 Good morning. So, we will quickly revise what we have learned in last lecture. We are looking at models for non-ideal reactors. You know non-ideality comes because of the non-ideal flow patterns and we have to somehow characterize this non-ideality. So, we are looking at one parameter models. Before this, in the earlier chapter, we looked at zero parameter models with two extreme situations, maximum mixingness and complete segregation. Now, in this particular chapter, we are looking at models with some parameters involved in it and these parameters have something to do with the flow patterns inside a reactor, the extent of mixing and so on. So, we will just moving away from zero parameter models. The problem with zero parameter models I told you that gives you bounds on conversion, does not tell you the exact conversion. Whereas, now we are looking at a model with a parameter. So, it is going to give us some idea about a conversion, exact conversion. The model that we are using is right. So, we are going for a simple model. Now, it is a one parameter model. You can have a model with many parameters, but right now you are just looking at one parameter model. The two types of models that we have looked at. The first is tank in series and the second one is dispersion. Of course, we have just looked at this one in detail and today we are going to talk about dispersion model. But just to quickly revise what is tank in series model, we have a tubular reactor, say tubular reactor and there is a flow that is taking place in this direction. Now, a plug flow reactor is a reactor where the mixing in radial direction is maximum, but there is no mixing in axial direction, right? No mixing in axial direction, but then velocity profile is flat and if I have a pulse injected here or a tracer injected here, then I will see the same tracer after a time tau here. That is the e curve and something that is known to you now. Now, if this pulse gets disturbed when it comes out, then it may be because of some extent of back mixing. So, if it comes out like this, there is a possibility that some back mixing is occurring here. It is not just a plug flow behavior, but it is hampered by some kind of back mixing. This may occur because of the concentration waves moving or there might be some disturbance in the flow, internals for example, say part column whatever, but I want to characterize this extent of back mixing. So, what I am going to look at is this as a series of small small CSTRs and this n number of CSTRs and this n characterizes extent of axial mixing. So, this n is a parameter, a given tubular reactor if I say that I have n number of CSTRs in series which are equivalent to this tubular reactor, then n becomes a parameter. This n can be 1, 2, 3, 4, 5, 6 and infinity. So, infinity means it is a plug flow reactor and n is 1 means it is a CSTR because complete mixing and in between these two you have partial back mixing and this n will n is large means this less mixing and this small means this good amount of back mixing that is occurring. Then we derived equation for n in terms of, now how do we get a value of n? n is obviously obtained by the tracer experiments. So, tracer experiments is a tool for any given reactor under the conditions whatever flow rates and all we are operating it at that tool gives you the parameter that is nothing but n. So, if I get E curve the variance in the E curve get tracer experiment the E curve and the variance in the E curve you know what is variance. Sigma is equal to 0 to infinity T sigma square sorry T minus tau square ET dt. So, variance now this variance is related to n what is the relationship n is equal to we already looked at it tau divided by sigma square. Now what is tau? Tau is a residence time total residence time volume of the tube divided by the volumetric flow rate volume of the tube do not talk about do not confuse it with the volume of small small CSTRs. So, given a tube somebody ask me whether this will behave like a plug flow reactor or not tell me this is flow rate that I am operating it at what will I do I will do a tracer experiment. I will see whether I get a nice pulse just getting repeated or all very close to something that I had injected after time tau if it is not happening it is not a plug flow reactor. If it is getting dispersed as I showed before it is a slightly dispersed or the pulse get distributed rather over time in that case there is definitely some back mixing occurring and there is a way to characterize this back mixing. I will do a tracer experiment I will get E curve from E curve I will get variance variance because in this just E curve is to be known otherwise everything else is known. So, I get E variance once the variance variance is known tau is nothing but volume divided by volumetric flow rate I get a value of n is number of CSTRs. Now once you have a value of n you have x that is conversion for series of CSTRs given by tau i k square sorry not square is raised to n. Now tau i is nothing but tau divided by n right and k is rate constant this is for the first order reaction do not forget this. And for second order again I can derive some equation if not I will just go on doing calculations step after step. So, it has allowed me to get a conversion for the given value of n sorry sorry should be. So, this is a conversion for a tubular reactor that corresponds to n CSTRs in series. Now one thing I forgot to tell you last time is this n. Now after doing this exercise and getting the value of n n need not be integer n can be 3.5 it can be 4.75. So, this value of n now as such like really number of CSTR number means what should be integer. But just ignore or others forget it whatever value of n comes just put it here. And once you do that whatever value of x comes that is a real value of x which is close to what you would get in that real reactor which is a tubular reactor neither PFR nor CSTR. It must be noted that this is possible that is using the non integer value of n only for the first order reaction. For other orders analytical expression is not available for the conversion after n reactors in such cases n will have to be rounded off to the next highest integer fine. We will solve some example or I will tell you the methodology to solve example as we go ahead. But before that let us look at dispersion model. So, this is where I stop as far as the tanking series model is concerned. You know n is a parameter n is to be obtained once you get n you get a value of conversion for a given reaction for a tubular reactor. Now, let us go ahead and discuss the other model that is dispersion model. Now, once we have one model why to have another model this is our convenience sometimes some models work well sometimes they do not work. So, I probably after discussing dispersion model we will have we will spend some time relating these two models as well. So, there are two models available just we should know both of them. So, that because they are both of them are popular in industry. Now, dispersion model in this case dispersion is what dispersion is again a synonym for axial mixing that is occurring of course, it can be in radial direction also. So, just dispersion it can be in any direction. But as far as tubular reactor is concerned we are talking about axial mixing. But it can be radial also can club these two together and get overall dispersion all right fine. Now, again you have a tubular reactor if you if you inject a tracer because of back mixing is going to get something like this if it spreads it becomes flat in that case extent of back mixing is is is very large right this is a behavior that I am going to see. Now, can we give a mathematical treatment to this the first model that we looked at was assumption that you have series of CSTR tanks in series. But here I am not going to do that I am just going to look at the way I do it for a plug flow reactor take a differential balance take a differential balance. Now, for unsteady state suppose I injected tracer how it behaves with respect to time I have to write unsteady state balance a PFR is a DFF by dv is equal to R is a steady state balance right. Now, I have a tracer injected R term that is reaction term is 0 there is no reaction taking place a non-reactive conditions ok. So, you have flow taking place but at the same time tracer concentration at any given point is going to change with respect to time ok. So, accordingly I will have an equation ok. So, what is the equation unsteady state equation for a PFR. So, it is going to be DFT F T is a flow rate of the tracer normally we say F A you know for A that is a reactor now instead of A now I have T here. So, DFT by DZ is equal to AC into DCT by DT. So, this is a unsteady state balance for a tracer in the tubular reactor right tubular reactor why I am writing this balance I am giving a mathematical treatment to this as time proceeds how the things are going to change with respect to Z and time as well. So, this equation describes everything now what is F T what is F T in a normal situation F T now this is very important for a plug flow reactor what is F T we say that is only convection that is V into C that is volumetric flow rate into concentration that is for a normal reactor. But in this case it is the flow is not taking place only because of convection, but there is an additional term which is caused by axial mixing which is caused by dispersion. So, this F T this let us try and write the expression for this F T this F T is of course, U into AC into CT what is this this is convection right, but it is not just this apart from this there is an additional term that is coming out of dispersion. So, let me call this as F T D and what is F T D this F T D is because of dispersion now normally it is written as some coefficient we will talk about this later D A into AC look at look at how it look at expression it is quite similar to Fick's law this is quite similar to Fick's law D A AC into del CT by del Z right. So, without AC it is flux. So, I am multiplying it by AC what is AC AC is a cross sectional area right. So, this is the expression for the dispersion or flux due to dispersion and why it happens it happens because of concentration gradient is a front that is moving of a tracer is a front that is moving. So, it is a concentration difference that causes the dispersion this is analogous to diffusion and this expression is analogous to Fick's law. I am not calling this as a Fick's law because the coefficient that I am using here that is D A is not a regular diffusivity or is not a normal diffusivity is not bulk diffusivity that we define in the case of Fick's law this is something that combines all the effects. Now, you may ask me whether it has the capability to combine all the effects it may not have in that case your one parameter model does not work that is it. But if for some reasons of somehow if for a given geometry and flow pattern if it is able to take care of all these dispersion effects in one parameter then you are done. That means your model is perfect go ahead and use it and try and predict the conversion of the reactor for a given volume. So, that is the meaning of it. So, I am I am expressing the dispersion by this particular term. Now, you you may ask me what kind of flow pattern laminar or turbulent. Now, it it can be proved that this expression works well even for laminar flows this works well for turbulent as well. So, this can be a diffusivity in the case of turbulent flow or it can be in the in the case of laminar it can be molecular jumping from one layer to another layer which are otherwise not mixing. So, one layer moves at one velocity the other layer moves at another velocity in laminar flow and there is exchange of some molecules this is non-ideality. So, it is not exactly laminar, but then that case this particular exchange can be incorporated here in this term diffusivity. If time permits we will try and derive the expression separately for laminar flow and show that it can be expressed in this way. But right now let us go ahead let us not worry much about whether it is laminar or turbulent flow we say that this is applicable to any of flow that is laminar or turbulent. So, let us put this term here f t is equal to u a c c t minus d a a c d c t by d z. Now, we have we already have this equation we have this equation and in this equation f t is given by this. So, let us substitute for f t here substitute for f t here remember f t otherwise was this only for a plug flow reactor now that we are considering dispersion we have an initial term. So, we have to put this here. So, if you substitute for f t what you get is this d a del c t by del z square. Now, you get a second order partial differential equation del u c t divided by del z is equal to del c t divided by del t. Second order is obvious because f t comes in we have d f t by d z and f t is now d c t by d z you have d c t by d z term in f t. So, that is why you have this expression. So, this is the expression that I am going to deal with and with this equation. Now, I have the tracer concentration changing with respect to time and z if I solve this equation I will get the response in sense rather this tracer how it spreads and all that at any given time and length in the reactor fine. So, let us remember this equation and in order to solve this particular equation I need boundary conditions. Now, these boundary conditions they are quite peculiar and we are going to spend some time in knowing what these boundary conditions are. Now, you have a tubular reactor you have the inlet and you have the outlet. Now, we are talking about b c that is boundary conditions differential equation needs boundary conditions otherwise you cannot solve this equation. Now, how do I look at it? You will say it is not that difficult what is the big deal is inlet concentration is outlet concentration and all that. So, if you see this is your second order partial differential equation how many boundary conditions you need here you need one initial condition and two boundary conditions right. So, there are some assumptions to be made here and some of the assumptions they work so well. So, you have a flow taking place you have a flow taking place dispersion is happening. Now, where is the dispersion happening? Dispersion is happening here that means d a value of d a is greater than 0 in the reactor. So, this is a this is a reactor this is a reactor. So, value of d a in this domain is greater than 0. Now, this is before the inlet and this is after the outlet. So, before the inlet I have a choice whether dispersion is happening here as well or not. So, let me let me let me consider a situation where d a is equal to 0 here and d a is equal to 0 here. The node dispersion is happening in the outgoing channel and incoming channel it all depends what kind of system you have what kind of nozzle you have here and here. But if I assume this then this is called as closed vessel and we have a corresponding boundary condition we will come to that. But this is a situation where I have a closed closed vessel that means there is no dispersion here and there is no dispersion here. There is another extreme this is another extreme where I have open open case or open open vessel where d a is greater than 0 here as well d a is greater than 0 here as well d a is greater than 0 here as well why not is sometimes like we showed the arrow like this and not straight why this is one indication where the flow is not ideal it is not like a plug flow rate. So, it all it all depends how we want to show it, but then the indication is like arrow is distorted means there is some back mixing happening all right fine. So, this is called as closed closed vessel. Now, you may have situation where you have open closed or closed open depending on what is happening. So, you have a choice and depending on that decide what kind of boundary condition is applicable in my particular case because every time you have a different expression. So, let us look at sorry did I say oh sorry this is not closed closed let me let me correct myself sorry this is not closed closed this is open open earlier was closed closed this is open open. So, you may have several possibilities and you decide what you want it can be closed closed it can be open open it can be closed open or it can be open closed all possibilities. So, depending on that you will get expression for the boundary condition. So, let us write expression for a closed closed vessel. So, let me draw it again at inlet you have flow taking place. So, whatever coming in is going out at this particular boundary right. So, I have expression f t tracer 0 minus here it is 0 minus and sorry minus and is 0 plus f t 0 minus at any given time is equal to f t 0 plus at any given time at that particular time of course, these two times are same all right. So, that is obvious no that is obvious there is no accumulation at the inlet. So, whatever comes in goes out. So, molar flow rate f t of tracer coming in is equal to coming in at a inlet is equal to coming inside a reactor fine. So, let us write expression for f t now f t in there is no dispersion here closed closed. So, there is no dispersion here. So, it is velocity into cross sectional area into concentration 0 minus t right concentration at inlet that is 0 minus at particular time I am multiplying it by u into a c what should I write here now what should I write here for this for this. Can I just write this u that is velocity a c c t 0 plus t is that sufficient the answer is no this is not sufficient because inside it is not just the convection or the velocity driven flow, but it is dispersion as well. So, we will have an additional term which is given by a c into d a right d c t by d z at z is equal to 0 plus right. So, f t coming in from here is this and f t here is this plus this right. So, these 2 concentrations are different these 2 concentrations are different concentration here is different concentration here why because this is an additional term which will make sure that is concentration changes. If this term was not there if this term was not there then u a c will get cancel in these 2 concentrations would be equal, but this term is responsible for making these 2 concentrations different alright ok. So, let us let us simplify this further c t 0 is equal to minus d a divided by u d c t by d z at z is equal to 0 plus plus c t 0 plus t. So, this is one boundary condition that is at inlet. So, inlet boundary condition I will say inlet pc at the exit or outlet what happens at the outlet. So, let me get back to this figure or let me draw it here again. So, d a is greater than 0 d a is equal to 0 here there is a continuous flow there is a continuous flow. So, at z is equal to L at z is equal to L c t at L minus is equal to c t at L plus the concentration here is same as concentration here concentration these 2 points are going to be same. So, what it means is d c t by d z is equal to 0. So, this is a boundary condition at the outlet. So, these are the 2 boundary conditions that I have alright. So, these 2 boundary conditions and where is our equation is the equation here. So, this is the equation this equation and these 2 boundary conditions what else do I need I need initial condition what is that initial condition at time is equal to 0 what is the value of concentration at all lengths. So, at time is equal to 0 initial condition I c initial condition at time is equal to 0 and z is greater than 0 that means at any length c t 0 plus 0 that means time 0 is equal to 0. Of course, it can be at any any length 0 plus means on word 0 initially the concentration 0 because I am injecting a tracer at time 0. So, at time 0 inside reactor concentration is 0. So, this is initial condition. So, initial condition then you have boundary conditions. So, these are the boundary conditions and this is your equation this is your equation main differential equation right. I can solve these equations together to get the tracer concentration how it changes inside a reactor. Why are we doing all this because we want to see how the tracer get tracer spreads. Why do you want to know how the tracer spreads because that is going to tell me the flow pattern inside a reactor. So, I will be able to characterize the flow pattern like what I did in the case of tanks in series. I characterize the flow pattern in terms of a number called number of tanks and similarly I am going to get a parameter here. We will see what parameter right that will characterize the flow pattern. So, let us solve this equation we will see right now see the main difference in what we did before and before means in the tanks in series model. There we just discretized it the entire domain the volume in different tanks. Now, I am looking at differential equations. So, there is a discrete dynamics there here it is a continuous dynamics. I am writing a differential equation here there I was writing difference equations. See the difference you have come across this particular way of formulating a problem in many cases the distillation column absorption column see number of stages packed column tray column similar. So, now let us go ahead before that we need to non-dimensionalize this equation because when we non-dimensionalize we will come up with the parameter that is going to decide everything. Now, what is this non-dimensional form of the equation? So, non-dimensional form of the equation is going to be like this. So, let us start with this sorry. So, let me this is a differential equation I want to non-dimensionalize this. So, let me define non-dimensional concentration psi C t by C t 0 lambda z by L theta. In fact, theta we will come back to this it will come on zone you know typically what happens when we do non-dimensionalization. If we do this. So, let us see you have d A d 2 psi by d lambda square. Now, here what we will see is you have L square minus u you can take it out assuming that velocity is not changing with respect to z and t. So, this is the of course is not a non-dimensional I am not d dimensionalize the time yet. Now, if I define theta equal to t u by L t u by L or in other words suppose I multiply this equation by L by u by L by u. So, let us multiply both the sides by L by u then what do I get d A divided by u L I am multiplying it by L by u L by u. So, u L this will go away L by u means what when it comes in the denominator you have this happening. So, L by you multiplied by L by u comes in the denominator 1 upon u by L and I take this here t into u by L. So, I get d theta del theta. So, this is the dimensionless equation theta have got right. If it is not clear you can do it on your own and check. So, dimensionless concentration dimensionless length dimensionless time what is unit of this second meters per second and meter. So, no unit u by L has a unit per second. So, that is why I am using u by L as a parameter here or other I have clubbed this 2 to get dimensionless time and d A by u L what is the unit of d A meter square per second this is meters per second and this is meter. So, this is again dimensionless number. Rest all on dimensionless quantity. So, this is the dimensionless number. So, let us spend some time on this particular number d A by u L d A by u L this is a dispersion coefficient velocity and length. This is a very famous number which this is this number that characterizes the mixing right is named after an engineer called Peclet. But of course, the Peclet number is inverse of this. That means, u L by d A. So, Peclet number is u L by d A or in other words 1 upon p is equal to d A by u L Peclet number that t is silent. So, it is a Peclet number the very famous number. So, let me put that number in the equation again you remember that equation. So, here right. So, d A by u L is 1 divided by p 1 by p right. So, this is my equation. So, this Peclet number before we go ahead and understand the meaning of it. If Peclet number is very large if Peclet number is very large that means the dispersion is small velocity is very large what does it mean? It means I am going towards what kind of flow plug flow because dispersion is less velocity is very large no dispersion or a very small dispersion Peclet number is very large if it is infinity then it is plug flow reactor. In other words if dispersion coefficient is very large if dispersion coefficient is very large then Peclet number will become very small and what is the meaning of large dispersion that means there is so much back mixing happening and what is that situation? It is nothing but a back mix reactor a CSTR when Peclet number is 0. So, 2 bounds see it is quite similar to number of CSTR in series number of tanks in series. So, this is a differential equation our continuous dynamics there it was discrete dynamics theoretically both are falling in line or we have got similar. So, Peclet number can be defined in words as Peclet number is equal to rate of transport by convection see those two terms U L in the or U in the numerator U U is convection is a velocity divided by rate of transport by dispersion because in the denominator you have dA here you have U and of course L that is characteristic length comes because of like you have to somehow dA means not dA is meter square per second. So, the dA by delta has to come. So, you have that L coming in the expression for packing number, but otherwise remember this U will come in the numerator and dA with the denominator and Peclet number will characterize the back mixing larger the Peclet number smaller is the back mixing and smaller the Peclet number larger back mixing Peclet number infinity means plug flow Peclet number 0 means CSTR. So, these were the boundary conditions I had if I non dimensionalize them. So, the first one looks like this the first one if I non dimensionalize this it becomes 1 by Pe into del psi by del lambda plus psi is equal to 1 you can derive it if I non dimensionalize this this is the expression that I get because this is I am just dividing both the sides by this number this number C t 0 sorry C t 0 I am dividing. So, C t 0 by C t 0 1 and then I get this. So, this is the non dimensional form of the boundary condition 1 then the boundary condition 2 this is the boundary condition 2 this was this was boundary condition 1 this is the boundary condition 2. Now, I get it is quite simple d C t by del C t by del z is equal to 0 means del psi by del lambda is equal to 0 fine. So, these two boundary conditions and of course, I will have the initial condition which says that concentration is 0 everywhere at time is equal to 0. So, psi is equal to 0 psi is equal to 0 let me write it psi is equal to 0 at all lambdas when theta what is theta time is equal to 0 I am working in terms of dimensionless numbers. So, with this now I have equation to be solved one equation ordinary sorry partial differential equation second order with three boundary conditions of course, two boundary conditions and one initial condition in dimensionless form what I will get is the plot of plot of tracer concentration say psi versus theta right how will I so what is the tracer concentration at the outlet tracer concentration at the outlet exit concentration because I am not looking at what happens inside of course, it is going to be in between what you get at the outlet and what you get in the what you injected in like. But normally see what do I realize I realize the outlet concentration because I do pulse experiment or I do tracer injection experiment. So, this is the outlet concentration when I solve this equation when I solve this equation at z is equal to l or at lambda is equal to 1 what is the concentration this is what I am going to see right. So, what kind of profile will I see here. So, it all depends on the Peclet number if Peclet number is very very large if Peclet number is very very large means dispersion is 0 right or say Peclet number is infinite dispersion is 0 what is the tracer concentration that I am going to see it is exactly similar to what I have injected I am going to see a plug flow behavior. So, what is that right this is what I am going to see this is what is this this is theta equal to the dimensionless time theta equal to 1 why dimensionless time it is t into u by l u by l or l by u is tau that is residence time. So, this time is tau this time is tau actual time. So, actual time divided by tau that is theta is equal to 1 right. So, this is when Peclet number is infinite or sometimes you will have like they talk in terms of d by u l. So, d by u l will be 0 or this corresponds to d by u l equal to 0 ok. Another extreme Peclet number is 0 famous e curve for another plotting e curve plotting concentration, but it is ok the qualitative behavior is say similar to the e curve and then in between these two see now this plot has to become this means the transition is like this right sorry this will move this peak will normally increase or like the height of so, because it is going to go up right. So, this is a typical behavior that I will see as the value of Peclet number increases. So, here Peclet number is 0 here the Peclet number is probably say some value 5 Peclet number what 100 and so on. So, this is a typical behavior and this is quite similar to what we have seen before before means what for a tangent series model. In tangent series model we looked at how this profile or how this e curve or the tracer concentration changes with respect to time sorry yeah with respect to time or e curve how does it look like with respect to n rather sorry number of tangent series. So, it goes from CSTR to PFR similar thing is happening here that means n in tangent series model is equivalent to Peclet number in the case of dispersion model right both are similar both are similar right. So, once you have the e curve now is other way round once you have the e curve you can calculate a Peclet number. I have drawn this for several values of Peclet number, but if you do laboratory experiment for tracer injection get some e curve I can calculate a corresponding Peclet number from this right and Peclet number will characterize the extent of mixing like the number of tanks in series in the previous model. So, as simple as that so do a tracer experiment get e curve or concentration versus time for the tracer determine the value of Peclet number what next then if you have value of Peclet number then you have to calculate a conversion when the actual reaction is taking place. So, far we just looked at a tracer experiment without any reaction we have characterized the flow patterns in terms of extent of back mixing and that is through the value of Peclet number. Once you get a Peclet number we have we go for the reaction model we go for the reaction model. Now, how to get a value of Peclet number so one can derive of course, one can solve this equation we have to solve this equation numerically by the way it is a differential equation. So, I have not given you analytical expression for this, but you have to solve this equation numerically or analytical expression is also possible it comes in the form of a series mathematical series but then that is not important see the expression is not important how it looks like what is its qualitative behavior is important and once I have the expression or I have the I have the profile or e curve rather how to get a value of Peclet number. So, one can get an approximate value of Peclet number from this. Now, once I have the e curve from this e curve or concentration tracer concentration versus time curve I know that sigma square that is variance divided by mean residence time which is nothing but tau that is our residence time in this particular volume divided volumetric flow rate tau square infinite 0 to infinite t minus tau square Et dt should be noted that Tm is equal to tau is valid only for closed closed system this is something that I know this is definition of sigma variance. Now, from this differential equation I can derive the expression for sigma in terms of Pe like what I did before in the number of tangent series sigma square was equal to sigma square was equal to tau by n or tau sorry tau square by n square. So, here again I am going to get an expression for sigma in terms of Peclet number. So, that expression happens to be sigma square tau m this is nothing but tau is equal to 2 by Pe minus 2 by Pe square 1 minus e raise to minus Pe slightly complicated no need to remember this equation, but then you know there is a relationship between sigma Pe and Tm or other tau and we had a similar relationship before as well between sigma tau and n there was n here that was relatively simpler. Now, this is slightly complicated but of course no need to worry about it earlier relationship was sigma square is equal to tau square divided by n sorry yeah. Now, instead of this relationship I have this relationship here I had n as a parameter now I have Pe as a parameter that is the only difference. So, once I get sigma square from my e curve that is mean variance I get sigma square from mean variance I can get a value of Pe that is Peclet number and once you have Peclet number you have characterized the flow pattern then the next job is to get a conversion for the reaction of interest. So, far no reaction similar to what we did once you get a value of n then for a first order reaction I say that x is equal to 1 minus 1 by 1 plus tau i k raise to n remember. So, this is what I did after getting a value of n now my next job here is after getting a value of Pe I need to get a conversion in terms of Pe. So, that is what we are going to see in the next lecture how to get conversion from Pe alright. Thank you very much.