 Friends, let us summarize what we learnt in the last lecture. We looked at catalyst deactivation, we defined what is catalyst deactivation, what is the activity of the catalyst, which characterizes the catalyst deactivation. We looked at different methods or different strategies or different ways by which catalyst can get deactivated. For instance, we looked at sintering slash aging process, which occurs because of long term exposure of the catalyst to the gas stream and the reactants. Then we looked at coking or fouling, which predominantly occurs because of the deposition of carbonaceous material, when there is a catalytic reaction that involves hydrocarbons. Then we initiated discussion on the poisoning of catalyst, which primarily occurs because of the poisons, which may be present in the feed or the reactant or the product itself may act as a poison. So, we started discussion on the strategies to characterize poison when it is present in the feed. So, let us continue with that. So, a good example of that is the presence of sulphur or lead in the petrochemical feedstocks, which actually acts as a poison for the catalyst. So, example are sulphur and lead, which may be present in the petrochemical feedstocks. So, poison actually competes with the reactants for the catalyst side, with reactants for catalyst side. This is because the poisoning process can actually occurs simultaneously along with the catalytic reaction. So the active sites, which are available for the reactants to go and adsorb onto the catalyst side and continue with the catalytic reaction, the poison also in a similar way is ready to go and bind itself, adsorb itself onto the catalyst side. So, therefore, it actually competes with the reactants for the available free active catalyst side. So, this poses a strong competition for the reactants and that determines the extent of the deactivation of the catalyst. So, let us look at this cartoon. Suppose if there is a reaction A going to B plus C, heterogeneous catalytic reaction. Now if the catalytic sites, suppose if these are the active catalyst sites, which are present at time t equal to 0 in the catalyst and after the reaction starts proceeding and if these are the catalyst sites at time t1, which is greater than 0, then the sum of these sites may actually be, some of the sites will be containing the species A, which is adsorbed onto the surface, so species A and some other sites may be containing the poison, which is present in the feed steam. Suppose if P stands for the poison, then this site may be contaminated with the poison, that is the active site is now blocked with the poison, which is adsorbed onto the active site and maybe some other site may contain species B, which is basically the product, which is formed because of the catalytic reaction. Now if I look at further larger time point t2, which is greater than t1, then you will have many more sites, which are actually filled with the poison. So we will have many other sites, which are filled with poison P and therefore the number of active sites, which are available for the reactant species to go and adsorb onto the active sites for the catalytic reaction progressively decreases. So if I look at further larger time point t3, which is greater than t2, then pretty much all the sites, all the active catalytic sites are now occupied by the poison, which is present in the feed and therefore the catalyst is completely deactivated at this time point t3. So it is very important to characterize how this deactivation process occurs because the catalyst normally used in the catalytic reactions are very expensive and they cost a million dollars and therefore it is, if one can avoid the poisoning of the catalyst then it saves a lot of money for the industry and therefore and also it helps in maintaining the certain desired conversion, which is required for, which is desired by the company. So therefore let us look at what are the different mechanisms that capture the poisoning process and deduce the rate law, which governs the poisoning process and try to incorporate that into the reactor design and this has a strong implications in terms of how to avoid the poisoning process in the reactor. So let us consider the reaction steps. So there are three steps, which is associated with this main reaction. So the first step is the adsorption step, where species A adsorbs onto the surface and A s is basically the species A, which is adsorbed to the catalyst site and then the species A, which is adsorbed onto the catalyst site undergoes a catalytic reaction and leaves out product B, which is again adsorbed onto the catalyst surface plus a product C, which goes into the gas stream. Then this product B, B.S, which is present in the, on the catalyst site gets desorbed and the product B goes into the gas phase and leaving one catalyst site vacant, one of the active sites. Now this particular reaction will typically have a rate law, which looks like this, which we have already seen in one of the earlier lectures. So if A is the activity of the catalyst, then the rate of consumption of the species A, because of this catalytic reaction per gram more of the catalyst per unit time is given by the activity of the catalyst A multiplied by the specific reaction constant and multiplied by the concentration C A divided by 1 plus the equilibrium constant for adsorption, desorption of species A multiplied by the corresponding concentration plus the adsorption constant for species B multiplied by the corresponding concentration C B. Note that the main reaction is assumed irreversible here. Now the poisoning reaction, the poison which is present, suppose P is the poison which is present in the gas stream, which is fed along with the reactants, then this poison is now going to participate in a poisoning reaction which competes with the main reaction for the active catalyst site. So the poisoning reaction which occurs simultaneously along with the main catalytic reaction is as follows. So P plus S, which leads to binding of that particular poison on adsorption of the poison on to the active site. And the rate law, rate of the decay which is because of the poisoning of the catalyst active site is given by minus D A by DT that is equal to the specific decay constant KD prime multiplied by C P M multiplied by A to the power of Q, where C P is essentially the concentration of the poison in the concentration of P in the gas stream, that is the concentration of the poison in the gas stream. So in order to estimate, in order to find out the activity of the catalyst, we need to integrate the rate expression, but before that we need to estimate what is the concentration of the poison in the gas phase. So for that we can simply write, find out what is the poison removal rate. We can find out what is the poison removal rate from the gas and that provides a mechanism to characterize the dynamics of C P. So D C P S by DT which is basically the rate of change of the concentration of the poisoned active sites and that is equal to RP dot S and that is equal to the specific reaction constant multiplied by C T 0 minus C P dot S into C P. So this C T 0, C T 0 is the total active site concentration at T equal to 0 and C P S is the concentration of sites in which P is present, in which P is present, in which P is present that is P is adsorbed onto that surface and C P is the corresponding gas phase concentration. Now if we define a new variable F, which is essentially the ratio of the concentration of the sites onto which the poison is adsorbed divided by the total concentration or the total number, total concentration of all the sites which is present at before the reaction is started. When the catalyst is fresh then we can redefine this differential equation as D F by DT equal to K D into 1 minus F into C P. Now by definition the activity of the catalyst is essentially the ratio of the reaction rate at a particular time when the catalyst is used up to a certain time divided by the reaction rate as if it were a fresh catalyst that is when the catalyst is unused but which is exactly equal to 1 minus F. So therefore the activity of the catalyst A T equal to 1 minus F and so we can rewrite this equation as minus D A by DT that is equal to K D into A into C P. So this equation captures the activity of the catalyst as a function of the gas phase concentration of the poison and the reaction rate on the decay rate constant. So C P is a measurable quantity so if one can measure the concentration of the poison in the gas phase we should be able to estimate the dynamics of the decay of the catalyst. So let us look at how the catalyst activity changes in a packed bed reactor. So suppose there is a packed bed reactor and suppose if this is the inlet of the packed bed reactor and this is the exit of the packed bed reactor and if we want to look at the activity as a function of the weight of the catalyst at 0 when the gas stream does not enter the catalyst activity is almost 1 and at time T equal to 0 when there is no reaction the catalyst activity and all the locations is approximately equal to 1 because the catalyst is not used and so all catalyst which is present inside is a fresh catalyst so it should be equal to 1. At some other time T1 the profile would be like this so this is T1 greater than 0. So now some of these catalyst which is present near to the inlet of the reactor has now experienced reaction and therefore these catalyst will now start deactivating and so the initial part of the reactor catalyst will be deactivated significantly and then this is the activity profile that one would get. Now if I further look at some other time at T2 which is greater than T1 then this is the kind of profile you will get where the reactor is exposed to reactor is now used for conducting the catalytic reaction for a longer period of time and therefore the more catalyst starting from the inlet of the reactor would actually get deactivated and so that is the kind of profile that one would get and then similarly you can profiles for a much larger time T3 which is greater than T2 would like this would look like this and at a much later time T4 which is greater than T3 the profile would look like this. So therefore progressively the catalyst which is present inside the reactor will start getting deactivated and that is what is captured in this profile here. So next let us look at so far we looked at the poisoning that is caused by the poison which might be present in the feed. So let us look at the poisoning which is caused because of the reactants or the products itself. So remember we mentioned there are two different ways by which the poisoning can occur. One is because of the poisonous compounds which may be present in the feed stream. Another one because of the nature of the reactants or products which may itself become a poison for the active science. So let us look at the second case of poisoning by reactants or products. So the main so which means that the react suppose let us consider the case of reactants and the framework for if the product is the poison the framework is exactly the same. So let us specifically consider the case of reactant being a poison. And so the reactants are now adsorbing onto the catalyst side for two purposes. One is for the catalytic reaction to occur and another one is basically to poison the catalyst side. So therefore the main reaction the main reaction would be A plus S which is the catalytic side which will basically lead to adsorption of the catalyst onto the catalyst surface. Catalytic side and then A dot S would lead to B dot S plus let us say some C and similarly B dot S can get dissolved to form B in the gas phase and release a catalytic side. So this can actually be captured as A plus S essentially leading to B plus S. So now the poison the reactant because it also behaves like a poison it now competes with the main reaction in order to block the active catalyst side. So therefore the main reaction which is A plus S leading to B plus S whose reaction rate can actually be represented as minus rA prime equal to kA into CA to the power of N let us say for simplicity sake then the poison reaction there might be a poison reaction which basically is A plus S which leads to A dot S which poisons the catalyst side and let us consider the rate of decay is equal to kD prime into CA to the power of M into A to the power of Q. Let us assume that this is the decay law and a very good example of reactant itself acting as a poison to the catalyst is the case of methane formation using carbon monoxide and hydrogen on ruthenium catalyst on a ruthenium catalyst it leads to CH4 and H2O. So this is an excellent example of a situation where the reactant which is the carbon monoxide which goes and poisons the catalyst side. So let us try to characterize this process particularly for this example reaction. So if the rate of reaction is given by minus rCO equal to specific rate reaction rate multiplied by the activity as a function of time and multiplied by the concentration of the carbon monoxide and the decay rate law is given by minus DA by dt that is equal to rD which is the rate at which the catalyst gets deactivated and that is the specific rate constant multiplied by the activity multiplied by the concentration of the carbon monoxide in the gas stream. Now if we assume that the kinetics is separable so if we assume separable kinetics and if we assume that the poisoning occurs at a certain constant concentration so if the poisoning at certain concentration Cpo then we can rewrite the rate law as decay rate law as DA by dt equal to Kd prime into Cp0 to the power of n into A to the power of n and that is equal to some Kd so we can now club these two constants together we can club these two constants together into one constant multiplied in that is equal to Kd into A to the power of n. So if n is equal to 1 that means that the activity is given by exponential of minus Kd into t activity is given by exponential of minus Kd into t so this is exponential of minus Kd into t so that is the activity as a function of time of this particular case when n is equal to 1. So there are various types of decay law that have been proposed in the literature and some of these are various decay law so common one is the one where the activity actually linearly changes with time so that is given by minus DA by dt equal to some beta0 and when you integrate this expression we find that A equal to 1 minus beta0t so that is essentially a linear profile, linear dependence of the activity on time and this is actually this kind of a decay has been observed for conversion of para-hydrogen of tungsten on tungsten with oxygen as poison. So this kind of a rate law decay rate law has been observed for a reaction of conversion of para-hydrogen on tungsten with oxygen as the poison that deactivates the catalyst that is tungsten here. Another rate law that has been observed is the exponential rate law so DA by dt that is equal to beta1a and integrating this expression one would find that A equal to exponential of minus beta1 into t so that is an exponential decay with respect to time and this has been observed in the ethylene hydrogenation on copper catalyst with carbon monoxide as one of the poisons. So this exponential decay has been observed in the ethylene hydrogenation which is a very fairly well very common reaction, industry scale reaction and then the third type is minus DA by dt equal to let us see second order decay the order of decay reaction with respect to the activity is 2 and so the dependence of the activity on time is given by 1 by A equal to 1 plus beta2 times t and this has been observed in the cyclohexane dehydrogenation on the platinum alumina catalyst. In addition to these decay laws corresponding to independent deactivation the rate of deactivation can actually depend on the reactant or product concentration for example during hydrogenation of ethylene on palladium catalyst ethylidine formation may reversibly block the active sites. And then the fourth type fourth class of deactivation kinetics is basically a power law behavior so where A power n into A0 to the power of m where n is given by beta4 plus 1 by beta4 so that is the exponent and this while integration one gets the expression for A which is basically some constant A into t to the power of minus beta by 4 so it behaves like a power law of time and this has been observed for cyclohexane aromatization on nickel aluminum catalyst. So now we have found the rate law for we have characterized the different types of deactivation of catalyst and we have found the rate law and for each of these cases so let us try to now implement the incorporate the rate law in a reactor design so let us consider a fluidized CSTR let us consider a fluidized CSTR and let us assume that a gas phase cracking reaction gas phase cracking reaction where A goes to B plus C is occurring in this fluidized CSTR and let us assume that the feed contains a sulfur which is which acts as a poison for the catalyst and also assume that the cracking is a first order process cracking is a first order process and if we assume that the decay catalytic deactivation or decay of the catalyst is first order with respect to first order in activity and its first order with respect to with respect to concentration of species A in the gas stream. So now we can actually so the schematic of CSTR fluidized CSTR it looks like this where you have a fixed volume fixed volume for the CSTR it is well mixed and you have a feed which goes in and let us assume that CA naught is the concentration of the feed at the inlet and V naught is the flow rate volumetric flow rate of the feed at the inlet and CA is the concentration of the product stream or concentration of species A in the product stream and V is the volumetric flow rate with which the fluid stream leaves the fluidized CSTR. Now a simple mole balance can be written for this system so the mole balance is so the mole balance is the rate at which species enters the molar rate at which the species enters the reactor which is V naught into CA naught V naught has the units of volume per time let us say meter cube per time and the CA naught which is the concentration in moles per unit volume minus whatever rate at which the molar rate at which the stream leaves the reactor which is V which is the volumetric flow rate with which the stream leaves the reactor and CA is the concentration at which it leaves the reactor plus the rate of generation of the species A that is equal to the rate of change of the number of moles of A with respect to time which is the accumulation term. Now for constant volume which is true for a CSTR for constant volume the rate multiplied by rate in terms of the volume that is moles per unit volume per time multiplied by the volume should be equal to the rate defined in terms of the catalyst weight that is the moles of A reacted per gram weight of the catalyst per unit time multiplied by the weight of the catalyst so these two have to be equal. So incorporating this equality into the mole balance we can rewrite the mole balance as V naught into CA naught minus V into CA plus rA into V and that is equal to DNA by dt. Now DNA by dt can actually be written in terms of the concentration of the species and so that will be V into dCA by dt. So now the rate at which the reaction occurs that is the rate of consumption of species A is minus rA equal to the corresponding a specific reaction rate multiplied by the activity of the catalyst multiplied by the concentration of the species CA and the corresponding decay law is given by dA by dt that is equal to the decay constant multiplied by the activity of the catalyst multiplied by the concentration of A in the gas phase. Now if we know the rates of this reaction that is the reaction rate then next step we need to do in terms of designing the reactors we need to use the stoichiometry and relate these terms here. So using stoichiometry we can find that the volumetric flow rate of the stream that leaves the reactor divided by the volumetric flow rate of the stream that enters the reactor should be equal to the molar flow rate ft with which the species leaves divided by the it is the ratio of the total molar flow rate of the outlet stream divided by the total molar flow rate of the inlet stream multiplied by the p0 by p which is the total pressure at the inlet divided by the total pressure at the outlet multiplied by t by t0. So if at constant pressure if p equal to p0 and at isothermal conditions t equal to t0 so v by v0 simply reduces to ft by f0 ft0 which is nothing but 1 plus the net fractional change in the number of moles because of the reaction multiplied by x. So now the conversion x is equal to 1 minus fa by fa0 so that is the conversion and that is equal to 1 minus ca into v divided by ca0 into v0. Now remember that the reaction that we are looking at is a giving b plus c so therefore the net fractional change in the number of moles is given by the overall mole fraction of a multiplied by this change in the number of moles which is equal to ya into 1 plus 1 minus 1 that is equal to ya0 which is given by ca0 by ct0 which is the ca0 is the concentration of the species in the inlet stream and ct0 is the total concentration at the inlet stream. So by using this expression we can rewrite v by v0 as 1 plus epsilon minus epsilon into ca into v divided by ca0 into v0 which is and from here we can easily deduce that the volumetric flow rate of this stream that leaves the reactor should be equal to v0 into 1 plus ya0 divided by 1 plus ca by ct0. So now so plugging in all these volumetric flow rate expressions in terms of concentrations etc into the mole balance we can rewrite the mole balance as we can rewrite the mole balance After defining residence time tau which is the time that the fluid element depends inside the reactor which is given by volume of the reactor divided by the inlet volumetric flow rate and so the mole balance will be dca by dt that is equal to ca0 by tau minus 1 by tau into 1 plus ya0 divided by 1 plus ca by ct0 plus k into a into tau into ca and the activity is given by activity a can be found from the decay rate expression for the catalyst that is dA by dt equal to minus kd into a into ca. So these two equations essentially characterize the simultaneous deactivation and the catalytic reaction which is happening in the reactor and so if one needs to obtain the dependence of concentration on time and the activity on time then one needs to solve these two equations, solve the equation simultaneously and one should be able to obtain the concentration profile as a function of time and the activity profile as a function of time. So now we observed that the catalytic deactivation is a very serious problem because it poses a strong major problem in terms of maintaining the conversion of the desired products and as a result it has a strong effect on the economy of the company which is actually producing that particular product. So therefore it has a strong effect on the economics of that particular product and as a result choice of reactor is extremely important which will help in minimizing or avoiding the deactivation process. So the choice of the reactor, so what we are going to discuss now is to look at how can we use a particular choice of how can we decide what should be the choice of the reactor that will help in minimizing the deactivation of the catalyst and with the objective of maintaining the conversion of the desired product. So we know that deactivation and reaction they occur together, they occur simultaneously. So therefore the performance is severely affected and as an offset one could actually use different reactor configuration to maintain the conversion. So what we are going to discuss now is to look at what are the different types of configurations that one has to use in order to minimize the deactivation of the catalyst or rather in order to maintain the desired conversion. So suppose if the decay is very slow, so naturally the choice of the reactor and the necessary steps that one has to take depends upon the speed with which the catalyst gets deactivated. So if the decay is very slow then what really works is basically using a temperature time trajectory where the feed stream is constantly preheated and the temperature is increased slowly in order to minimize the deactivation and thereby maintain the conversion of the reaction which is happening inside the reactor. Suppose if the decay is moderate, it is a moderate decay then it is suggested to use a moving bed reactor and we are going to look at the design of this. So moving bed reactor can be used in order to minimize the deactivation and then if it is a rapid decay of the catalyst then one needs to use a straight through transport reactors. This classification is based on the deactivation time scales as dictated by the corresponding decay constant. So let us look at the slow decay, first case of slow decay. Suppose if the catalyst is decaying very slowly, so if it is a slow decay system then in order to maintain a constant conversion the strategy is to increase the reaction rate by slowly increasing the temperature, slowly increasing the feed temperature. So this by slowly increasing the feed temperature one could actually increase the reaction rate and thereby maintain the required conversion. And so the classical setup which helps in such kind of a design is basically a fluidized bed. In fact in many fixed bed reactors as well such as strategy is employed. Suppose if this is a fluidized bed, it is a fluidized bed reactor and then a preheater is added to a preheater is added to the inlet stream. So suppose the inlet stream is entering the preheater at certain initial temperature T0, then as the reaction proceeds the preheater constantly increases the temperature of the inlet stream so as to maintain a certain fixed conversion x. So which means that the reaction rate at T equal to 0 that is at the start of the reactor at the temperature T0 should be equal to the reaction rate at certain other time and certain other temperature T and in terms of using the rate law that we have derived for catalyst deactivation rate law for the reaction when the catalyst deactivation is simultaneously occurring. So that will be A into T, T. So now the activity is a function of both time and temperature and so minus rA into T equal to 0, T0. So these two quantities have to be equal in order to obtain the desired conversion. This quantity must be equal to this quantity in order to obtain the desired conversion in order to maintain the desired conversion. So now if we assume that it is a first order reaction, if we assume that it is a first order reaction and also if we neglect the concentration variations, if we neglect the concentration variations then we can write K of T0 into CA should be equal to the activity of the catalyst multiplied by the K that is the specific reaction rate at a particular temperature multiplied by the corresponding concentration CEM. So if we neglect the variations in the concentrations then we can write that K T0 should be equal to A into K of T. But we know from the Arrhenius dependence of the specific reaction rate that we can rewrite this expression as K into T0 that is equal to A into K T0 multiplied by exponential of EA which is the activation energy for that reaction divided by R into 1 by T0 minus 1 by T. So cancelling these two we can now find an expression that relates the local temperature that is the temperature at which the reactor is being operated with the activity and the temperature when the reaction was started. So 1 by T should be equal to R by EA which is the activation energy of that particular reaction into ln A minus plus 1 by T0. So this is the relationship between the temperature at which the reactor has to be operated and the activity as a function of activity and the initial temperature T0. So now if we look at the decay law which basically characterizes the deactivation of the catalyst. So if you look at the decay law so minus DA by DT minus DA by DT is given by K D0 multiplied by exponential of ED by R which is again the specific decay constant is also depends on temperature using the Arrhenius type expression into 1 by T0 minus 1 by T multiplied by A to the power of N. But we know using the rate law we can relate the temperatures using the expression 1 by T minus 1 by T0 that is equal to R by EA into ln A. So by substituting this expression we find that minus DA by DT that is equal to the specific deactivation constant multiplied by exponential of minus ED which is the activation energy for the deactivation of the catalyst multiplied divided by EA into ln A into A to the power of N. So this is equal to K D0 into A raised to the power of N minus ED by EA. So this expression provides a way to find out what is the dynamics of the activity as a function of time. Now if N is not equal to 1 that is the that is the order of the decay or deactivation as a function of activity is not equal to 1 then with assuming that the activity is 1 at time T equal to 0 the expression can be integrated and found that T equal to 1 minus exponential of EA divided by EA minus N into EA plus ED that is the activation energy for the decay divided by R into 1 by T minus 1 by T0 the whole divided by K D0 into 1 minus N plus ED by EA. So this provides this expression provides a relationship as to how the temperature has to be changed as a function of time. So this provides a relationship of how the temperature has to be increased in order to maintain a certain fixed conversion or desired conversion. Note that this relationship is valid only for independent deactivation. So this kind of an expression provides an industry operating person as to how much the temperature has to be increased or how much preheating has to be done in order to maintain the conversion of the desired products. So now if I look at the if I plot this as a function of time and temperature so this is the kind of profile that I will get. So if the catalyst deactivation is very slow then I can maintain the conversion by actually increasing the temperature of the gas that comes into the reactor and the kind of profile and the kind of temperature in raise as a function of time is captured in this graph here. So suppose if it is a first order decay process if the deactivation kinetics is directly proportional to the catalytic activity and if the exponent is 1 then the relationship between the temperature and the time at which the temperature has to be changed is given by ea divided by kd0 into ed multiplied by 1 minus exponential of ed by r into 1 by t minus 1 by t0. So that is the expression one would get and that sort of tells you what is the how the temperature has to be increased as a function of time when n equal to 1 that is basically the first order dependence of the deactivation kinetics on the activity of the catalyst. Now as a next step we look at what is the what kind of configuration that one has to use if the decay is moderate or a rapid deactivation of the catalyst. So if you look at the moderate decay case what is required here is because the deactivation is fairly quick, fairly rapid then there has to be a continuous regeneration or replacement of the catalyst inside the reactor in order to maintain a certain conversion, certain desired conversion of the products, certain desired conversion and therefore it is important to have a continual regeneration or replacement of the catalyst and there are two types of configurations, two reactor configurations have been used. So one is the moving bed reactor and the second one is the straight through, second one is the straight through transport reactor. These are the two types of reactors which is useful when there is a moderate or rapid decay of the catalyst which is used in the catalytic reaction. So the moving bed reactor in the way the moving bed reactor is basically a reactor where the catalyst is constantly replenished and the way the catalyst is replenished is suppose if this is the reactor, it is a cylindrical tube reactor, suppose this is the reactor and the feed to the reactor is provided from the top and the catalyst leaves the reactor and enters another chamber which is called a kiln and this is used as a regenerator, it is used as a regenerator which regenerates the spent catalyst which is deactivated and air is passed through this regenerator and once the catalyst is regenerated, the regenerator catalyst is circulated back, the regenerated catalyst is airlifted into another chamber which is called the separator chamber, so it is a separator into the separator chamber and once the catalyst goes into the separator chamber, it pumps the catalyst back into the reactor. So this is how the constant replenishment of the catalyst occurs in a moving bed reactor and this helps in maintaining the conversion and the products are removed from the bottom of the reactor. So the catalyst pellets that are typically used is about 1.8 inch or 1.4 inch, that is the typical size of the catalyst pellets that is used in these kinds of reactors. So now let us try to write model equations in order to characterize the moving bed reactor and thereby maintain the conversion of the reactants into products. So suppose if this is my reactor and if the catalyst enters the reactor at a rate of US, that is the mass of catalyst per unit time and if it leaves at the same rate, then one can write a molar balance and obtain an equation for the molar flow rate of the catalyst, molar flow rate of the species as a function of weight of the catalyst. So this equation molar balance actually captures the dependence of the molar flow rate on the weight of the catalyst and so this can be re-written as FA0 into dx by dw equal to minus rA prime. Now the rate of the reaction is now can be written as minus rA prime is equal to some activity of the catalyst A multiplied by the rate of the reaction at time t equal to 0. So therefore it can be written as A of t into the rate constant k multiplied by some function of the concentration CACB etc. And the deactivation kinetics is given by dA by dt is equal to the deactivation rate constant multiplied by A to the power of n. So let us assume that this is the decay model, this is the decay rate, that is the decay rate. This decay rate law is a simplified form and in general the decay rate can depend on reaction slash product concentration. And so the contact time needs to be related with the weight of the catalyst in order to characterize this particular reactor. So the contact time of the reactor, contact time of the catalyst t is given by w by us and from here we can estimate that at a differential time the amount of catalyst that enters the reactor is given by dw by us. So from here we can now rewrite the mole balance in terms of the, in terms of the catalyst which enters the reactor and also the decay rate law accordingly and the decay rate law is now given by dA by dt into dt by dw equal to kd divided by us into A to the power of n. And that is nothing but dA by dw. So that relates the decay of the catalyst as a function of the catalyst that is entering the reactor and dx by dw is given by A by fA0 into minus rA prime at t equal to 0. So one has to solve these two equations simultaneously, these two equations need to be solved simultaneously to obtain x versus the weight of the catalyst and A versus the weight of the catalyst. So that characterizes the moving bed reactor. So once we know this we can use this in design parameters in order to appropriately design the catalyst. So what we have seen in today's lecture is we have summarized the different types of deactivation of the catalyst that is sintering, aging, fouling or coking process and poisoning of the catalyst and we looked into the details of the mechanism of two types of poisoning. One is poisoning if the poison is present in the feed or if the reactant or the product itself acts as a poison. And then we looked at fluidized CSTR to incorporate the decay law which is based on the poisoning of the reactor and then poisoning of the catalyst and then we proceeded to look at what are the different reactor configurations have to be used because it depends upon the nature of the deactivation. If the deactivation is slow then one can use a temperature time trajectory reaction system and if it is moderate or a fast reactor one has to fast decay then one has to use a moving bed reactor or a straight through transport reactor. Thank you.