 We go on to advanced reaction engineering. We now look at residence time distribution. Now, why are we interested in looking at residence time distribution? Of course, it comes from common sense that every reaction equipment, the time for which the reagents sit inside the equipment determines the extent to which the reaction would occur. And that is a prime motivation to understand the distribution of residence times inside the equipment. So, typically if you have an equipment like this and then we have fluids coming in and fluids going out. And you would like to know what is the time of residence of different fluid elements inside the equipment? We provide what is called as a tracer. We mentioned earlier that the tracer can be a solid, can be a liquid, it can be a gas depending upon the systems that you are looking at. And only important thing is that the tracer must be able to mingle you know with the streams. And then should have properties same as that of the stream and it should be measurable very accurately. So, that we can get the information of our interest. So, we said that if you put a tracer and measure the composition at the exit with the function of time. If you put a pulse of tracer by pulse we mean is that we put a certain milligrams or grams of tracer very quickly and that is called the pulse. And we look at the response of the system with respect to time. So, and then if the flow is v we can measure this, we can measure c, we can measure v c and this will be something like milligrams per some time say minute versus. So, we will get a curve like this, we do not know what the curve is, but whatever the curve is from here we said our e function or residence time distribution function is given as v time c d divided by m 0 where m 0 is the total amount of tracer that we have put into the system b t. So, what is it mean this term v time c t represents the material that is leaving the system at time between t and t plus d t. Well this m 0 represents the total amount of material that we have put in therefore, this fraction refers to material with time of residence in the equipment between t and t plus d t or shall we say a time of residence t. And this is the information of our interest. Now, we can do this for any kind of equipment and we said little earlier that if you have a stirred tank, we said little earlier that if you have a stirred tank that means if you have a stirred tank and you have a material coming in and going out. And the residence time distribution function for the stirred tank we said is something like this. We derived this where tau is the residence time which is given as v divided by v 0 v 0 is the flow. Now, we also said that suppose you have a sequence of stirred tanks, sequence of stirred tanks flow is coming in here going out here out here and this is tank 1 2 and n. We derived this and using first principles to say that the time of residence time distribution of the n tank sequence is given as t to the power of n minus 1 divided by factorial n minus 1 tau i to the power of n e raised to the power minus of t by tau i where we said tau i equal to tau i equal to v of any tank divided by v 0 that means the assumption is that tau 1 equal to tau 2 equal to tau n that is the assumption. Therefore, the total residence time tau which is v divided by v 0 equal to n times tau i. Based on this what we have derived and therefore, this function is a result of the experiment that we have done which means that we take put a put a tracer here put a tracer here and then measure at the end of this time this is what we will find if it is a number of stirred tanks of equal volumes. We also said that if you want to understand the performance of a real vessel you want to understand the performance of a real vessel suppose you have a real vessel. We do not know how this and this fluid is coming in fluid is coming out going out and we want to see how this system is performing as a chemical reaction suppose the reaction of this type occurring what is the expected performance of this equipment. So, this equipments because of this tracer study we have done we know the sigma square we know this mean and variance both mean and variance are known because we know the residence time distribution. So, what is our expectation the way to do this is that let us say we have this particular system consist of n tanks as an assumption suppose it is it can be approximated as an n tank sequence. So, that the first time this is first second and so on first time will perform like this therefore, the n th tank will perform like this therefore, or the performance C n by C naught which is essentially 1 minus of x n this is given by 1 by 1 plus k tau i to the power of n. And we also said in our analysis of n tank sequence that this that sigma squared we mentioned this if you recall n equal to mu squared by sigma squared we mentioned this. So, this is known on other words just to summarize if you have an arbitrary vessel and we have done the RTD test of this vessel for from which we have found out sigma squared and more square. Therefore, this arbitrary vessel can be approximated as a sequence of stirred tanks where n is given by mu n squared by sigma n squared. So, once you know that an arbitrary vessel can be described in terms of an n time sequence if it is the first order reaction we know that 1 minus of x is given by C n by C n 0 therefore, this given by 1 by k tau i to the power of n. Now, we can replace this tau i tau i is tau by n or it is same as mu n by n therefore, and this you can replace this n from the fact that n is equal to mu n squared by tau n sigma n squared sorry sigma n squared. Therefore, n everything can be replaced in terms of measured numbers the measured quantities are mu n and sigma n therefore, 1 minus of x equal to C n by C 0 which is equal to 1 by 1 plus k tau i to the power of n is also equal to 1 by 1 plus k what is tau i tau by n to the power of n that is equal to 1 plus 1 by k what is tau tau is mu n correct mu n. And what is this is mu n squared by sigma n squared and then what is n it is mu n squared divided by sigma n squared or simplifying we get this as 1 plus k 1 plus k sigma n squared by mu n to the power of mu n squared by sigma n squared equal to mu n squared by mu n squared C n by C 0 is 1 minus of x what is it that we achieved what we are saying is that if you have an arbitrary vessel we do a tracer test on the arbitrary vessel then that tracer test gives you sigma squared and mu n squared. And once you know sigma squared and mu n squared you know the number of tanks which is given by number of tanks can be represented as mu n squared divided by sigma n squared. So, in view of this relationship and if it is a first order reaction we can now say that the performance of this n tanks of this arbitrary vessel approximated as an n tank sequence can be given by this equation 1 minus of x given to the right hand side where all numbers are known because rate constants are known sigma n mu have come from the experiment. So, this RTD test now helps you to determine the performance of an arbitrary system approximated using the parameters of the tanks in series model. So, this is where the advantage of this tanks in series model comes in we are in a position to tell what is the likely performance of an arbitrary vessel. We go on further the second type of descriptions that we want to do say conversions directly from RTD. What you want to do now? You want to directly take the RTD data and say what is the likely conversion? Let us look at this problem again. We have an arbitrary vessel, we have fluids coming in going out and one fluid element goes like this, another fluid element does this, does this, does this, goes out like this and so many fluid elements seem to spend various lengths of time inside the equipment. We know from our understanding that the extent to which the fluid elements would react depends upon how long they spend inside the equipment. Therefore, the time of residence inside the equipment is crucial to the extent to which the reaction occurs. Keeping this in mind, let us say our inlet fluid consists of number of fluid elements. All those fluid elements enter at some concentration C A 0. So, fluid element 1 enters like this and then goes around comes out that is I call this is C A 1. There is another fluid element C A 2 which goes around comes out C A 2 and so on. So, different fluid elements enter start with the same concentration, but go through the equipment and come out with different concentrations. Why? Because they have spent different lengths of time. So, let us say we have C A 0 which is this element, this C A element I will call this as C A element. So, one C A element 1 which starts at C A 0 and let us say the chemical reaction is A going to be with rate constant k first order. Therefore, this is C A 0 e raised to the minus of k t. This is the extent to which because it is spent time of residence t 1. This spends t 1 seconds, this spends t 2 seconds and so on. So, this fluid element 1 starts at C A 0, but emerges as C A 0 e to the minus k t 1. This is what we would expect because it is spent t 1 second. Therefore, it will be reacted and become C A 0 e to the minus k 1 t. What is the fraction? What is the fraction of material that belongs to this residence time? If you know the e function, we think it is e t 1 d t 1 is the fraction that belongs to time of residence in the equipment which is t 1 seconds. This is something that comes out of your RTD analysis. There are another fluid element C A 0 which is 2 which starts at C A 0, but reacts for a time of residence t 2 and the fraction of fluid elements in the equipment that belongs to time t 2 is e t 2 d t 2. In other words, you have various fluid elements entering, but it is going through the equipment for different lengths of time in emerging. Therefore, the C A exit or what comes out must be some average of whatever goes in, multiplied by the e function. Whatever goes in, multiplied by the e function, what are we saying? What we are saying is that, if you have a first order reaction and if you have an equipment in which fluid elements go through and then come out at different lengths of time and each of those fluid elements react as per a first order law, then the fraction that belongs to t 1 is e t 1 d t 1. Fraction that belongs to d 2 which spends time t 2 seconds in the equipment is e t 2 d t 2. Similarly, so many other fluid elements. Therefore, the average concentration of material that emerges from this equipment will be what happens to each fluid element multiplied by the fraction of material that belongs to that time of residence inside the equipment. So, this will be the average concentration that comes out at the exit. So, in other words what we are saying is that RTD theory directly helps you to tell you what is the likely concentration of fluid elements at the exit. If you can tell what happens to each element and what is the fraction of time each of those elements spend inside the equipment. So, this general theory helps us to tell what happens to the chemical reactions taking place. So, let us take an example. So, what we are saying is that as per RTD theory the average concentration that comes out is C A 0 e to the power of minus k t multiplied by the fraction that belongs to this time of residence. Example, let us take an example that this reactor which we are trying to understand is a stirred tank. Let us say it is a stirred tank which means the e function for this tank is 1 by tau e to the minus t by tau. This we know because if it is stirred tank this therefore, let us see what is the concentration that comes out from this reactor as per the RTD theory. So, it should be what minus of k tau k t this multiplied by the e function which is 1 by tau e raise to the power of minus of t by tau and d t. What are we trying to say? What we are trying to say is that if you have an arbitrary vessel of e function given in this case if that arbitrary vessel is a stirred tank then the e function is 1 by tau e to the minus t by tau. Therefore, if it is a first order reaction each element that enters with emerge with this kind of concentration. Therefore, the average that you will see at the exit from the equipment is the average of all elements between time of 0 to infinity that goes in and comes out multiplied by the e function of the equipment. Now, we can integrate this I mean it is quite common sense please you can check it yourself when you integrate this you get the output as c a 0 1 plus k tau s where tau is the residence line. I have not done the integration, but because fairly straight forward showing that the average concentration of material comes out is c a 0 divided by 1 plus k tau s. Now, this equation is something that we have seen for a very very long time this is the output from a stirred tank. Sturred tank gives you this kind of output what have we now saying then what we are saying is that if we know the e function of a system. If you know what how the reaction takes place then we have a way of telling what is the concentration of the exit. And the proof is that moment you put E t as a C S T R we get results that we have been using for a long time. It is a way of validating the way our new procedure of understanding stirred tanks it is validates the same understanding we already have based on our previous knowledge. Now, let us look at another model called the dispersion model. We have so far talked about what we call as the tanks in series model for describing an arbitrary vessel. Then we talked about what we call as the general RTD model we call also call as the completely segregation completely this particular model directly from RTD is also called completely segregated model completely segregated model. What are we saying here when we say it is completely segregated model what we mean is that every fluid element that enters goes through the equipment without recognizing the existence of other equipment it emerges. It is here that we do an averaging of the concentration and find out what is the exit concentration that means as per the model each fluid element moves through the equipment without recognizing the existence of other fluid elements and emerges. That means it is completely segregated as it goes through the equipment and only at the exit the mix and we measure the concentration. Some people also call this as a late mixing model late mixing. Why is it called late mixing model because we assume that the mixing occurs after the exit from the reaction equipment completely segregated model is also called as late mixing model. What is that model the models assumes that the fluid elements move through the equipment without mixing with each other they emerge outside and mix and we measure that average concentration. So, that is the completely segregated model the previously we looked at the tanks in series model what is the tanks in series model is an instance where as soon as the fluid enters the tank it mixes. So, it is an instance of completely micromixed model. So, this something that we did earlier this is the completely micromixed model for which we got our answers here this is the completely micromixed model completely micromixed. So, the completely micromixed model completely segregated model these two we have used already and we know roughly what kind of answers they give. Now, we are looking at a third model called the dispersion model let us understand what this model is. What are we saying here is we have a reaction equipment fluid comes in fluid goes out. Now, let us see what happens at any interface at any volume what is it that is going on here inside an elemental volume. Now, we know from our basic understanding there is convective flow and plus there is always a diffusive transport because of the molecular conditions at that interface. If we take these into account we can say that u a c at some x u a c at some x plus b. So, this is delta x plus you have the diffusive transport which is del c del x at x and then you have minus of a d del c del x at x plus delta x. What are these this is convective entry into control this is x and this is x plus delta x this is the volume delta v. So, this is convective entry input output diffusive entry input output and then you could have what is called equal to if there is generation reaction then you put a reaction term if it is not there then you have accumulation term which is del x of del c of del t. What are we saying what we saying is that if you look at an elemental volume inside a reaction equipment we can visualize that there is a convective transport and as well there is a diffusive transport of material entering that control volume. Now, we can divide throughout by a delta x and simplify and so on. So, that we will get minus of del c del x times minus del c del x I can cancel of a here, a here, a here and a here and a here delta x divide throughout. So, you will get u del c del x the minus sign and here we will get equal to d del squared c by del x squared plus plus equal to del c del t. This is del c del c where I have got to write minus x to the delta x. So, this minus sign is this is plus so this is this is fine. So, this is the differential equation that governs the motion of this fluid inside this. Now, let us just make some small simplification to make it little easier. So, let me make a small simplification simplification I make is that theta equal to t by t bar t bar equal to l by u and z equal to x by l. So, that this our equation now becomes del c del theta del c del theta this left hand side. So, it will become a t bar will come in the denominator equal to d by l squared del squared c by del z squared. So, I have taken the second term. So, this I have taken the second term here second term here I replace x in terms of z minus of u by l del c del z. So, u del c this x I put in terms of z. So, it will l comes here. So, it is I have done nothing new has been done. So, you can simplify this and say this del c del theta equal to d by l squared t bar minus of u by l t bar del c del z. Now, t bar by definition is u l by u sorry is l by u. So, this goes off. So, this becomes l by u therefore, this becomes del c del theta equal to d by u l minus. So, del squared c by del z squared I forgot del squared c by del z squared I forgot this del squared c by del z squared minus of del c del z. So, this is this is the differential equation that governs. So, our system is this this is the system and this is the control and then we have fluid entering and leaving to which at the here we are putting a tracer. We are trying to see what is the concentration of tracer at the exit. Let me restate the problem. The problem we are trying to understand and solve is that you have a reactor. There is an elemental volume to this whole reactor we have a pulse input going in. We want to know what is the concentration of the pulse at the exit here. What is this concentration? This is the question that we want to solve because moment you know that we have better understanding of the effect of this parameter diffusion coefficient and dispersion coefficient on the performance of the system. Now, what do we do now? Now, we recognize that our arbitrary vessel which is described by this equation del c del theta equal to d by u l del squared c by del z squared minus of del c del z to which we are having a pulse input the pulse input. And we want to know what is the concentration that we will measure at the exit at z equal to 1. Now, you notice here this is the linear problem. There are no non-linear is involved here and d by u l is called as the dispersion number, dispersion number. Some people call u l by d as Peclet number. This is a more common nomenclature u l by d whichever the way we look at it. This differential equation has one parameter which is called the Peclet number u l by d. Now, if you have the response to this curve to this to an arbitrary vessel going through here. So, you will have data on c t versus t. You will have data at position z equal to 1. So, there is some response. Therefore, once you have this response in principle you should be able to solve this equation. And then adjust the value of the parameter d by u l and fit this data parameter which is Peclet number which describes the arbitrary flow inside this equipment. On other words what we are saying is that if you are going to describe your system in terms of dispersion model you require what is called as the Peclet number which describes this system. To understand the value of Peclet number you have to do a pulse tracer test which you have done and found out the concentration as a function of time. Having done that all you have to now do is to try and fit this function to you will have a solution to this and then adjust the value of d by u l. So, that it fits the data. So, that way you can find out what is the value of d by u l that describes your equipment. So, essentially all we are saying is that if you know d by u l or u l by d whichever you like then you know what is the state of mixing inside the system. Now, in the literature solution to this equations are available for various types of boundary conditions. For the moment we will try to look at some of these things in our tutorial class. For the moment we can let us assume that we know the solution and the solution that is available in the literature is of this form. So, what people are saying is that the solution to this differential equation is available in the form of sigma square or the variance of this of the response divided by the mean value of the response. Second moment divided by first moment square this is given as in terms of Peclet number. On other words all that is being said is do a tracer test find out the variance mean and variance of the response that you will get. Moment you know the mean and variance the ratio is given by this representation 2 by Peclet number minus of 2 by Peclet number square into 1 minus e to the power of minus Peclet number. On other words if you know mu square sigma square and mu square then you can find Peclet number from this equation. So, you have now done a tracer test and on that tracer test you have determined what is the Peclet number that is appropriate for your kind of mixing in your equipment. Now, please refer at little earlier you said in this arbitrary vessel can also be thought of as a series of stirred tanks from sigma square and mu square. We also determine number of tanks that are required to describe this flow field. On other words the number of tanks is a measure of mixing of the equipment. Similarly, Peclet number is also a measure of mixing. On other words in these 2 models in 1 model the parameter is the number of tanks in another model the parameter is Peclet number. Or number of tanks is a measure of mixing in the equipment similarly Peclet number is a measure of mixing in the equipment. So, different parameters have different way of understanding mixing in the equipment. Now, that we know the Peclet number in the equipment or in other words which is the state of mixing as captured by Peclet number. Now, we are now in a position to use this information to understand how our reactor as per this model is called the axial dispersion model will perform. Let us try to do that now. Let us look at the dispersion model once again. What we said this is convective flow input convective flow output and this is the diffusive flow input diffused flow output equal to the accumulation. Generation term we removed suppose you want to put the generation term as minus of k and then concentration c times a delta x. Suppose, I put this term as the term corresponding to the generation. So, in the limit as a tends to 0 remove this this here you will get a term which is minus of k c. What are we saying now? What we are saying is that now that we know the Peclet number that describe the state of mixing in the equipment. Now, we are in a position to understand how this reactor will perform therefore, we have written the unsteady state equation taken into account the presence of chemical reaction. We want to understand the performance in system under steady state therefore, we remove the unsteady state term. So, what are we saying now? We want to now look at the performance of this chemical reactor at steady state in the presence of a chemical reaction. The steady state description of our equation is the following. So, we have d del squared c by del x squared minus u del c del x minus of k c equal to 0. This is what we have to solve to understand the dispersion model behavior of our system. Now, since it is only a first order differential sorry since only a difference equation in one variable. So, I will replace this like this d squared c by d x squared minus of u d c d x minus of k c equal to 0. Let us look at the system once again this is our system and we have taken an element and then we have written our balances and come to this equation. The question is how do we understand the boundary conditions for this problem? So, tremendous amount of literature in this for this situation, but let us try to understand the boundaries properly. This is boundary 0 boundary 1 z equal to x by l. How do we understand the boundaries? Now, a good way of understanding the boundary is to see what happens to flow in and flow out. So, what we have said here is that at x equal to 0 x equal to 0 minus you have u c a 0 times a at 0 minus must be equal to u c a times a minus d a del c del x at 0 plus d v. What are we saying now u a c a 0 this is the flux that means this is the flux at this is 0 minus this is 0 plus. So, at the boundary whatever material that comes in it goes out it goes out due to convection as well as due to diffusion. On other words what seems to be important to recognize here is that c a 0 if you make a plot this is distance versus concentration. You find that concentration at 0 minus is c a 0 concentration at 0 plus it is not c a 0 because of the del c del x is negative. Therefore, this whole term is positive therefore, this whole term u c a would be less than u c a 0 a. On other words it starts concentrations are like this suddenly drops and then it does this. This is the point that we should recognize the dispersion model explicitly recognizes the fact that at the boundary 0 there is a sudden drop in concentration because of the effect of dispersion. It is this effect which is not been appropriately accounted for in other models which makes this model so superior to others. Now at x equal to l what happens that is once again write u c a times a minus of d a del c del x this is at l minus must be equal to u c a times a at l plus. What are we saying? What we are saying is that our system which is our tubular reactor the actual dispersion problem exists inside and as soon as it emerges outside there is no such dispersion problem. On other words this term is appropriate to be used only inside the equipment and not outside therefore, that term has been deleted. So, this equality of material balance whatever is coming in at l minus this is convection this is diffusion and whatever is going out is by convection only because there is no diffusion term outside the chemical reactor that is the way we are understanding the chemical reactor. How do you make this equality? I mean this equality must hold at l minus and l plus. Let us look deeply at this we notice that del c del x in a chemical reactor is negative del c negative and between l minus and l plus we also know that this concentration c a cannot have changed because there is no reaction equipment. Therefore, we should expect that these two should go away therefore, applying the del c del x must be equal to 0 at del c del x equal to 0 at z equal to sorry x equal to l r z equal to 1. What we are saying is to understand the boundary conditions for solving this problem of axial dispersion where this chemical reaction occurring is to recognize the fact that there is there is convective flow outside at 0 minus inside the equipment there is convective flow as well as diffusion flow therefore, at 0 plus and the other end there is convective flow and diffusion flow inside the equipment at 0 l minus but outside the equipment there is no diffusion flow and therefore, that term has been deleted. In order that this equality holds and since there is no reaction happening between l minus and l plus we would expect that c a is the same at l minus and l plus implying that this term del c del x must be 0 at the other boundary. So, this is this or in other words del c del x equal to 0 at z is equal to 1 is also called as a celebrated dank words boundary condition. This issue was brought out and explained beautifully by dank words many many years ago and this is now very well recognized as we have understanding tubular reactors with axial dispersion. So, come to I mean summarizing this whole thing what we are trying to say is that our differential equation d square which describes the tubular reactor with axial dispersion axial dispersion. Now, can be solved solved with boundary conditions what are the boundary conditions the boundary conditions are this is the flux condition u c a u c a 0 at a minus equal to the flux at the other this one condition other condition del c del x equal to 0 at z is equal to 1. The solution to this to this equation I mean this huge amount of literature we just looked at this and then solve this problem. And we will probably have time to look at this once again during our tutorial classes, but as for now what we will do we will take the solution because it is not necessary to spend too much time. So, I am just writing down the solution just. So, that we can go forward our solution looks like this also where q squared equal to that the differential equation the differential equation that describes axial dispersion with chemical reaction inside is given by this. And the boundary condition that is appropriate to describe the problem is the flux equality condition at 0 0 minus and 0 plus. And then the no del c del x equal to 0 condition at x equal to l this is the two boundary conditions with these two kind of boundary conditions the solution looks like this. There is lot of literature in this available. So, we will not do this once again we will do it in our tutorial class. What is it that it is saying? What it is saying is now if we did not have dispersion on other words if dispersion was nil which means what it is a plug flow reactor we know how to solve the problem. Because the differential equations are much much simpler one term disappears this term disappears. So, you can solve this two straight away that is what we have been doing for a long time. What we have now down is added this term and try to understand the importance of this term. This differential equation solutions are available in terms of two parameters one is q which is which involves what is called as the Damkohler number the other one is the Peclet number. So, solution to the actual dispersion problem is available to us in terms of two parameters one is called as the actual dispersion coefficient or Peclet number u l by d inverse of dispersion number other is called the Damkohler number which is essentially k times t bar. Now what is it that we expect? What we expect is that when the when the dispersions are large or then our performance approaches CSTR when dispersions are small or performance approaches a plug flow. On other words real equipments either perform between plug flow and mixed flow. So, we will have great variations of performance between plug flow and mixed flow and our real reactors lies somewhere in between and to be able to capture the performance of real reactors we need to have an understanding of the flow field or understanding of mixing. And to be able to understand mixing we said we should have a model which quantitates mixing first we had one model which is called as the tanks in series model and where number of tanks was the way of understanding mixing then we had what is called as the dispersion model where the dispersion number u l by d which is Peclet number is a measure of mixing. So, depending upon the model the parameters vary, but they all describe mixing in their own way. So, we have looked at 3 models one is a completely segregated model based on RTD where there is no there is no other modeling parameter excepting the raw data directly plugged on with the reactant reaction kinetics. So, we have the completely segregated model based coming entirely from the RTD data there is no other modeling parameters. Then we have the tanks in series model which involves the number of tanks as a modeling parameter or you have the axial dispersion model where the dispersion coefficient or the Peclet number which is the inverse of dispersion coefficient it is used as a modeling parameter. There are 3 ways of understanding what is called as the real reactors. Please recall what we have been saying. So, far that if you have a dispersion model then this psi C A C A at l divided by C A 0 we already derived this which is equal to 4 times q exponential Pe by 2 divided by 1 plus q whole squared exponential Pe q by 2 minus 1 minus q whole squared exponential minus of Pe q by 2. We have derived this where q squared is given by 1 plus 4 k tau bar divided by Pe where this is residence time this is the residence time. We have said all these this is not new to us where Pe is the Peclet number. So, on other words what we are saying is that for a dispersion model the performance of the equipment phi which is C A by C A 0 is given by the right hand side where our parameters are q and Pe both q and Pe are numbers that we know about our reactor. That means if you have an equipment whose Pe and q are known then the dispersion model will be able to tell us how this system will perform. So, this is one way of understanding that moment q and Pe are known we know how to characterize our system. Now instead of dispersion model suppose we look at a recycle reactor model what we have been trying to say here is that we can look at a real equipment in various ways dispersion model is one way of looking at it of which we have already said what that model is all about. Now similarly we can look at the same equipment as a recycle reactor model where this is the recycle reactor this is our reactor this is our reactor. We can look at this reactor as a recycle reactor where R is the recycle ratio and the performance that means what is the conversion that you will get at the exit x 3 can be given by this equation for a first order reaction this is for a first order reaction all this we are doing it for a first order reaction. We can do for others but it is what is important is to understand the fundamentals therefore we are done for a first order reaction. So it says k t bar by R plus 1 is given by the right hand side where x 3 is the conversion at position 3. On other words what we are saying is that if we know the rate constant k t k t bar which is the rate constant multiplied by residence time and R is the recycle ratio there are two parameters what are the two parameters k times t bar and then R if you know k t bar and R and then we can say how the reactor will perform that means we can find out what is the extent to which the reaction takes place moment k t bar and R are determined through an appropriate experiment this is another way of understanding a real vessel. On other words the point we are trying to put across is that suppose you have real equipment which is giving you a conversion of x 3 for which you know what is k t bar from our experiment then we can tell what is the recycle ratio that is appropriate to this process that is another way of understanding the real equipment. There is another way of understanding the real equipment is the following in tanks in series model what we do is that we have a real equipment which we think can be modeled as a series of stirred tank which means we have done experiments on the real equipment and we find that the conversion is x and as per this tanks in series model which we already derived 1 minus of x is given by 1 plus k tau i to the power of n where n where n can be given by this equation which we already derived n is equal to mu n square by sigma n square and tau i tau i equal to and we also know that tau i equal to tau by n and what is tau? tau is equal to mu n by n where mu is the first moment of the tracer response. On other words what we are trying to say here is that if you have a real vessel and you want to model this as a tanks in series model so if you understand from your experiments what is the extent of reaction that you observed then using that data using the data you will be able to tell how many tanks are required to model this equipment. So if you look at the final expression 1 minus of x equal to this if you look at this if 1 minus of x equal to 1 by all these parameters k sigma n and mu n. So how many parameters are involved here? The number of parameters involved here is sigma n k mu n are the 2 parameters. So if you know these two then we can tell how this tanks in series model will perform. So we talked about three models now dispersion model the recycled reactor model tanks in series model. On other words if you have a real equipment you can use any of these models to understand how real equipment is performing and that is the idea of trying to develop models to describe real equipment. So just to summarize what we are saying is what we are saying is that suppose we have a real equipment giving conversion of x corresponding to different Peclet numbers corresponding to different tanks in series number of tanks or corresponding to different recycle ratio. For example suppose let us say our conversion that is expected that is been observed from experiment is x this is the conversion. Then you can notice here if it is a dispersion model it will give us a Peclet number corresponding to this which means this particular Peclet number would be able to describe that equipment or if it is tanks in series model this particular number of tanks will be able to describe the equipment or if it is a recycled reactor model this recycled ratio will be able to describe the particular equipment given that what is the value of this k t bar which is the residence time multiplied by the rate constant. On other words if you know the residence time multiplied by the rate the residence multiplied by the rate constant if this number is given then you can tell what will be the conversion or you will be able to tell what is the Peclet number corresponding to given extent of reaction or what is the number of tanks required to describe it under the tanks in series model or the recycle ratio that is required to describe it under the recycle reactor model. On other words what is important to recognize here is that a real equipment can be described by many models and then we can choose model that we think is most appropriate for our description and three models have been described here the dispersion model tanks in series model and recycle reactor model having said this there are few point that we want to take it forward a little. Let us say we have an equipment and which we want to describe by what is called as a completely segregated model what is this completely segregated model the complete segregated model tells us that fluid elements that is entering the reaction equipment they go through the equipment and emerge another fluid element goes through the equipment and emerges and these fluid elements as per this model it is assumed that they do not recognize the existence of each other as it goes through the equipment that means that completely segregated as they move through the equipment and they mix only at the exit this is the model what happens this is how we are visualizing what happens in the equipment. Therefore if as per this assumption if this fluid element goes through the equipment and emerges without recognize the existence of the other element then we can tell that if it enters at CA0 it would undergo chemical reaction depending upon the time that it has spent in the equipment. So what this completely segregated model tells us is that if you have a fluid element 1 then it goes through the equipment starts at CA0 and reacts as per this first order model if it is the first order reaction and emerges like this another fluid element enters at CA0 but spends time T2 therefore emerges with this kind of concentration therefore the mean that we see at the exit from the equipment will be what happens to each element multiplied by the time of residence of each element inside the equipment. So this is this is where we understand what is called as completely segregated model that we each element goes through and emerges without recognize the existence of the other therefore what emerges at the exit is the value of each element multiplied by the time of residence of that element this is now if it is the first order reaction then the CA element is given by the first order law if it is the half order reaction the CA element is given by the half order law if it is the second order reaction it is given by the second order law so what is what we are trying to say here is that if it is a completely segregated model is the description that we have chosen to understand this equipment then the average that you would see at the exit is the CA element as described by whether it is the first order reaction or second order reaction or half order whatever the case may be multiplied by the residence time distribution fraction that spends the time that is assumed okay that is the that is how we understand the completely segregated model. Now we can also look at the same the same phenomena through what is called as through what is called as the completely micro mix model so the completely micro mix model the completely micro mix model what do we do we have this material entering material leaving and material reacting equal to 0 this is the completely micro mix model okay material entering material leaving and then material reacting equal to completely micro mix model once again we can solve this and find out what is x versus the reaction parameter okay so what we are saying is that whether it is completely segregated model or completely micro mix model in both cases we can describe this reaction equipment in terms of conversion versus the reaction parameter okay now when we do this for the completely segregated model you find that for both for completely segregated model and for completely micro mix model for first order reaction the results are the same on other words whether you use five completely segregated model or completely micro mix model you get the same result if it is a first order process okay now if it is a second order process we find that if it is completely segregated and completely micro mix there is a difference roughly about 7 to 8 percent and if it is order of reaction less than 1 we also find a difference once again I have shown the difference once again 7 to 8 percent on other words what is being said is that when we try to describe a real equipment by completely segregated model or if you describe the real equipment by completely micro mix model the results are not the same if the order of reaction is different from one the results are slightly different but that difference is about 7 to 8 percent and typically the kind of variations that we might expect in an experiment on other words what we are trying to say is that you can use the completely segregated model or the completely micro mix model to describe the equipment you will get answers which are satisfactory because the fact that the difference between the two models are generally between 7 to 8 percent which might be in the acceptable range of the experimental errors that we will observe just to cut this long story short what we are saying is a completely segregated model assumes that the fluid elements mix only at the exit and therefore the average at the exit is the what happens to each element as per the order of reaction multiplied by the residence time distribution that gives us the average if it is the completely micro mix model the assumption is that as soon as in the completely micro mix the assumption is that as soon as it enter it mixes it is called early mixing in complete segregation it is called late mixing in the early mixing model or complete micro mix model what we find is that the extent of reaction against reaction parameter when plotted you find that the results are not the same there is a slight difference if the order of reaction is different from one but in both cases whether the order of reaction is less than one or order of reaction is greater than one the difference between the two models are 7 to 8 percent and therefore in the range of experimental errors that we would anticipate therefore for describing real equipment we might like to use either complete segregated model or complete micro mix model.