 Good morning. So, in the last lecture, we looked at how to incorporate effectiveness or rather the effective diffusivity or pore diffusion effects on the reaction kinetics. We again I will repeat like we have the reaction taking place in 6 steps. First is external mass transfer of the reactant, then internal diffusion, adsorption, reaction, desorption of the product, internal diffusion, back diffusion of the product and then the external mass transfer of the product. So, out of which we have already considered 3 steps adsorption reaction and desorption. Now, we are trying to incorporate the effect of internal diffusion. So, we have this catalyst particle and we are looking at the diffusion inside and this is external surface r is equal to capital R and this is center. I told in the last lecture that it is a porous particle. You have pores, actual moment of the particle will be torches, but all these effects the constriction, torchocity, porosity everything is incorporated in a term called effective diffusivity, DE effective diffusivity and when we write when we use this term effective diffusivity then we are free to write a normal fixed law for diffusion in the radial clock coordinates if the particle is spherical. So, in that case the flux becomes DE into DC by or DCA by DR of course, the minus sign. So, this becomes a flux. Now, what is the difference between this flux and the flux that is taking place in a non porous particle or other not non porous or as a continuous medium rather the continuous medium the D is an is a different diffusivity is a bulk diffusivity that we talk about normally. There is no effect of porosity, there is no effect of walls, no effect of the torchocity as well. So, this is a diffusion that is taking place in the radial coordinate, but then we are incorporating the effect of the solid matrix there in a porosity material effects. So, DE takes into consideration all those effects and then I can write as if it is a continuous medium inside a particle. We already discussed this at length in the last lecture. Now, what are the equations because if like why what we are trying to do here is to get the concentration profile inside a particle the concentration profile inside a particle. Now, we define dimensionless concentration psi is equal to CA divided by CAS where this is the external surface concentration it can be CAB as well. So, I can use these terms sometimes like either CAB that is bulk concentration or CAS that is external surface concentration because right now we are not considering external mass transfer effects. So, 5 so this is psi and then there is this lambda which is nothing but R by R is a radial distance R R is the sphere radius alright. Now, if we do this I get an equation for the concentration dimensionless concentration as d 2 psi by d lambda square plus 2 by lambda d psi by d lambda minus psi sorry not psi phi m square remember this psi raise to n is equal to 0. Now, what is this? This is a very important term this is very important term that tells you how important the pore diffusion effects are. Now, we call this as a Thiele modulus. Now, Thiele modulus is given by phi n square k n S A rho C R square CAS raise to n minus 1 divided by d e. So, we know the meaning of each and every term here k n is a rate constant per unit area right per unit area n denotes the order of the reaction and I am defining it for the order here n right S A surface area per unit weight of the catalyst rho C is the density of the catalyst mass per unit volume. So, this entire quantity k n S A rho C is rate constant per unit volume this is per unit area this is per unit these two together per unit weight and these three together per unit volume of the catalyst. And then we have these terms these are radius then your concentration order and effective diffusivity and this we can further simplify this or other further play with this particular expression and show that phi is the ratio of the intrinsic rate to the diffusion rate. So, it represents the relative magnitudes of intrinsic reaction rate and diffusion rate. And if value of phi is very large value of phi is very large in that case the diffusion resistance is significant. So, Thiele modulus if it is significant or substantial value or the very high value diffusion effects we cannot neglect them on the other side like if you have phi value very small say 0.001 in that case diffusivity effective diffusivity is likely to be very large compared to the intrinsic reaction rate or the rate of the reaction rate constant is very small. So, that it is diffusion is not important at all it is like open surface or the part or the molecules are getting access to every side without any problem concentration gradient inside a particle is negligible it is almost a flat profile. So, that is the meaning of phi all right. So, now we so, let us try and solve this equation now we go further we want to see the concentration profile and why we want concentration profile is to further calculate the overall reaction rate because you have not yet got the expression for the reaction rate to be used in the reactor design. So, we are heading towards it fine all right. So, this is the equation and this is the meaning of psi n boundary conditions we already looked at boundary conditions in terms of concentrations dimensional concentration now let us try and write them in terms of non dimensional concentration. So, psi is finite where at lambda equal to 0 very important right at the center psi is finite or d psi by d lambda is equal to 0 d psi by d lambda is equal to 0. Then next psi is equal to 1 at lambda is equal to 1 right why because at external surface at external surface lambda is equal to 1 r is equal to capital R and psi is equal to 1 because c a is equal to c a s right. So, lambda is equal to 1 lambda is c a by c s all right. So, we have these this differential equation with these boundary conditions now we need to solve this equation not so easy, but there are ways to work with it. So, if you assume the reaction to be say first order reaction solve it analytically first we have a variable called say y and which is defined as psi into lambda. Now if you do that and substitute for it in this equation then you can do some mathematical jugglery and get a equation in terms of y and lambda like this. So, d 2 y by d lambda square minus phi 1 psi by d lambda square minus phi 1 psi by d lambda square y is equal to 0. Now why 1 here because order is 1. So, this n here represents the order do not forget that. So, I am writing it for a first order reaction I am writing it for first order reaction I am making assumption that y is equal to psi into lambda the product of these two variables. So, what happens is I can get rid of psi here you can write d y by d lambda just differentiate with respect to lambda and then substitute here you can do that mathematical manipulation and what you get is this and this is a very simple equation to solve and there is a solution for this equation the general solution for this equation which can be given as y is equal to a 1 cos phi 1 lambda this is this is mathematics b 1 psi n h phi 1 lambda. So, this is a general solution for this equation with the boundary conditions of course, I have not applied those boundary conditions yet because now I have the equation in terms of two constants a 1 and b 1 which has to be found out with the help of boundary conditions and you know the meaning of phi 1 and lambda. So, what is y is psi into lambda. So, let substitute for y now and I get my equation that I want because y is a dummy variable there. So, substitute for y. So, what you get is psi is equal to a 1 divided by lambda cos phi 1 into lambda plus b 1 divided by lambda sin h phi 1 lambda all right. Now, we have to make use of the boundary conditions to get a value of a 1 and b 1. So, let us talk about the first boundary condition where at lambda is equal to 0 at lambda is equal to 0 lambda is equal to 0 you have lambda is psi finite. Now, look at this equation if this is lambda if lambda is equal to 0 then of course, this factor becomes 0 sin h you do not have to worry much about it, but then here this becomes 0. And this term is finite why? So, this term because cos 0 cos 0 tends to 1 right when lambda becomes 0. So, this is 1 this is 0 this is finite this term of course, taken care of because this is 0 this is 0 probably you do not have to worry much about it, but if this is 0 this is sorry yeah if this entire term is tending to 1 this is 0 and if you I am saying that this is finite then what should be the value of a 1? It is quite obvious if a 1 is finite a 1 is finite then this becomes infinite because this is 0 right, but this is finite. So, this is not true. So, a cannot be finite means a 1 is 0. So, that implies a 1 is 0 I will repeat this tends to this cos becomes 1 this becomes 0 this is finite this is anyway 0 by 0 tending to 0 by 0 that means this is this term is finite, but then if a 1 is not finite then this would become infinite that is not correct because psi is finite. So, that means that a 1 has to be 0 it cannot be any finite number I hope this is clear. So, once a 1 becomes 0 I can get a value of b 1 from the another boundary condition. So, let me first write psi is equal to b 1 by lambda psi in h phi 1 into lambda. So, let us apply the second boundary condition what is that second boundary condition? Second boundary condition is a 1 where you have at lambda is equal to 1 psi is equal to 1. So, this equation gives me the value of b 1 what is the value of b 1? So, 1 is equal to b 1 divided by lambda psi in h phi 1 right. Now, b 1 is nothing, but which implies b 1 is equal to lambda divided by psi in h phi 1. So, if you substitute for all the value of b 1 in this in this then I get a final expression in the form psi is equal to which is nothing, but c a divided by c a s we already defined it which is equal to 1 by lambda psi in h phi 1 psi divided by psi in h phi 1 psi in h phi 1. So, this is the concentration profile this is the concentration profile that we wanted of course, for the first order reaction for the first order reaction that is phi 1 first order reaction because it was easy for me to solve for this equation solve equation for first order reaction. How am I going to use this? See this is lambda sorry the way I have shown lambda here it looks like tau, but actually it is lambda all right. So, this is the equation for concentration profile I can substitute for lambda r small r divided by capital R and get the equation in dimensional form c a relation between c a and small r that gives me the concentration profile inside a particle. Now, before we go ahead and define certain parameters later we will look at a profile how it looks like. Now, we can always qualitatively draw a profile a concentration profile from this equation. In fact, we do not need equation initially to just imagine or the guess the profile inside a particle. How will it look like? You have r right r increasing in direction that means r is equal to 0 here r is equal to capital R here and then you have this concentration or you can write psi c a by c a s is better we do that because then I can I know what is the maximum value that is 1 all right. Now, this distance as I go from center to the external surface how will it look like where exactly it is going to be maximum it is going to be maximum at the external surface at the external surface it is going to be 1 here because it is c a is equal to c a s here right and inside. So, this is where I have bulk and particle started from here and it goes towards the center right. This concentration is going to go down. So, typical concentration profile is going to be like this and at r is equal to 0 at r is equal to 0 at center d c a by d r or d psi by d r is 0. So, it is going to be flat it is going to be flat look at this at this the condition is satisfied it is finite, but d c a by d r is going to be 0 symmetry all right. So, this is a typical profile that I am going to see this is a typical profile that I am going to see inside a particle. Now, this profile will depend on which parameter this profile will depend on just one parameter look at a differential equation there is one dimension less number that defines everything inside a catalyst which is nothing, but Thiele modulus phi ok phi n for nth order reaction phi 1 for first order reaction right and depending on the magnitude of phi this profile will change qualitatively it will look like this, but you can imagine now tell me if value of phi is very very small what does it mean value of phi is very small means that the concentration variation inside a particle is negligible ok. The profile is flat ok there are no gradients why because there is no pore diffusion effect the pore diffusion is. So, fast that a particle before the sorry not particle molecules before they react they immediately get into the inside a pores inside a particle and some of the concentration is leveled off ok it is exactly or which is quite close to what you have at external surface ok. So, for phi is very for very small values of phi this profile is flat ok. So, this is very small value of phi. So, let us say phi 1 or phi let me call general value of phi n ok which is very very small this is some intermediate value of phi and if the value of phi is very large then the gradients are significant ok. Then you can imagine something like this this is for phi n very very large very large value of phi n or phi 1 whatever depending on order of reaction importance of pore diffusion how it changes ok and how it is characterized it is only one parameter which tells you about importance of pore diffusion that is phi ok. I hope this is clear pore diffusion effects you should remember this no pore diffusion effects this is the one somewhere in between both are important reaction and pore diffusion I cannot neglect both of them. But suppose I am dealing with case like this then pore diffusion can be ignored whatever exercise we have done so far Thiele modulus of course, you have to calculate it first to know whether it is important or not not. But then later on you do not have to worry about profile inside reaction is taking place at external surface concentration ok and you decide your reactor like what you do for a normal case. But then form for a catalysis person he should think why even if so it is like this he has prepared a good catalyst say platinum on alumina ok. And he has nice porous catalyst with very large surface area and it is nicely dispersed that means it is concentration is uniform everywhere inside a catalyst particle the concentration of p t is uniform what is p t platinum and I am using alumina as a support and platinum species are nicely distributed inside. Now imagine if value of phi is very very large a situation like this is it a good catalyst no why because the particle the platinum particles somewhere in the core somewhere in the core that is near the center are not seeing reactant molecules at all. Because you see the profile here is flat and almost zero concentration all the reactant molecules are somewhere near the external surface. So, what is the use of putting platinum there is useless unnecessarily I am taking efforts on preparing the catalyst dispersing it very well. But some of the platinum species or the sides are inaccessible why it happens because effective diffusivity is very small why the effective diffusivity is very small look at a expression of the effective diffusivity bulk diffusivity into porosity sorry Knudsen diffusivity into porosity into constriction factor divided by tortuosity. So, if the tortuosity is very very large effective diffusivity will go down if the porosity is very small pore diameter that matters the pore diameter is very small effects will be significant. So, the diffusivity so I need to design a catalyst now such that I will get rid of the pore diffusion effects. So, that this platinum is accessible or otherwise if I am happy with the platinum that I have put at a surface here and reactions taking place at a rate that I want then do not do this do not take efforts on dispersing the catalyst nicely just put a catalyst only in the exterior part or the part where it is very close to external surface only in this part. So, that is the meaning of it to this analysis helps the catalysis person the catalysis or the person who designs the catalyst very well. So, he needs to have the knowledge of pore diffusion effects. So, that is the meaning of all this. So, let us go ahead now. Now, we understood how the concentration changes inside a particle we are going to make use of this and get a rate equation that we want for the reactor design see the ultimate aim is to design a reactor ultimate aim is to design a reactor. So, before you go ahead. So, let us get back to our CSTR. So, suppose I have a CSTR in that I have this catalyst particles remember what did I write F A 0 minus F A plus R A into W is equal to 0 W is the weight of the catalyst and I express sometimes I put a dash here if you see notation in Fogler's book use R A dash here it is pouring it weight of the catalyst. Now, this equation I told you last time before we started our discussion on diffusion effects this term is nothing but the one that is obtained from either Langmuir initial mechanism or Euler ideal mechanism whatever taking into consideration adsorption, desorption and reaction. So, this particular term now what we are looking at is how this term is going to get modified if you have pore diffusion effects also present. So, that is the exercise we are doing say this is a design exercise I am just talking about CSTR can be applied to a plug flow reactor as well. But then what is the objective of this entire exercise that we have been doing is to look at this particular parameter the rate of reaction this variable how does it change now? How do we incorporate effect of pore diffusion effects? So, let us go ahead I have this equation concentration profile size is equal to C A divided by C A S is equal to 1 by lambda sin phi 1 lambda divided by sin H phi 1 for the first order reaction. Now, I am going to define a factor which is very very important and it is going to help us later a lot for the reactor design. So, what is that factor which is effectiveness factor? So, let me call this is effectiveness factor I am going to define this first and then get an expression for it this is effectiveness factor as a name says it tells me how effective is the catalyst how effective is the catalyst. Now, I have this particular factor is going to incorporate effect of pore diffusion if the effectiveness is very large that means, pore diffusion effects are insignificant. If the effectiveness is less then the pore diffusion effects are more. So, it works exactly opposite to Thiele modulus. Thiele modulus means if the effectiveness sorry if Thiele modulus is large then the pore diffusion effects are significant the resistance is significant. If Thiele modulus is small the resistance is small whereas, effectiveness factor is exactly opposite. So, we are going to see an inverse relationship between these two eta and phi that relationship is going to be inverse, but not so straight forward we will see how it works. There is an advantage you may ask me why you are defining two parameters or another parameter. So, it will be clear as and when we go and solve equations later because effectiveness factor tells you many things which are like quickly you can estimate the importance. Now, phi value can vary from 0 to infinity whereas, effectiveness factor will varies from 0 to 1 and sometimes it can be greater than 1 we will talk about it. Now, this effectiveness factor is nothing, but now listen to me carefully this is the observed rate this is the observed rate of the reaction or actual rate observed or actual rate. That means, if I do experiments whatever rate I get divided by now this is very important the rate calculated at external surface rate calculated at the external surface. Conditions how do you calculate the rate at external surface what do you need to calculate the rate at external surface you need external. So, you need concentration. So, this concentration is not a concentration inside, but the external surface concentration in our case it is C A S or C A B whatever you need temperature also we are not talking about non isothermal reactions so far, but then later on we will talk about that as well. So, I am just calculating rate at the external conditions it can be concentration can be temperature because for rate calculation I need temperature also because rate constant is function of temperature. So, I am calculating at external surface what does it mean let me write it down in words because this concept is quite important let me write down in words rate calculated at the external surface. It means that let us assume or let us consider a hypothetical situation where you have the surface which is present inside a catalyst the surface which is present inside a catalyst is opened up to the external environment to the external environment. So, this is your C A S what was happening before that the diffusion taking place some of the sides inside we are seeing less concentration than external surface concentration. Now, I am imagining a situation where entire surface is opened up. So, let it be say 500 meter square per gram let it be a football ground whatever a large external surface, but this is open to the external conditions. So, that C A S so, all the sides on the surface are going to see this. So, the rate calculated under those conditions for this concentration is going to be much larger compared to what it would be when calculated for inside particle when the concentration goes down in the presence of pore diffusion effects. So, the rate here is smaller because the concentration is less whereas, the rate here is large because all the sides are seeing higher concentration or larger concentration which is there at the external surface. So, imagine two scenarios. So, what I am doing is I am comparing this with this. So, rate calculated at the external surface is nothing, but rate when the area or all the catalytic species is exposed to the external condition. Conditions means C A S and T S I hope it is clear I am comparing these two scenarios comparing says in terms so, through eta. So, eta now eta is a rate for this divided by rate for this and this rate is going to be smaller. Most of the cases there are exceptional situations where this rate is higher will come back come to that, but otherwise in normal situation this rate is smaller than this rate. So, I am going to see the value of eta to be less than 1. So, value of eta is less than 1. Now, eta is R A since it is reactant let me call it is minus R A divided by minus R A at surface. So, it depends like it can per unit volume it is per unit weight it can be R A dash divided by minus R A dash is it is all units that matter. So, dash is taking a per unit area then there is another thing called as per unit per unit surface per unit surface per unit volume per unit weight per unit surface sorry it would be other way round it would be per unit volume per unit weight and per unit area. So, whatever so, what we are doing here is we are looking at a rate which is observed rate actual rate divided by rate calculated at the external surface. So, we can say that say let us take this particular ratio that is minus R A dash which is per unit weight of the catalyst into mass of the catalyst divided by minus R A dash S into mass of the catalyst. What is the unit of this this is per unit weight of the catalyst into weight. So, it is the unit is going to be moles per second divided by moles per second. So, I am going to evaluate this or get the expression for this in terms of moles per second and substitute here and get a value of eta. So, let us let us try and get a numerator first. So, let me call this is m A divided by m A S divided by m A S that is moles per second reaction rate actual reaction rate divided by moles per second reaction rate calculated at the external surface. So, let us get the expression for m A let us get the expression for m A that is or rather m A S first that is easy to understand that is calculated at the external surface. So, it is rate per unit area into surface area per mass of the catalyst into mass of the catalyst right. So, rate per unit area is unit surface area is say k rate constant into C A S into surface area I am calculating at external surface. So, C A S into surface area S A right in terms of density mass of the catalyst is rho C into fourth third pi r q alright. So, this is m A S I am going to use this later. Now, let us look at m A let us look at m A I will bring back this particular slide again fine what is m A m A is the observed rate. Now, how do I calculate the observed rate? Now, I know the concentration profile inside the catalyst let us assume first order reaction now I know that psi is equal to whatever that psi in h 5 1 lambda divided by psi in h 5 1 right. So, that concentration profile I know now you have this catalyst particle inside which there is a concentration gradient right and then I know what is the gradient at external surface. So, if I am sitting at external surface I know how many molecules are going inside how many molecules are going inside at steady state there is no accumulation inside. So, if I am at the external surface and just watching number of molecules going inside a particle calculate the rate there calculate the flux there then that is nothing but the rate of the reaction. Because whatever happening inside the gradient that is generated will is based on the reaction that is taking place inside is because the reaction takes place inside that you have a gradient that you have the gradient inside is because of the reaction. So, I am taking into effect or taking into account effect of reaction. So, if I am sitting here and just observing number of molecules going inside then it is the rate of the reaction. So, how do I quantify it? How do I quantify it? It is nothing but a flux at the external surface into area into the area what is the area? Area is this external surface area and flux at the external surface. So, it is nothing but 4 pi r square d e into d c a s 4 pi r s by d r or in a dimensionless form or other if I just want to get a gradient in the in terms of psi that we have defined earlier it is going to be d psi by d lambda at lambda is equal to 1 not sorry here it should be c a and at the external surface that is r is equal to capital R. So, I have this particular expression for the actual rate. So, this is the actual rate number of molecules going inside per unit time. What is the unit of d e into d c a by d r? This is moles per second per meter square per unit area and I am multiplying it by area. So, the unit is moles per time let me write it. So, this has a unit moles per meter cube that is concentration into meter that is radius this is unit of d c a by d r what is unit of d e unit of d e is meter square per second. So, what is the unit of this product moles per this meter square into second and if I multiplied by area in which you have r square. So, meter square so, moles per second the unit of this entire thing is moles per second unit of m a that is what I want. So, look at expression for eta that we have got. So, this is eta eta is m a divided by m a s and this is nothing but moles per second divided by moles per second. So, I have got expression for m a and m a s. So, I am just going to substitute for m a and m a s in eta and see what I get. So, eta is equal to m a that is 4 pi r square c a s d divided by m a s k 1 c a s s a rho c 4 third pi r cube. So, this is eta. Now, I can get expression of for eta if I know this and this is something that I have already calculated or rather I know d psi by d lambda I substitute for lambda is equal to 1 and I get the expression for this. So, how do I get that psi is nothing but 1 by lambda sin h phi 1 lambda divided by sin h phi 1. This can give me d psi by d lambda and if I substitute for lambda is equal to 1 what I get is d psi by d lambda at lambda is equal to 1 is equal to phi 1 cot h phi 1 minus 1 phi 1 is Thiele modulus for the first order reaction. You can do it just a mathematical. So, you have you can get a value of d psi by d lambda at lambda equal to 1 substitute for this in the expression for the effectiveness factor. Substitute for this in the expression for the effectiveness factor here. What do you get? It is going to be a big explanation, but easy to understand just doing it systematically. So, you know the meaning of each and every term there. So, eta is equal to 4 pi r d e c a s phi 1 lambda sin h phi 1 lambda sin h phi 1 cot h phi 1 minus 1 divided by k 1 s a rho c 4 third pi r cube into that is it. So, this is the expression for eta. Now, let me just play with this 3 into 1 divided by k 1 s a rho c r square divided by d e into phi 1 cot h phi 1 minus 1. And this what is this? This is nothing but the Thiele modulus. So, I have eta equal to 3 divided by phi 1 into square into phi 1 cot h phi 1 minus 1. So, I have got expression for eta in terms of phi. This is what I wanted. I told you there is a relationship between phi and eta and this is slightly complicated relationship. We will continue with this in the next lecture and I will tell you more about it. But remember as phi increases, eta decreases. Effectiveness factor is more means catalyst is effective, porosity or pore diffusion effects are less. So, we will continue with this in the next lecture. Thanks.