geometry is not in your hand then that case will have to make assumptions or it can be a more complicated or a complex network of different ideal reactors and bypass dead volume and so on. What is the key is a tracer experiment the E curve the C curve tracer experiment you should learn to read the tracer experiment results properly if that happens then your job is easy of course mathematical treatment is something that you can definitely do, but then solving the equations numerically and all you may need to take help of software and all, but a method law is very clear. So, it is a tracer experiment that gives you an idea about a reactor behavior as far as the flow patterns is concerned. So, this is all about how we use multi parameter models for real reactors I have shown you two parameter model here, but you can very well imagine scenarios where you can have now the same case here like for example, in this case like you have two streams giving, but you may have another rather some part of the reactor where you will have an intermediate velocity. So, you will have third PFR in series. So, and then the parameters would increase as simple as that how number of parameters will grow that will depend on the complexity. So, first thing is you need to come up with the proper reactor or multi zonal model or network. How do you do that? The two things that are required for this first is your E curve tracer experiment and the knowledge of the system as I said only E curve can give rise to multiple networks giving the same E curve. Out of those multiple networks whether it is first CSTR and then PFR or first PFR or then CSTR that I would know based on some knowledge of the system of the reactor that I am going to use. These two will tell me or these two will help me formulate a network. Once you have a network write down balance equations get the expression for the conversion and from in that expression you will have the parameters for the network those parameters are to be determined through tracer experiments. So, I will just write down quickly what it is formulate a model formulate a model this is going typically going to be a network which will have CSTR, PFR, bypass, dead volume. We have not seen many other cases a recycle exchange many features possible. Who will this is based on what? This is based on based on E curve or tracer experiment and knowledge of the reactor system. Why do I say system? Because not just reactor geometry, but where is the inlet nozzle, where is the outlet nozzle all these things will matter. Once you have once you formulate a model network write balance equation and get x conversion. You may get expression for this or you have to solve these equations numerically. In terms of model parameters for example, alpha, beta and so on. Now, third step would be get parameters from the tracer experiments and then evaluate x. Now, this step may need use of optimization solvers say least square or whatever. So, this is overall procedure. I hope this is clear like how to incorporate or how to consider non-ideality in real reactors and predict the right conversion with the help of E curves. Thank you very much.