 We have been looking at predicting the conversion in an non-ideal reactor provided the RTD function as function is known. So there are two types of models that we are going to discuss, one is the segregation model, the other one is the maximum mixedness model and these are two extremes which actually provide a bound for the conversion depending upon which is the, what is the order of reaction and the kinetics of the reaction which is conducted in the reactor. So these two extremes are the early mixing, early mixing model, a regime, early mixing that is one extreme and the model that represents early mixing is the complete segregation model while the late mixing which is the other extreme that represents the maximum mixedness model. So these two extremes essentially represent the two levels of mixing where the, two levels of mixing of the macro fluid globules that we actually defined in the last lecture. So let us look at the first one, the segregation model of a reactor. Now suppose let us consider a CSTR, so let us consider a CSTR or tank reactor, it could actually be any reactor and the, let us assume that the fluid elements of different ages they do not mix that is the, that is the segregation model and which also which means that they remain segregated all through, they remain segregated all through and the flow is essentially like a series of globules where one globule enters the reactor at a certain time and that gain that spends some time, another globule which enters at a different time the age of that particular globule is going to be different and which means that the globules of different ages they do not mix with each other. So, so flow is essentially a series of, series of globules and we may assume each globule as a batch reactor because it does not mix, so whatever reactant which is present in this globule it continues to undergo reaction as long as all the species which is present in that globule is gradually going to complete conversion. So therefore, each globule is considered as a, we may consider that as a batch reactor. So now how do we model this? So we can actually depict this, depict the segregation, depict this particular aspect in a CSTR as basically a tank which contains several globules and each of these globules are now going to have different ages, so that is going to have a different age, this is going to have a different age and this is going to have different age and each of these globules can actually in principle be of different sizes as well. Properties that are of these globules will be that they will be of different sizes and then more importantly because they do not mix each of these globules as actually going to retain their identity, they are going to retain their identity and there is no exchange or exchange between globules. So there is no exchange of molecules or matter between different globules and each of these globules will continue to have maintained its own identity. Now one may also depict the same picture, similar model in a plug flow reactor. Suppose we have a continuous flow plug flow reactor, so suppose if we model a continuous flow system of which is a continuous flow system of the non-ideal reactor as a plug flow reactor, suppose if we model this as a plug flow reactor then if you want to incorporate a segregated model for this non-ideal reactor this may be depicted as follows. Suppose if there is a tube and if there is a fluid which is flowing at a certain volumetric flow rate V0 into this tube and if the volume of the tube is V, now instead of the fluid leaving from the other end of the reactor directly the fluid can actually be taken out from the side of this tube at different locations along the side of the reactor and they can all be joined together and together they actually leave the reactor. So now each of these streams if because they are withdrawn at different locations each of the globules will actually have different residence time. So the location where the fluid is actually withdrawn from the reactor can actually be decided based on the residence time distribution of the actual non-ideal reactor. So suppose if this is the residence time distribution, if that is the residence time distribution and that is the E curve which is the RTD function which is actually measured experimentally of a real reactor then based on this RTD function such kind of a withdrawing of the fluid from the reactor can actually be designed. So the fluid which is actually withdrawn from the reactor right at the entry location is going to have the shortest residence time and the one which is actually withdrawn at the other end of the reactor is going to have the longest residence time. So that is going to have a long residence time. So therefore what we have done is we have removed batches of fluids from different locations inside the reactor and the location from where it is withdrawn is actually specified by the residence time distribution function E of t which may be measured experimentally for a non-ideal reactor. And because we are withdrawing from the side there is no interchange of molecules between each of these globules when they are actually present inside the reactor where the reaction actually occurs and each of these can actually be considered as actually a batch reactor. So what it suggests is that the mixing for this kind of system as actually the fluid appears as a well mixed system when it enters the reactor but when it leaves the globules are actually completely segregated. So therefore the reaction time in batch in each of the batch reactors that is each of the globules which may be which is basically withdrawn from different locations along the side of the reactor that time the reaction time of batch reactor of each of the batch reactor is equal to the time spent in the reactor. So this is the reaction time spent by each of the batch reactors. Remember that every location from where the fluid is actually withdrawn from the side of the reactor that globule can actually be considered as a batch reactor and the time that is spent by the reaction time of each of these globules that is each of these batch reactor is equal to the time that it actually spends in the reactor itself. So therefore the mean conversion if x bar is the mean conversion that is equal to the average conversion over all globules. So what is of interest is essentially this mean conversion. So we want to predict the conversion of the reactor and what we essentially require to need to predict is this mean conversion from the reactor. So how do we find this mean conversion? So the mean conversion of globules with residence time of a certain time interval dt. So let us say that the mean conversion of globules with residence time between t and t plus dt in that small interval of time if that globule has a residence time in this small time interval then that is essentially given by the conversion achieved by a globule after spending t amount of time in the reactor and that multiplied by the fraction of globules with residence time between t and t plus delta t. So that is the abstraction of how to get a mean conversion of globules with a certain residence time between t and t plus delta t which is essentially the conversion achieved by the globule after spending that time t amount of time inside the reactor multiplied by the fraction of the globules with that residence time between t and t plus delta t. So now if you put the corresponding expressions corresponding to each of these terms in the mean conversion we will find that dx bar which is the mean conversion of the globules whose residence time is between t and t plus delta t. So that should be equal to the conversion achieved by a globule after spending that time in the reactor multiplied by the fraction that has a age between t and t plus delta t. So therefore from here we can write the dx bar by dt which is the mean conversion that is equal to xt into e of t and therefore x is equal to integral 0 to infinity x of t e of t dt. So remember that the x of t which is actually present inside the integrand that is essentially the conversion in batch reactor because we consider each of these globules as actually a batch reactor. So that is the conversion as though it were a batch reactor and that multiplied by the corresponding residence time distribution integrated over all residence times will give the mean conversion. So let us consider a first order reaction where A goes to products with the rate which is the specific reaction rate k and for a batch reactor the performance equation is d minus dna by dt that is equal to minus rA into v where v is the volume of the reactor and let us assume that it is a constant volume system and so now we can use the relationship between the number of moles with the corresponding conversion and we can rewrite this equation as na0 into dx by dt that is equal to minus rA into v which is equal to k CA0 into 1 minus x into v and so from here we can easily decipher that the conversion is equal to exponential of minus k into t and remember that the CA0 into v is essentially equal to that is equal to na0. So therefore the conversion as though a conversion in each of these globules is given by 1 minus exponential of minus kt and so from here we can now find out what is the mean conversion we know the mean conversion is equal to integral 0 to infinity that is over all residence time of the product of the conversion in the batch reactor multiplied by the corresponding residence time. So therefore x bar is equal to 0 to infinity x of t E of t dt and so that is equal to integral 0 to infinity 1 minus exponential minus kt E of t dt. Now suppose if it were suppose if the reactor is actually a plug flow reactor so suppose if it is a plug flow reactor then the residence time distribution E of t is simply given by the delta function of t minus tau where tau is the space time of the reactor which is the ratio of the volume to the volumetric flow rate of the reactor. So now from for a plug flow reactor it will simply be 1 minus integral 0 to infinity exponential of minus kt into delta function t minus tau dt and that is nothing but 1 minus exponential of minus k into tau and k into tau is essentially the dimensionless quantity called the Damkohler number this is the Damkohler number which is the ratio of the space time to the reaction time and so we can write this as exponential of minus da. So that is the mean conversion that would be achieved if it were to be an ideal plug flow reactor and what is interesting is that the model that we actually obtain from the segregated model is actually same as that of the mole balance of the plug flow reactor. We know that from the mole balance of the plug flow reactor the performance is actually performance equation suggests that the conversion is actually equal to 1 minus exponential of minus da and let us see how that is the case. So the mole balance of a plug flow reactor on species A is nothing but dx by d tau that is equal to k into 1 minus x and therefore x is equal to 1 minus exponential of minus k into tau which is equal to 1 minus exponential of minus da. So therefore the conversion that is up that is actually achieved by using a completely segregated model is actually exactly equal to the conversion that is achieved by an ideal plug flow reactor if it were to be a first order reaction. In fact we observed this in one of the lectures before that we mentioned that if it is a first order reaction then it does not only RTD function is sufficient to estimate the conversion and the level of mixing actually does not play any role. So we will see in a short while as to why that is the case. Now before we look at that let us consider if it were to be a CSTR. So the residence time distribution function for a CSTR is given by 1 by tau into exponential of minus t by tau. So therefore x bar the mean conversion plugging this into the integral and integrating the expression shows that x bar is equal to tau into k divided by 1 plus tau into k which is equal to the Damkohler number divided by 1 plus Damkohler number. So now if we write the mole balance for a CSTR so the performance equation for the CSTR is the mole balance on A is FA naught which is the molar flow rate of the species at inlet that multiplied by x is equal to minus rA multiplied by V that is equal to k into CA naught into 1 minus x into V. And minus rA is the rate of generation minus rA is the rate at which the species is actually being consumed. So from here we can see that x is equal to tau into k by 1 plus tau where tau is the space time of the reactor 1 plus DA. So clearly you can see that the conversion achieved through a segregated model is actually exactly same as the conversion that is achieved from the performance equation of the ideal CSTR. So this suggests that the for a first order reaction the information of RTD function is actually sufficient and the degree of mixing is not going to add any additional information and the RTD function itself can be used to predict the conversion mean conversion of the reactor. Now the question is why is that the case? So the reason is the complete mixing or segregation actually makes no difference for first order reaction. This is because the rate of change of conversion actually does not depend upon the concentration of the reacting molecule. So the rate of change of conversion independent of concentration of the reaction reacting molecule. So this is the reason why for a first order reaction the RTD function alone is sufficient to predict the conversion that is achieved by the non-ideal reactor. So the rate of change of conversion is independent of the concentration of the reacting species that explains why for a first order reaction RTD function is sufficient to estimate the conversion of the reactor. So now let us extend this to a laminar flow reactor LFR stands for the laminar flow reactor and the residence time distribution is given by 0 for t less than tau by 2 and tau square by 2 t cube for t greater than or equal to tau by 2. So that is the residence time distribution for a laminar flow reactor which we have actually derived in the previous lecture. In the normalized form E of theta is equal to 0 1 by 2 theta cube and this is theta less than 0.5 and theta greater than or equal to 0.5. So now we can plug this distribution into the conversion equation and we can find that will be equal to 1 minus 0 to infinity exponential of minus k into t into E of t dt and that is equal to 1 minus integral 0 to infinity exponential of minus k tau theta into E of theta into d theta. So on performing the integration by substituting the corresponding distribution one can find that the mean conversion in a laminar flow reactor is essentially given by 1 minus 1 minus 0.5 into the space time multiplied by the specific reaction rate into exponential of minus 0.5 k tau minus 0.5 k tau the whole square integral 0.5 to infinity exponential of minus tau k theta divided by theta into d theta. So that is the expression and if one solves this integral one will be able to find out what is the conversion in a laminar flow reactor. So let us now compare the mean conversion that is achieved using these three different types of reactors. So if we plot as a function of the Damkohler number which is the ratio of the space time to the reaction time x bar. So the CSTR would actually be like this and the plug flow reactor would actually predict a much higher conversion for a first order reaction remember this is for a first order and the laminar flow reactor would be somewhere in between. So the plug flow and the CSTR they sort of provide a bound for the predict conversion of the first order reaction in a non-ideal reactor and such kind of graphs actually can be generated for such kind of plots can be generated for reactions of other orders and other types of kinetics. So what it suggests is that for a first order reaction the extent of mixing not required while for other reactions other kinetics extent of mixing plays an important role. So extent of mixing actually plays an important role and it is required in order to predict the conversion of the non-ideal reactor. So now let us move to the next model which is the maximum mixedness model. Let us look at the maximum mixedness model. So the segregated fluid is one where the mixing between the fluid globules actually does not occur. So there is no exchange of material between the globules which are present inside the reactor. So the flow is essentially like a series of globules which are actually flowing through the reactor. On the other hand that is called the minimum segregation, minimum mixedness model where the globules do not actually interact with each other and each of the globules behave like a batch reactor. On the other extreme is a maximum mixedness model where the globules, the matter which is present in different globules they are allowed to actually mix and interact with each other and therefore the molecules which are of different ages they all mix with each other and that is that kind of representation or that kind of a situation is called the maximum mixedness model. So let us look at how to estimate the conversion for that kind of a situation. So maximum mixedness is achieved when there is complete mixing as fluid enters. So as soon as they get into the reactor all the globules can actually exchange matter with each of them and so therefore there is complete mixing. So the maximum mixedness is the complete mixing of the fluid right at the entry point of the reactor. So how do we depict such kind of a situation is we can consider a plug flow reactor with side feed. So where the feed is actually fed through the sides of the plug flow reactor at different locations and that can be used to depict the situation of maximum mixedness in a non-ideal reactor. So suppose if we know the residence time distribution function so if we know the E of t of a real reactor then we can actually mimic the reactor by using a plug flow reactor and instead of providing a feed at the entry to the plug flow reactor whose volume is V we can actually split the we can actually feed them through the sides. We can feed them through the sides and the feed through the side can actually be according to the we can split the feed and feed them through the side and this feed could be according to a certain distribution function which is the residence time function of the real reactor. So the residence time distribution function could be something like this where the side entrance is actually according to this distribution function. So which suggests that the mixing actually occurs as early as possible and then they actually go into the reactor. So mixing earliest possible which corresponds to the maximum mixedness situation in the reactor. So now suppose we define lambda as the time to move from a particular point to end of the reactor. So that is the time taken by a fluid element to move from a particular location inside the reactor to the end of the reactor. So remember that we have now represent the non-ideal reactor using a plug flow with the side stream in different locations in the side of the plug flow reactor. So now this also reflects the life expectancy at that point that is the amount of time that actually the fluid particles are going to spend inside the reactor which is actually fed into the reactor at that point in the side. So now we can now draw schematic of this reactor. So suppose if this is the plug flow reactor with a volume V and then we now make a feed, we feed the fluid, we feed the reactor with fluid along the sides and according to a certain residence time distribution function. Now if we assume that this is lambda equal to 0 because the time that is actually spent by the fluid that is pumped into the reactor near the exit of the reactor is almost equal to 0. So therefore lambda equal to 0 is the life expectancy of the fluid that enters the reactor in this location is going to be 0. So lambda equal to 0 starts from here and then lambda equal to infinity which is the maximum time that is taken inside the reactor is at the entry of the reactor and if the volumetric flow rate of the fluid is V and V equal to 0 is this location and V equal to V0 that is the full volume of the reactor. Now if we now identify a small element and if the volume of that element is delta V and the flux with which the fluid actually enters that element is given by V into CA that is the volumetric flow rate at that location and if this point is lambda in the life expectancy dimension and this is lambda plus delta lambda. So that is the difference in the life expectancy from this point to this point. So this will be V into CA at lambda plus delta lambda and whatever is leaving from here will be VCA at lambda. Now what is the amount of fluid that actually enters through the side? So that amount of the volumetric flow rate with which the fluid is actually going to enter is let us say is given by V at that location and we will be calculating that in a short while. So what is the flow rate with which the fluid actually enters a small element delta V? So the flow rate in at delta V so that is equal to the volumetric flow rate V0 that is the overall volumetric flow rate of the reactor. So we are essentially trying to calculate what is the volumetric flow rate with which the fluid is actually entering in this small element delta V. So flow rate in at delta V should be equal to V0 which is the volumetric flow rate with which the fluid is being pumped into multiplied by the fraction of fluid with life expectancy. So let us call this life expectancy, life expectancy between lambda and lambda plus d lambda. So that is equal to V0 multiplied by the corresponding E lambda times d lambda where E lambda is the essentially the RTD function which says what is the residence time distribution of the fluid element inside the reactor. So now once we know this we can now write a flow rate balance, we can now formulate a flow rate balance and the flow rate balance is volumetric flow rate at lambda should be equal to the volumetric flow rate of the fluid at lambda plus d lambda plus whatever is actually added through the side. So that will be equal to V0 into E lambda d lambda. So this is the flow rate in through the side. So that is the flow in through the side of the plug flow reactor and so now we know, so now we can actually take the limits of delta lambda going to 0. So limit delta lambda going to 0, this essentially becomes d V lambda by d lambda that is equal to minus V0 into E lambda. So that is the differential equation which captures what is the flow rate at a certain life expectancy lambda. So now V0 is the flow rate with which the fluid is actually flowing at the entrance of the reactor. So which means at entrance that is when conversion is actually equal to 0. So before V0 is the overall volumetric flow rate of the fluid that is actually flowing through the reactor. So now we can actually integrate this expression as V lambda equal to 0 at as lambda tends to infinity. So the flow rate of the fluid that is actually at the entrance is V0 and the conversion at that location is equal to 0 and so therefore the amount of fluid that is actually right at the entry point of the reactor, remember that it is a feed that is coming at different locations in the side. So at the volumetric flow rate of the fluid whose age is almost equal to infinity is equal to 0 and V lambda is equal to V lambda at some lambda equal to lambda that is at certain age let us assume that V lambda is the corresponding volumetric flow rate and so using these two as limits we can now integrate to find that V lambda equal to V0 into integral 0 to integral lambda to infinity E lambda d lambda which is equal to V0 into 1 minus f of lambda. So that is the volumetric flow rate with which the fluid is actually flowing at any location lambda. So now the objective is to find the overall conversion need to find x. So that is the objective. So how do we find x? We need to write a mole balance of the species in order to find the conversion of the species in the reactor. So before we write a mole balance we need to know certain aspects of the reactor certain aspects before we write the mole balance. For example, what is the amount of species? What is the rate at which enters the small element delta V? So this can actually be found by using what is the volume of the fluid whose life expectancy is actually between the between lambda and lambda plus d lambda. So the volume of the fluid with life expectancy between between lambda and lambda plus d lambda. So if we know this volume this volume multiplied by the concentration will tell us what is the number of moles that is actually entering that particular element delta V. So that is equal to so delta V will be equal to V0 into 1 minus f of lambda. So that is the volumetric flow rate multiplied by the corresponding age delta lambda will tell us what is the volume of the fluid with that certain life expectancy that is equal to that somewhere between lambda and lambda plus d lambda. So now what is the rate of generation of the species? What is the rate of generation of species that is actually given by the rate at which the species is being consumed multiplied by the corresponding volume delta V. So that is equal to Ra into V0 into 1 minus f lambda into delta lambda. So we now have all information that we need to write the mole balance. So let us now write the mole balance for this particular species. So the mole balance for the species is so mole balance on a between with life expectancy of lambda and lambda plus d lambda. So let us write the mole balance for this. So what is the rate at which things are coming inside at lambda plus d lambda? Remember that the age of the fluid is actually decreasing from the exit of the reactor increasing from the exit of the reactor while the positive direction is actually increase of volume from the entry of the reactor to the exit of the reactor. So in that lambda plus d lambda plus the introduction through the side what is the rate at which things are actually introduced into the reactor through the sides minus what leaves the reactor, what leaves that element at lambda plus whatever is generated by reaction. So that should be equal to 0. So that is the mole balance on a for age between lambda and lambda plus d lambda. So we know all these quantities. So V0 into 1 minus f lambda. So that is the volumetric flow rate at lambda, lambda plus d lambda into CA at evaluated at lambda plus d lambda will tell us what is the rate at which the species is actually getting into that element plus the whatever is introduced through the sides that is given by V0 into E lambda d lambda multiplied by CA0 where CA0 is the concentration of the species in the feed stream minus V0 into 1 minus f lambda into CA evaluated at lambda plus r a into V0 which is the volumetric flow rate of the feed into 1 minus f lambda multiplied by d lambda equal to 0. So that is the mole balance on a between the age lambda and lambda plus d lambda. So now we can actually divide this expression by V0 into delta lambda. We can divide this expression by V0 and d lambda and take limit as d lambda goes to 0. So that will be CA0 into E lambda plus d by d lambda into 1 minus f of lambda into CA lambda plus r a into 1 minus f of lambda equal to 0. So that is the expression for that is the mole balance. And so now we can open up this differential here and we can rewrite this expression as CA0 into E lambda E of lambda plus d CA lambda by d lambda into 1 minus f lambda minus CA lambda into d f lambda by d lambda plus r a into 1 minus f lambda equal to 0. Now if we stare at this expression, this d f by d lambda is nothing but the RTD function E lambda where f is the f curve or the cumulative distribution function. So using this property we can actually write the mole balance. So the final mole balance essentially is d CA of lambda by d lambda that is equal to minus r a plus CA minus CA0 into E lambda by 1 minus f lambda. So that is the mole balance for the species for a maximum mixedness model. And so in terms of conversion we can actually write this expression as minus CA0 dx by d lambda that is equal to minus r a minus CA0 into E lambda by 1 minus f of lambda. And so that can actually be written as dx by d lambda equal to r a which is the rate of generation of the species divided by CA0 which is the concentration of the species in the fluid inlet stream into E lambda by 1 minus f lambda into the conversion x. So while solving this equation we will be able to find out what is the conversion if we know the residence time distribution function. So what are the boundary conditions for this equation? The boundary conditions are very simple. So lambda goes to infinity when CA equal to CA0 that is at the entry point into the reactor the age of the fluid is actually approximately infinity. So how do we integrate this? We have to integrate this equation from backwards starting from very large lambda. So we have to integrate this equation by starting from large lambda and move backwards till lambda equal to 0. So that is the method to integrate this equation and once we integrate the equation we will be able to find out what is the conversion under the situation of maximum mixedness. So now if RTD is known then the conversion for the maximum mixedness situation can actually be, model can be estimated. So this conversion provides a bound for the conversion of the species in the non-ideal reactor and so for n greater than 1 it is been observed that for n greater than 1 the maximum mixedness model, maximum mixedness model gives the lower bound on the conversion. So the maximum mixedness model actually gives the lower bound on the conversion and the complete segregation model gives the upper bound on the conversion. So now we have looked at the single reaction case. Now it is possible to extend it to multiple reactions. In reality many reactions can actually occur simultaneously in parallel. So there can be sequence reactions, there can be sequential parallel reactions etc. So several reactions can actually happen simultaneously in a reactor. So is it possible to predict conversion when there are multiple reactions happening inside the reactor and the answer is yes it is possible. So it is very simple to extend the segregation and the maximum mixedness model for multiple reactions. So if there are multiple reactions which are actually happening and let us say A and B are the reactants and P is let us say the products which is formed. Let us say the products which is formed and if it is a segregation model, if it is a complete segregation model then if we assume that each globule has different concentrations of A and B and if we assume that each of them behave like a batch reactor which is one of the assumptions of the segregation model each of these globules then C A bar which is the concentration of the species, the average concentration of the species. Remember that if you are looking at multiple reactions and multiple species it is actually better to work with concentrations rather than conversion. So the average concentration of species A will simply be integral 0 to infinity C A of t E of t dt where E of t dt is the residence time function distribution function of that reactor and similarly C B is given by integral 0 to infinity C B t into residence time of the reactor and the C A t and C B t are essentially the concentrations can be achieved from a batch reactor because this is the concentration of the species in each of the globule and we assume that each of these globules actually behave like a batch reactor. So now if we write the batch reactor performance equation for each of these species, so if there are Q reactions occurring simultaneously, so the reactor volume is V and Q reactions are occurring and there are Q reactions which are occurring simultaneously then for batch reactor we can write the performance equation as d C A by dt that is equal to the rate of generation of R A so that is equal to sigma 1 to Q that is sum over all the reactions and the reaction rate of the individual reactions which is leading to the formation of species A and similarly we can write for the species B d C B by dt equal to R B which is equal to sum 1 to Q R I B. Now this actually has to in order to find the concentration of the species A and B in this model in this reactor for following this aggregation model. So these two batch reactor rate expressions has to be solved simultaneously with the other two reactions which represent the overall concentration of the species in the reactor. So d C A bar by dt that is equal to C A T into E of T so that defines how the concentration of the species overall concentration of the species in the reactor that changes with time and the corresponding equation for species B that is equal to of T into E of T. So by solving these four equations simultaneously this one equation 1 equation 2 equation 3 and 4 so these four equations have to be solved simultaneously and we need to find C A of T and C B of T so that gives us the concentration of the species as a function of time which actually follows the segregation model. So next let us look at the multiple reactions for the maximum mixedness model. So for a maximum mixedness case so if it is the maximum mixedness model once again if we assume that there are Q reactions which are actually happening simultaneously then the model equation is d C A by d lambda is just an extension of the single reaction case. So d C A by d lambda is minus summation of reaction rate over all reactions which is actually happening simultaneously plus C A minus C A naught where C A naught is the concentration of the species in the feed stream of the reactor multiplied by E lambda which is the distribution function for that particular reactor divided by 1 minus F lambda and similarly for d C B by d lambda that is equal to minus sum I equal to 1 to Q R I B plus C B minus C B naught into E lambda by 1 minus F lambda. So where E is the RTD function for that particular reactor and F is the cumulative distribution function. So now once we know the rate law for all of the reactions so if we know the rate law so we can simply have to plug in this rate law and then solve for the concentration. So solve for C A and C B from large value of lambda to lambda equal to 0. So once we solve this equation we will be able to find out what is the concentration of C A and C B as a function of different age. So this is the set of equations and this can actually be extended for many other species even if n species are participating one can actually write the maximum mixedness model for all n species and similarly for the segregation model. So let us summarize what we have actually discussed in the last several lectures in the residence time distribution problems. So first we looked at the ideal versus non-ideal reactors we looked at ideal versus non-ideal reactors and then we looked at the RTD functions we looked at the RTD functions what is RTD function why do we need RTD function etc and then we looked at measurement of RTD functions measurement of the RTD distribution function in real reactors where we looked at the pulse tracer input and we looked at the step tracer input and we looked at how to perform these experiments and how to actually estimate the RTD function what to measure etc etc and then we looked at RTD properties of distribution properties of RTD function so particularly we looked at the mean we looked at the variance and then we looked at the skewness of the distribution which actually tells us how skewed is the function around the mean of the distribution and then we looked at RTD that is the residence time distribution in ideal reactors we looked at plug flow reactor we looked at single CSTR and then we looked at the laminar flow reactor these are the three cases that we looked at for RTD in ideal reactors and then we observed that the RTD function can actually be used for diagnostics purposes in order to estimate whether the reactor is operating under perfect conditions usually it never is perfect but how close is it to a perfect operation whether there is bypassing of the fluids that is actually entering the reactor or and if there is a dead volume which may be present inside the reactor and then we looked at the combination of reactors we looked at combination of reactors particularly we looked at the PFR-CSTR combination and we looked at how to estimate the residence time distribution and we also observed that if the residence time distribution for PFR followed by CSTR and CSTR followed by PFR is actually same however the sequence actually dictates as to what is going to be the performance of the combination of reactor so which is suggested that the RTD function alone is insufficient to actually predict the complete conversion or it is not the complete picture of the performance of the reactor additional piece of information is required and from there we marched on to the next topic of looking at the predicting the conversion so in this case we looked at the segregation model we looked at the segregation model and we looked at the maximum mixedness model we looked at maximum mixedness model and then we extended we we looked at these models for first order reaction and we also extended this to multiple reactions thank you.