 Friends, in the last lecture, we looked at different operations of CSTR and we stopped at what is the residence time distribution curve for different operations of a tubular reactor modeled as a plug flow reactor. Specifically, we looked at what is the residence time distribution if the plug flow reactor was operated under normal conditions that is if the residence time distribution is actually a delta function. And then we looked at what happens if there is a bypass in the reactor and then we looked at what happens when there is dead volume in the reactor. So, now if we compare the, let us now compare the distributions of these three modes of operation for a tubular reactor. So, suppose if this is the, if this is time axis and this is f of t which is the f curve and if it is a normal operation, then the f curve will start at the space time of the reactor tau which is essentially given by the volume of the reactor divided by the volumetric flow rate with which the fluid is actually flowing through the reactor and then it actually is a, it looks like a step function and this is 1. So, that is for the perfect operation. Now suppose if there is a bypass in the reactor, then what happens is that there will be a jump in the f curve, so the jump will be start at VB by V0, so that tells you the extent of bypass which is actually present in the reactor and then the, because the space time is now larger than the space time of the reactor if it was conducted under perfect operation or if there was no bypass, therefore the fluid stream that enters the entry of the reactor is now going to take longer time to leave the reactor as compared with the perfect operation. Therefore the f curve will essentially look like this if it is a bypass operation, if there is a bypass inside the reactor and then if suppose there is a dead volume which is present here, then the net volume which is available for, net active volume which is available is for the fluid stream to access is actually smaller than the actual volume of the reactor. So, therefore the residence time, the space time of the, the apparent space time of the reactor is actually going to be smaller than that of the, of the space time of the plug flow reactor which is actually operated at a perfect conditions, so therefore the fluid stream will actually leave faster than the normal or perfect operation of the reactor, so this is tau st, so the, so the method is first we should actually, first one should actually estimate the f curve experimentally, estimate the f curve experimentally and compare with the perfect operation case, compare with the perfect operation case and this actually will provide a clue whether there is a bypass in the reactor or whether there is dead volume which is actually present inside the reactor. So, this is the story for the single CSTR and the single plug flow reactor, so now let us look at the combination of reactors, so we observed earlier that RTD function is actually used to model the real reactors as combination of ideal reactors, so let us take a very simple example of a combination of two ideal reactors, so let us consider the plug flow reactor and CSTR in series, so what will be the RTD function, RTD function for this case, remember that the real reactors can actually be modeled as a combination of a plug flow and a CSTR and let us assume that there is a real reactor which is actually modeled as a CSTR plug flow as a CSTR and plug flow operated in a series mode, now where is it possible, when is it possible to approximate it as a CSTR plug flow series model, so consider the situation of a real stirred tank reactor, so one could intuit that the, there will be highly agitated zones which will be present in the real stirred tank reactor, so these zones will typically behave like a CSTR, so they behave like a CSTR and then there will be situations where the fluid may take tortuous path before it actually, tortuous path before it actually leaves the reactor, so which means that it sort of behaves, so that particular fluid stream which actually takes such tortuous path can actually be modeled like a plug flow reactor, modeled like a PFR, so this kind of a situation one can actually use a combination of CSTR and plug flow, plug flow reactor in order to model the real reactor, now the question is what should be the order in which they should be combined, so should it be CSTR that follows the plug flow reactor or it should be the plug flow reactor which follows the CSTR, so when it is in series combination question is which one comes first and which one comes later, so let us analyze this by considering one by one, so let us consider the first case of CSTR being the first reactor and the plug flow reactor follows the CSTR, CSTR followed by a plug flow reactor, so suppose if there is a CSTR and let us assume that the concentration with which a certain tracer is actually flowing into the reactor is actually CA0, let us assume that there is some species and then if CAI is the concentration with which the species leaves the CSTR and remember that it is a CSTR, it is well mixed therefore the concentration of the species inside the CSTR which is uniform because it is well mixed will be equal to the concentration of the species in the effluent stream and then now this is followed by a plug flow reactor and as it goes through the plug flow reactor let CA be the concentration of the species that actually leaves the plug flow reactor, now if tau S is the residence time of the CSTR and if tau P is the residence time of the plug flow reactor then let us say that the tracer that is introduced is actually a pulse tracer, let us assume that it is a pulse tracer, the objective is to find the RTD function for this combination, eventually the objective will be to find the conversion if there are these two reactors are actually operating in series mode, so let us start with a pulse tracer to find out what is the RTD function, now we know that the CSTR output concentration we know what is the mole balance for the tracer, we have seen many number of times, so the CSTR output concentration based on the mole balance is basically CAI by solving the mole balance is given by CAI as a function of time that should be equal to CA0 into exponential of minus T by tau S, where tau S is the space time of the CSTR. Now the output of the CSTR is actually fed into a plug flow reactor and therefore it simply brings a delay in actually taking the species from the entry of the reactor to the exit of the reactor, so therefore the output from the will be just delayed by the space time of the plug flow reactor tau P, so therefore the output will simply be delayed by the space time of the plug flow reactor at the output of the plug flow reactor itself, so therefore based on these observations we can actually intuit what is going to be the residence time distribution function E of T, so the residence time distribution function will simply be because the output is now going to be delayed, there will be no output that will actually come out of the plug flow reactor till the residence time of the plug flow reactor which means that E of T is going to be 0 for T less than tau P, which is the space time not the residence time the space time of the plug flow reactor and for other times it will simply be whatever RTD function for the tracer that actually goes through the CSTR that is simply delayed by the space time of the plug flow reactor, so therefore we know what is the RTD function for the CSTR, we simply have to introduce the delay of in the plug flow reactor that of the space time of the plug flow reactor, so therefore that will be equal to minus T minus tau P which is the space time of the plug flow reactor divided by tau S which is the space time of the CSTR divided by tau S and this is for tau T greater than or equal to the space time of the plug flow reactor, so that is the RTD function and so now we can simply plot this function it will start at tau P because the plug flow reactor is now going to introduce a certain delay and this will be 1 by tau S and there will be an exponential fall and then similarly we can draw the f curve and it will start at tau P because the plug flow reactor is going to introduce a certain delay and the delay is actually the delay time is equal to the space time of the plug flow reactor and then there will be an exponential increase in the f curve and goes all the way up to 1, so that is the residence time distribution for CSTR followed by a plug flow reactor. Now let us look at the other option where the plug flow reactor comes first and the CSTR comes later, so let us look at the PFR for CSTR which actually follows the plug flow reactor, so the depiction is you have a PFR and then there is a CSTR and there is a the final outlet stream actually comes from the CSTR, so the pulse first enters the plug flow reactor and then there will be a pulse when enters the plug flow reactor because of the nature of the plug flow reactor there is simply a delay of the delay time of space time of the plug flow reactor is actually introduced, so there is a delay by the space time of the plug flow reactor and then after this delay the tracer actually appears at the entry of the CSTR and then it follows the CSTR the residence time distribution function of the CSTR, so the RTD function can simply be E of t, so nothing is going to appear till the space time of the plug flow reactor is reached because from the RTD function of the CSTR the E curve starts from term t equal to 0 that is the tracer will actually start appearing immediately after the tracer is actually pulse tracer is fed into the CSTR but because it is presented in series the delay that is introduced by the plug flow reactor of the time period of the space time of the PFR that will actually be the delay in the overall residence time distribution function as well, so therefore there will be no output in the plug flow reactor CSTR series combination till the space time of the plug flow reactor is reached and after that it will simply be the this will be controlled by the residence time of the CSTR, so therefore the residence time function will be minus t minus tau p divided by tau s where tau s is the space time of the CSTR and this is for t greater than or equal to tau p, so what one can clearly see is that if you compare the residence time function of the plug flow reactor and CSTR which is appearing in series with the residence time function of the situation of the CSTR and the plug flow reactor that is the plug flow reactor follows the CSTR, so this is for CSTR and plug flow reactor actually follows the CSTR, these two residence time functions are exactly one and the same, so therefore what one can actually one observes is that the RTD function is same for CSTR followed by a plug flow reactor in series and plug flow reactor followed by a CSTR, so it does not matter what order in which the CSTR and the plug flow reactor are placed the residence time function of the series combination that is CSTR and plug flow reactor will continue to be the same, now does it mean that the properties of this combination is same, so no that is not true actually the properties will not be same, in fact the performance actually differs significantly and it completely depends upon what is the order in which the CSTR and the plug flow reactor combination is actually placed, so therefore RTD function being same for both reactors does not necessarily mean that the performance of these two reactors will be same, so in fact in this case the performance will be different, first order reaction case is actually an exception only for first order reaction the conversion will be same irrespective of the order in which the two reactors that is CSTR and PFR are actually placed, so let us take an example of that, so let us take a second order reaction, let us take a second order reaction of A going to products and the corresponding specific reaction rate is K and so now, so this can be depicted as let us consider the combination of CSTR followed by a plug flow reactor, so if CA0 is the concentration of the species fed into the CSTR and if CAI is the concentration with which the species leaves the CSTR and if CA is the concentration with which this the species actually leaves the plug flow reactor and the space time of CSTR and plug flow reactor of tau S and tau P, so now we can write a mole balance, write a mole balance for CSTR and the mole balance is that V0 into CA0 minus CAI, so that is the rate at which the species actually enters the reactor and this is the rate molar rate at which the species actually leaves the CSTR and that should be equal to K into CAI square into V where this is the rate at which the species A is actually consumed multiplied by V which is the volume of the CSTR, so simply by dividing this by V0 which is the volumetric flow rate, so volumetric flow rate with which the fluid is actually entering the combination of reactors, so if you assume that V0 is constant, if you assume that the volumetric flow rate is constant then we will find that this equation can be simply rewritten as tau S into K into CAI square plus CAI minus CA0 equal to 0, so that is a quadratic equation in CAI, CA0 is a measurable quantity which is known and tau S is again a measurable quantity, so from this we can solve this quadratic equation and we will find that CAI is given by square root of 1 plus 4 tau S into the specific reaction rate multiplied by CA0 minus 1 whole divided by 2 times tau S into K, so that is the concentration of the species that actually is in the effluent stream that is leaving the CSTR, so now let us write similar mole balance for the plug flow reactor, let us write the mole balance for plug flow reactor, so that is given by DFA, FA is the molar flow rate of the species that is flowing into the plug flow reactor divided by DV which is the volume of the reactor that should be equal to V0 into DCA by DV and that is equal to DCA by D tau P assuming that the volumetric flow rate is constant and tau P is the space time which is the volume of the reactor divided by the corresponding volumetric flow rate and that should be equal to RA which is minus K into CA square, so that is the mole balance and by solving this we can actually obtain that the solution of this equation is 1 by CA minus 1 by CAI that should be equal to the space time of the plug flow reactor multiplied by the specific reaction rate, so from here we can see that CA can actually be expressed as 1 by tau P into K plus 1 by CAI where CAI is essentially given by, CAI is given by the quadratic function that we actually derived a short while ago, this is the concentration with which the species actually leaves the CSTR, so that is actually given by square root of 1 plus 4 tau S K CA0 minus 1 divided by 2 tau S into K, so that is the specific reaction rate, so this is the concentration of the species that actually leaves the plug flow reactor. Now we want to compare the performance of the combination of CSTR and plug flow reactor where the plug flow reactor follows the CSTR and the other combination where the CSTR actually follows the plug flow reactor, now we found that the residence time for these two are same and we are attempting to find what is the conversion of the species for a second order reaction for both these combinations, so let us now take the second combination where the CSTR follows the plug flow reactor and the depiction is CA0, this concentration which we see species actually enters the plug flow reactor and if V0 is the corresponding volumetric flow rate, so that is the plug flow reactor and CAI is the concentration with which the species actually leaves the plug flow reactor and then it enters the CSTR and CA is the concentration with which the species actually leaves the concentration of the species in the effluent stream of the CSTR, so now from the plug flow reactor mole balance we can find that the 1 by CAI minus 1 by CA0 that should be equal to the space time of the plug flow reactor multiplied by the corresponding specific reaction rate. So from here we can estimate that CAI is equal to 1 by tau pk plus 1 by CA0, now from the CSTR mole balance we can actually find that from the mole balance of the CSTR we can actually find that CA is equal to square root of 1 plus 4 tau sk into CAI minus 1 divided by 2 times tau s into k, so where CAI is actually given by 1 by tau p into k plus 1 by CA0, so clearly the expression that you get for the concentration of the species that is actually leaving the CSTR if there is a plug flow reactor which is preceding the CSTR is completely different from the expression that you get for the situation where the plug flow reactor actually follows the CSTR, so what it suggests is that the permock performance of the reactor depends on how the CSTR and plug flow reactor are combined, so it is important how they are combined which means that whether CSTR appears first or CSTR appears later actually matters when we actually estimate the performance of the combination of reactors. So therefore this clearly suggests that the overall performance of the combination of reactors depends on overall performance of combination reactors actually depends on the combination sequence and not just the combination even if it depends on the combination sequence even if the RTD is the same, so even if the RTD function is same for different combinations the overall performance of the combination actually depends on the sequence actual sequence in which the reactors are actually placed, so this clearly shows that RTD function actually does not completely characterize, RTD does not completely characterize the real reactor situation, in fact it only says what is the nature of the residence time distribution and it cannot directly tell what is the actual overall performance of the reactor, in fact additional piece of information is actually required and this also suggests that RTD the residence time distribution function is unique to a reactor slash reactor system combination, reactor system but vice versa is not true, that means that the reactor power reactor system is not unique to a given RTD whereas an RTD can actually be unique to a given reactor slash reactor system and what we demonstrated just now is that depending upon the combination in which it is placed the overall performance could be different but both combinations have exactly the same RTD function. So now what all this points to is that in order to determine the performance of the non-ideal reactor which is the original objective of the whole topic of residence time distribution is to determine overall performance, so determine the performance one needs to know what is the RTD but RTD function alone is insufficient, RTD alone is inadequate, in fact one needs to know additional piece of information if one needs to characterize or one needs to find out what is the performance of a real world reactor. So in addition to RTD adequate model of the non-ideal reactor is required one needs to know what is the appropriate or the correct model of the non-ideal reactor without which the performance can actually not be estimated, so just the RTD function is alone is not adequate one needs to know what is the adequate model and in addition to that more importantly even if the model is known the knowledge of the extent of segregation needs to be known which means that what is the degree of mixing of the fluid elements inside the reactor that information or that knowledge has to be available in order to estimate the overall performance. So what all this points to is that simply knowing the RTD function is insufficient there is there is clear need for adequate model in addition to the RTD function and more importantly what is the degree of mixing or what is the extent of mixing of fluid elements inside the reactor that needs to be known and that is what we will see for the next few lectures. So now in real reactors the real reactors are actually not very well mixed they are not very well mixed and in fact they do not even behave like a plug flow. So far we have seen in the first course of reaction engineering and also in this second course of reaction engineering there are actually CSTR models and what happens if there is different order reaction in CSTR what happens if it is in a plug flow but unfortunately it turns out that the real reactors are not well mixed and there they do not behave like a plug flow. So which means that one needs to come up with various different kinds of approach a different approach in order to model the non ideal reactor in order to estimate the performance of such real world reactors. So we know that the RT residence time distribution function is actually used for diagnosing various aspects of the non ideal reactor for instance RTD for diagnosing the mixing extent of mixing and we will look at it in a little bit more detail in this lecture and in the next lecture and then it is also used for detecting bypassing in the reactor if it is present and it is also used for detecting dead volume once again if it is present. But it does not give any clue about what is the conversion of the species which may be undergoing a certain type of a reaction maybe it is a first order maybe it is a second order or some other type of kinetics. So it does not tell us what is the so the question is what is the exit conversion what is the exit conversion of the species. So can we use RTD to actually find out what is the exit conversion and suppose let us assume that the kinetics of a particular reaction is known then are there methods or are there ways by which we can use RTD function which is used for diagnosing various aspects of the functioning of the reactor but can we extend that and do something develop a method in order to estimate what is the exit conversion of a particular species. So the next topic that we are going to look at starting from this lecture is actually going to be predicting the conversion of a real reactor. So predicting conversion actually has a strong impact. So knowledge of conversion of a particular reaction actually has a strong economic impact because the design of things around it can actually be planned accordingly and therefore if one is able to predict the conversion it actually goes a long way in terms of designing the reactor and designing other processes which is associated with this particular with the reaction of interest and this has a strong economic implications for the industry which is actually involved in conducting that reaction and generating synthesizing a certain required product certain desired product. So let us look at how to predict conversion if we know what is the residence time distribution function. So there may be two systems, two reactor systems which have same RTD functions but will it be possible to estimate the conversion in these two reactors even if the RTD functions are same and the answer is yes it is possible and there are methods to do this. So the information that one needs to know is of course one needs to know what is the RTD function and one needs to know what is the model that represents the non-ideal reactor that captures the process which is occurring inside the non-ideal reactor or real world reactor and one needs to know what is the kinetics of the reaction that is actually being conducted inside the reactor. So if we know this then one can actually devise methods to find out what is the exit conversion of the species. So there are different models which are available. So the next couple of lectures we are going to look at a couple of these models and other models will actually be dealt with in some of some other future lectures. So first we are going to look at the zero adjustable parameter models and these are essentially two types of models. The first one is called the segregation model and the second one is the maximum mixedness model. So we are going to look into details of these two types of models to predict the conversion of a real world reactor. So the residence time distribution what does residence time distribution function give? What does it give? It actually gives information about time spent by various fluid elements inside the reactor. But it does not provide any information about the mixing. There is no information about the mixing of the fluid elements inside the reactor. Now what is mixing? Mixing is essentially it is the exchange of molecules or matter which is actually in the fluid stream. So the exchange of matter between different sections of the fluid stream is what is called as mixing. What does it do? Why is mixing so important? Because it actually controls the behavior of the reaction and it actually strongly controls the conversion of that particular reactant in the reactor. So it actually controls the reactor behavior. It has strong implications on the performance of the reactor and now an exception to this is that the first-order reaction is actually a special type where only RTD information, RTD function is actually enough to predict the conversion and in fact we will see in one of the future lectures that how is it that the first-order reaction for a first-order reaction just the knowledge of RTD function is sufficient to predict the conversion and what is the reason why this is the case and this is not true for other kinetics. So the degree of mixing is actually the information about the degree of mixing is actually required for all other kinetics other than the first-order reaction. So for reactions which are not first-order, for reactions which are not first-order what is required is actually a complex model of the reactor. So reactions which are not first-order the complex model of reactor is required, complex model of the reactor is actually required to find to predict the conversion of the species which is undergoing the reaction and also in addition to that the degree of mixing, degree of mixing both macro and micro mixing information is required. So these terms macro and micro mixing will actually be defined in a short while. So let us first look at the macro mixing. So there are two types of, two levels of mixing which one is the macro mixing and the other one is the micro mixing. So the macro mixing is essentially is the extent of mixing which is actually characterized by the RTD function itself. So that is the, it is characterized by the RTD function, the residence time distribution function and an important aspect of macro mixing is that the information about mixing at the molecular scale is actually does not occur if the fluid is actually under a macro mixing state. So the no information on mixing at molecular scale. Now it is important to pay attention to what the scale means here. So remember that the fluid particles, fluid elements which actually is flowing through the reactor essentially consists of atoms and molecules. So now there may be exchange, there may be interactions and exchange of matter at the molecular scale and in other extreme there may be situation where there may be lumps of molecules which actually do not interact with each other. So there are different scales which are actually present inside. The microscopic scale is where the molecule, we look at what is the interaction or how one molecule of the species it encounters another molecule which is actually present next to it. So that is the microscopic scale and the macroscopic scale is actually a situation where the microscopic scale is actually completely ignored or the information is not available. So let us next look at what is the, what is micro mixing, let us define micro mixing. So micro mixing that describes the how molecules with different age. Now suppose we consider a reactor where there is some fluid which is already present inside the reactor and then there is new set of fluid elements which are actually flowing into the reactor. Now the age that is the time that is spent by the fluid particles which is already present inside the reactor is now going to be larger than the time that is actually spent by the fluid which is freshly entering into the reactor. So now the question is how these molecules which has spent a larger time inside the reactor, how do they encounter when a fresh set of particles are actually, fresh set of fluid elements are actually pumped into the reactor. So that the aspect which actually characterizes this is what is micro mixing where we look at how molecules encounter other molecules which are actually having a completely different age or the time that they have spent inside the reactor is different from the ones which are actually entering a fresh into the reactor. So now there are two aspects associated with micro mixing. The first aspect is where all molecules of same age group remind together. So irrespective of how the molecules actually are flown inside those molecules which actually spend certain amount of time they all remind together and freshly entering molecules that is new set of molecules which have which are currently entering the reactor they also cling together and they all remind together. So the first state of micro mixing is a situation where all molecules of same age they are actually reminding and they are glued together in one group. So this kind this actually means that there is no mixing between groups and this state this situation or the state of the fluid where there is no mixing between groups is what is called as a complete segregation system where fluid particles of different age are actually completely segregated and they are actually not interacting between each other. The second situation is where molecules of different age are completely mixed. So consider a situation where there is a reactor in which the fluid is already present and then there is new set of fluid particles are actually added into the reactor. So therefore now the reactor is now having fluid particles which are of different age and so these fluid particles which are actually pumped in a fresh they actually interact with the they are actually encountered by the fluid particles which are actually already present in the in the reactor in the reactor and so therefore these two systems they mix with each other and they get completely mixed. So therefore one cannot virtually distinguish where the particles of which has which has a higher age is actually located as compared with the particles which are actually freshly pumped into the reactor. So that kind of a situation is what is called as a complete micro mixing. So these two states of complete segregation and complete micro mixing they have strong implications on the performance of the reactor. Now for a given residence time distribution function, so remember that the residence time distribution it only characterizes the macro mixing inside the reactor. For a given residence time distribution the upper and the lower bound of the conversion that is actually achieved in the reactor is controlled by these two limits of micro mixing that is the complete segregation and the complete micro mixing. So the for a given RTD given an RTD function for a real reactor the upper and the lower bound conversion actually are specified are actually specified by these two extreme cases that is complete segregation and complete micro mixing. So the complete micro mixing is also called as maximum mixedness. So the two zero parameter zero adjustable parameter model is essentially the model that corresponds to the complete segregation and the other model corresponds to the maximum mixedness model. So how do we incorporate this aspect of complete segregation and complete micro mixing in the reactor model and combine that with the RTD function is what is going to be the next step which is next step towards predicting the performance of a real reactor. Now suppose if n is larger than 1 that is the order of reaction n here stands for the order of reaction. So if the order of reaction is greater than 1 or if the order of reaction is less than zero that is if it is a negative reaction negative order reaction then it turns out that the segregation model the complete segregation model predicts highest conversion. On the other hand for reaction orders between zero and one that is even for partial fractional order reactions it turns out that the maximum mixedness maximum mixedness model predicts highest conversion. So now there are a few definitions that we need to define before we attempt to consider the segregation model and the maximum mixedness model. So the first one in the definition is what is a globule? So a globule is basically a group of molecules millions of molecules which are actually essentially of same age. So it basically is millions of molecules of same age. So that is what is a globule and then a macro fluid a macro fluid is essentially the one where the globules of a given age they actually do not mix with the other globules. So in a reactor suppose there is a fluid which is already present and it consists of many different globules and each of these globules will actually have contain millions of these molecules all of which are actually of same age. Now macro fluid is one where the fluid in which these globules which are actually distinctly and independently present and they do not interact with each other and that is what is called as a macro fluid. So let us look at what macro fluid it is a globules, it is globules of a given age they do not mix with other globules which means that these are essentially non coalescing globules and so one can actually visualize this as suppose if there is a reactor and there may be globules which are actually present here and these globules will continue to maintain their identity and they will not interact with each other. Now on the another definition is another type of fluid is the micro fluid where the globules are not constrained they are not constrained and in fact they can freely mix with any other globules which are actually present inside the reactor. So they mix with other globules which means the molecules present in one globule actually moves to another globule and there is exchange of molecules between different globules. So this can actually be depicted as suppose if there is a reactor then if the fluid that is present inside the reactor is a micro fluid then one can actually virtually not distinguish between the particles which are actually of different ages. So they will all be present together they will all be completely mixed inside the reactor. So if this is the inlet stream and the effluent stream so the fluid which is present inside is now going to be mixed and so one will not be able to identify the globules independently. So let us summarize what we have learnt in this lecture. We have looked at what is the, we have compared the residence time distribution for a plug flow reactor that is operated under different conditions and then we looked at the combination of plug flow and CSTR in series mode and we observed that the residence time distribution function is the same whether the plug flow reactor appears first or the CSTR appears first. However the performance of these two reactors combination, performance of the combination of plug flow and CSTR depends upon which one comes first whether the plug flow or the CSTR comes first and the other reactor follows the first one. So this we demonstrated for a second order reaction and then we observed that the this example sort of suggest that the residence time distribution function is not sufficient to actually predict the conversion or the performance of the non-ideal real world reactor and additional piece of information such as a model and also the knowledge of the extent of mixing is actually required in order to predict the performance of the reactor. And so we looked at different types of mixing that is the macro mixing and the micro mixing and introduced a certain definitions and we will continue from there in the next lecture. Thank you.