 Now, I know the method to calculate minimum reflux ratio. Next step would be to get the actual reflux ratio and then calculate actual number of stages. And that is going to go as input to my simulator. That is going to go as input to my simulator. So, calculation of actual number of stages. So, the next slide tells you about the calculation of actual number of stages for given reflux ratio. Now, I have a reflux ratio with me because in the last step I calculated minimum reflux ratio. The actual reflux ratio is typically 1.3 to 1.5 times the minimum reflux ratio. Now, I have the reflux ratio given and we are going to see how to calculate the total number of stages required for the given system. So, this is the algorithm given here. Given X F, X F is known that is your feed composition. Q that is the feed quality is known. Also R, what is R? Now, the reflux ratio. I have denoted it as R R somewhere in earlier slides, but R is the reflux ratio. And then X D 1 and X B 1. So, these are the important compositions. I have already started solving design problem. The top composition, bottom composition is the given. But I keep one composition floating. I will come to that. So, once you know R, you can calculate S that is the reboil ratio. There is a direct relationship between the reflux ratio and the reboil ratio if you just do overall material balance. If you have constant molar overflow assumption, you can calculate reboil ratio which is V by B in terms of reflux ratio. So, there is an equation. Now, I want to start solving problem for getting actual number of stages. I know X D 1, I know X B 1, but I need to know all the end compositions. So, these are the important compositions which are already given by the process requirement. So, next step is I will just say that my X D 3 that is what is X D 3? The composition of the least volatile component in the distillate stream X D 3. It is 0 in the beginning. Let us make this assumption in the beginning that X D 3 is 0. If X D 3 is 0, then X D 2 will be 1 minus X D 1. In reality, it will not be 0. I am going to come to this point later. But in order to start the calculations, since X D 3 is anyway trivial composition, nobody is interested in that because it is the composition of the least volatile component in the top stream. So, I am just making the assumption that it is 0. So, that gives me the starting point for my calculations. So, my X D 2 is 1 minus X D 1. So, I know X D 2, I know X D 1, X D 3 is 0. So, I can calculate the bottom composition by overall material balance. So, all 6 compositions are known now. All 6 compositions, top and bottom compositions are known. I can start solving a design problem. Once X B is known, all X B's calculate the stripping profile or stripping section profile from X B all the way to the stable node. What is stable node? It is field pinch. So, that is what I am doing here. See, I have X B, the bottom composition. I go on solving this and then solve it all the way till you get a field pinch. That is what I do. Step by step calculation. You have a stripping section equation and you do this calculation. Any doubt? Straight forward. Once you have this point, you have stripping section equation. You do this calculation, get this profile. So, I have this stripping section profile. Now, in actual case, a feasible case where my reflux ratio is greater than minimum reflux ratio, what is going to happen? Two profiles are going to intersect. Now, I am going to look at rectifying section profile. Rectifying section profile will start from this point. That is close to the most volatile component. But then, I am not doing that. I am not actually solving it from here because anyway I have assumed X D 3 to be 0, then it would not come inside a triangle at all. It will treat it as a binary mixture. So, I am not going to solve it from this point. I am going to adopt a very different method. What am I doing here? See the next step. Locate the point X 0 on the stripping section profile as shown here. Now, we have the stripping section profile. I have located X 0 point. Now, stripping section profile at two different parts. One part is going towards the intermediate boiling component and other part is going in this particular direction. So, locate a point, take any point. Any point on this particular part because the intersection is likely to be here. So, I have identified this point. I will change this movement. Initially, it is at point A. Point A where I say my X 0 is lying on point A. I have just located that point. Now, the next step is to solve for the rectifying section profile in opposite direction. Let us say, let us see whether this point A is the intersection or not. Intersection of stripping section profile and rectifying section profile. Let us say this point is the intersection. If that point is intersection, then if I solve the rectifying section profile equation from this point onwards in opposite direction, I should be able to meet X D. But what happens here? If I start solving rectifying section profile equation in opposite direction from point A, I go this way. I go this way. So, that means there is something wrong. Point A is not the intersection of both the profiles. Then I move this point X 0 from A to position B and again start solving rectifying section profile equation in opposite direction. Again the same thing happens. That means again point B is not the intersection of these two profiles. I move it further. I get this point. What happens? Now, it goes very close to the saddle of rectifying section profile and then moves in this direction. Moves in this direction which is what I want. Because my X D is lying here, my rectifying section profile if solved in opposite direction should meet my X D. And this is what is happening at this particular point. What happens later? If I move my point in this direction, suppose it is at point C, it is at point C. Then also the rectifying section profile goes in this direction and moves towards X D. Not exactly falling on X D. Where is X D? X D is on the binary edge because X D 3 is 0. So, it may not be exactly falling on that particular point, but going close to it. So, on all the points here after this point, after this point I have the rectifying section profile or trajectory going in this direction towards X D which is what I want. So, that is what I have said in the next step. For various values of X 0, that means X 0 I move it on the stripping section profile. Calculate the rectifying section profile until it intersects the desired distillate composition. Goes very close to it. Then count the number of stages in each section and also note the distillate composition of the component 2 and 3. If I start counting stages, what will happen? I have shown it here. Up to this point, up to this point, there is no point calculating number of stages because it is not feasible design. We have to only look at the points after this where there is a possibility of a feasible design. So, this is my first point from which I start doing calculation. I start counting number of stages. So, this is that point. X 0, this is my X 0. I go on counting stages. How will I count stages? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then I move to rectifying section profile. 11, 12, 13, 14, 15. You can count these stages and I plot this here. This is the total number of stages. So, for X 0 here, I have some point here. Total number of stages. If I move this X 0 inside, what happens to total number of stages? This number goes down. Why? Look at this. When I am here, my rectifying section profile is going very close to the saddle. And at saddle, I have many stages. That is why I have almost infinite stages at this particular point because I am going very close to saddle. The moment I bring this point inside or rather move this in this direction X 0, the number of stages would come down. At particular point, you will see N T going to almost minimum. But then after certain value of X 0, that means if I take it to this particular part of the stripping section profile, again the total number of stages would increase. Why? Why do you see this? Sorry? That is true. That is true. So, you see feed pinch here. There are many stages. And because, when I am talking about X 0 here, that means I am here. If I count number of stages, you have feed pinch here and there are many stages. There are many stages there. That is why it goes up again. So, what is a good design? Good design is where I have less number of stages. So, I should select my X 0 in such a way that I am in this particular range. I am in this particular range. That is the total number of stages. But I have not answered one question. We started with what? We started with X D equal to X D 3 equal to 0. But that is not correct. In this case, what is going to happen? I am going to select some value of X 0. I am going to draw rectifying section profile. It is going to hit this point and for that particular point, X D 3 will not be equal to 0. It will have some finite value. But all calculations that I have done here for material balance are based on the value of X D 3 to be 0. But with this procedure, I am going to get some value of X D 3. So, there is some discrepancy. So, what am I going to do later? I am going to repeat the calculations with new X D 3 now. So, X D 3 I have got here. I am going to put it here in this equation. Now, the new X D 2 is going to be 1 minus X D 1 minus X D 3. I will repeat that calculation. I go on repeating that calculation and we will have many such plots. And I will see, I will do this calculation till there is no change in my X D 3. This is called as convergence of course. And normally it is observed that within 2-3 iterations, you see there is no change because X D 3 is a very small value. It does not have much impact on the material balance. So, we do not have to really worry about it. So, this procedure seems to be working well to calculate actual number of stages. So, once you have minimum reflux ratio, you get operating reflux ratio and calculate actual number of stages with this method. And this is applicable to even multi-component systems. You can write this program for multi-component systems a 4-component, 5-component and this will work well. So, what is the design method? Determine end compositions by process requirement. X D is X B is D by B, degrees of freedom analysis. Identify whether the split is direct, indirect or transition. That is very important. Determine field pinch and saddles of the concerned sections independently without actually doing step by step calculations. Vary the reflux ratio and generate R versus determinant E 1, E 2. If it is multi-component system, then if it is 4-component system, then it will be E 1, E 2, E 3. So, this plot is generated. Get R mean the reflux ratio at which this determinant is equal to 0. Determine actual number of stages. The method that we just seen for R is equal to 1.3 to 1.5 times R minimum. Once you have actual number of stages, now you are ready for simulation, rigorous simulation. Then I can go to the commercial simulator like Aspen, Heises, Proto, ChemCAD whatever. Then I give this data to the simulator because simulator will ask me the input. How many stages? What is the reflux ratio? Because it is a virtual experiment. Is that all decision variables? Design is in front of you. So, you give this input to simulator and see the X D that I started with. The X D, the design requirement that I started with is same as that given by the simulator. It may vary because now you are incorporating energy balance, you are incorporating probably the pressure drop, efficiency of stage because now in this entire exercise I have assumed a stage to be an equilibrium stage. So, we will relax all those assumptions in simulation because anyway like I am going to take help of computer. In design exercise also we are taking help of computer to some extent, but not the computational requirement is not as high as that in simulation because simulation is solving all the equations simultaneously. You have many equations, sometimes dealing with 1000 equations depending on how many components you have, how many stages you have and depending on the complexity of the model. So, that is the procedure and more or less the same procedure is applicable to non-ideal systems. Now, whatever we have seen so far is for ideal system. Yesterday there was a question that can I use Underwood's equation? Can I use Fansky equation? How is it different from this method? Of course, in terms of methodology it is totally different. Underwood's equation they have already derived it and you have to solve those equations for getting the root and just put that in the equation and you will get a minimum reflux ratio. It is a very simple method. So, why to go for this? As far as ideal system is concerned you can use Underwood's equation. You can use Fansky equation to calculate minimum number of stages, but if the system is slightly non-ideal then it fails. Underwood's equation uses alpha. Moment I use alpha and assuming that the system is ideal. So, whatever I have explained so far is for ideal system, but when you have ideal system in front of you like say benzene toluene xylene, ethanol, propanol, butanol. If you have ideal system in front of you you may not go for this. We need not go for this particular method, but if you are not sure whether the system is ideal or most of the time the system is not ideal. If you are in refinery fine, but if you are in a chemical plant, fine chemicals, you are always dealing with polar components and components are of different nature. You are doing a reaction and then separation. Reaction and separation. So, in reaction you get the mixture consisting of components with different polarities. So, there will be non-ideal interactions system is quite likely to be non-ideal. So, the separation is most of the time you will deal with non-ideal systems may or may not form azeotropes, but it will be non-ideal. That is why Underwood's equation, Pansky equation or that FUG method which is a semi empirical method may not work for such systems. That is a limitation. That is a limitation those methods. Now, what we have learned here we can extend it to non-ideal systems. So, this particular methodology has a potential to incorporate non-ideal systems under its umbrella because as far as vapor liquid equilibrium is concerned, fine. What is the difference between this and non-ideal system? Only the vapor liquid equilibrium methodology is going to be same. So, where do I use vapor liquid equilibrium in all these calculations? When I do that stage by stage calculations rectifying section lines, stripping section line. So, I go and get those points by simultaneously solving material balance and vapor liquid equilibrium. So, only I have to replace that model. Instead of using y i is equal to alpha i x i divided by sigma alpha i x i, I will use some other model, thermodynamic model. It may be based on EOS approach, equation of state approach or it can be that activity coefficient based approach. So, for activity coefficient I can use say Wilson model, Morgan model, Uniquac model, Unifac whatever. So, if I just replace my VLE model by the one which is mean for non-ideal system, same method can be used. So, this method has wide applications. You can very well appreciate that because in this case I am not really taken help of VLE anywhere in devising or in formulating the methodology. So, let us see what happens in non-ideal systems. Not so easy. I cannot say that now it is up to you. Just use this method for non-ideal systems because the main problem in non-ideal systems is the formation of azeotron. So, when you have formation of azeotron you should be very careful. While calculating number of stages and everything it is fine, but the main problem with non-ideal system is the feasibility. Feasibility is an issue in non-ideal systems. All these calculations are possible when I know the operation is feasible. That means I can separate pure A, I can separate pure B. That is the assumption I have made here. Direct split, indirect split. I say x d 3 is 0. I can assume that my point is in the corner like very close to most volatile component. When can I say that when there is no formation of azeotron? But when you have formation of azeotron you should be very careful. So, the feasibility issue would come in picture. So, next part we are going to see how this method can be extended to non-ideal systems. In non-ideal systems you have non-azeotropic and azeotropic. Non-azeotropic again is not much of a problem except you have, once you have tangent pitch of course you will face some problems. But normally we do not realize such systems in reality where you have multi-component system with tangent pinches and all that. But there is a method. We are not going to cover that in the interest of time. But most important system is where you have formation of azeotrops. When you have formation of azeotrop, how this method would be extended to non-ideal systems. Let us go back to MacArthur again. Of course, here as I said before in what way the non-ideal systems would be different from ideal systems, presence of tangent pinches or presence of azeotrops. When you have as I told you before this is not so common multi-component systems. But still there are methods to deal with the tangent pinches and I am not going to cover that part in this particular program. It is given in this particular book and if you have any queries you can get back to me on this. But what is important is this, where you have presence of azeotrops. What is the typical feature of azeotropic system? You have distillation boundary and feasibility issues. Let us go back to Macapthil method, binary system. There is so much to learn from Macapthil method. Though it is for binary systems, we can extend all those concepts to multi-component systems. This is your azeotropic system where you have formation of azeotrop, a minimum boiling azeotrop, minimum boiling azeotrop. This is your XF, this is your XD, top composition, XB, bottom composition. Now can we design a simple column? I am not talking about azeotropic distillation and all. Can I design simple binary distillation column to achieve this separation? Can I do that? Is it possible? It is not possible. It is not feasible. That is a feasibility issue. I am talking about feasibility, feasibility and all. This is what it is. It is not feasible design at all. Thermodynamics or vapour liquid equilibrium puts a constraint on you. So, that I cannot cross this boundary. This is the distillation boundary. This is the distillation boundary defined by the azeotrop, defined by the azeotrop. So, if I am in this particular region, suppose my XB is here, I cannot get XD in this region. So, there are two regions, region 1 and region 2. The answer is no because there exists a distillation boundary. Now what is distillation boundary? It divides the composition space in two regions. The two regions as I said before, the operating lines from one region cannot cross the boundary to enter another region. And that is why I do not have direct connection between these two, XD and XB. This is very well known. It is quite straightforward. We know all this. But then I am going to extend this concept to multi-component systems. Same would be true for multi-component systems, presence of azeotrops. Now in this case, unfortunately you have possibility of formation of many azeotrops. As I said before, if your terminal system, can you imagine how many possibilities? You have three binaries. In terminal system, you have three binaries. So, you may have three binary azeotrops. You may have ternary azeotrop, all three components forming an azeotrop. The example is say butyl acetate, butanol water, benzene ethanol water. You have ternary azeotrop also. And then you have various combination of this. That means one binary, one ternary, two binaries, one ternary, only two binaries. Then all three binaries plus ternary, only three binaries. So, many possibilities and every system will be different. Very interesting situation. So, we have to be very careful while dealing with azeotropic multi-component system. I am only talking about ternary system. Imagine if you have four component, five component, how many azeotropes possible and many possibilities. Why do we learn residue curve map? The answer is very simple to know whether the distillation boundaries are there or not. Once I know the distillation boundary is there, then I should be very careful. The first issue of feasibility would be solved. And later on it is just a calculation of minimum reflux ratio and number of stages. The same procedure will be valid, what we have done. So, while we are learning all this, the reason is to just know whether the design is feasible or not, whether there are distillation boundaries or not, whether there are distillation regions or not. That is the only purpose. It is an effective tool to assess the feasibility of separation. It is a representation of vapor liquid equilibrium on the ternary diagram. So, it is nothing but just the representation of vapor liquid equilibrium curve. We will see what it is actually, but it is one way of knowing the vapor liquid equilibrium curve in the residue curve map. Now I told you XA versus XB diagram. The first point that I made when I introduced this new reference frame, I said that I cannot visualize the vapor liquid equilibrium on ternary diagram. Because I am looking at XA versus XB, there is no Y there. But still this concept, residue curve map tells you about the vapor liquid equilibrium on the same diagram XA versus XB. Let us see how. That is the purpose and once I can see vapor liquid equilibrium on XA versus XB, then I know the feasibility. I know whether the distillation boundaries are there or not. Like in Macapthil, I know it is intersecting the diagonal. So, there are two different regions. Y versus X, it is intersecting the diagonal. I can identify the location of azeotrop in Macapthil. How do I identify the location of azeotrop in the ternary diagram that I have defined? Let us see. Residue curve map helps you. Following set of equations is solved and mapped on the composition, which are these equations. Let us see. I have a still liquid mixture present. Then I heat it. It is boiling. I take the vapors out. Very simple system. Bad system. I charge the still. I vaporize it or boil it. I take the vapors out. These vapors are taken out at constant flow rate V. Now you may ask me like if I start heating, how to maintain this flow rate? Because the latent heat of vaporization would change because the composition is changing as the time progresses. So, I have some control such that I take care. I adjust the heat input in such a way that this vapor flow rate is constant. So, this is V and it has certain composition Y i. Let us say Y i means there are many species. I represent a species. The composition in the still is X i. The composition in the still is X i. Now I want to write a model for this system. I want to write some equations. Material balance, the overall material balance and the component material balance. Can I write equations? Overall material balance, how the things will change with respect to time? Suppose m is the hold up. m is the hold up. I have d m by d t is equal to negative of V. Correct. Because hold up is going to change and it is going to decrease with respect to time. It will depend on how much vapor I remove, the rate of removal of vapor or material through vapor phase. There is no outgoing steam. Only vapor is going out. This is one equation. Overall material balance. Now I will write component material balance. For species I, I have d X i m by d t equal to V into Y i. I am writing for a particular component I. I varies from 1 to 1 to c or c minus 1. Because if I use summation constraint, I do not need to write this equation for all the components. I will write it for only c minus 1 component. For ternary system, I will have two equations. The third will be calculated by the summation constraint. Now can I simplify this? I just expand this m d X i by d t plus X i d m by d t is equal to minus V into Y i by V i. I have an expression for this. I will just substitute for that m d X i by d t is equal to V X i minus Y i. Can I do that? I have just taken one jump, omitted one step. I can do this. I can write this. Can I simplify this further? I non-dimensionalize this. If I non-dimensionalize this, then it becomes d X i by d zeta is equal to X i minus Y i. What is d zeta? d zeta is equal to v by m d t. Yeah, zeta is a dimensionless time. Zeta is the dimension less time. I will talk about it in detail what it means. But you have this equation. You have this equation, very simple equation. d X by d zeta is equal to X i minus Y i and this is nothing but residue curve map. Why it is called residue curve map? Because I am looking at a composition of the residue. I am looking at the composition of the residue. How it changes with respect to time? This particular equation tells me how it changes with respect to time. Because X i, if I solve this equation, this is a differential equation, ordinary differential equation. How many such equations I have? I have two equations for a ternary system or C minus equation for a C component system. And I solve this equation. There are many techniques to solve these equations. Simple initial value problem, ODE, ordinary differential equation. And if I solve these equations with respect to time or non-dimensional time, what do I get? I get the relationship between X and T. How the different compositions they change with respect to time in this particular still. But look at this equation. What do you have here? It depends on what? Now, this Y and X are they related to each other? I have proper contact here. There are no mass transfer limitations here. Y and X are related to each other through equilibrium, vapor-equilibrium. So, Y is related to X. So, once I know X, I can calculate Y. So, this particular moment of X with respect to time will depend only on vapor-equilibrium. So, the solution of RCM, the solution of RCM will depend only on vapor-equilibrium. Or in other words, I say that residue curve map is indirect representation of the vapor-equilibrium curve. Or it is one way of representing vapor-equilibrium. I hope it is clear. Now, what happens at azeotrop? Now, what is the condition of azeotrop? X i equal to y i. What happens in this equation? In x i becomes equal to y i. Moment is, right hand side is 0. That means, d x by d zeta is 0. That means, your residue curve does not move. Or it goes very close to that point and then goes away from it. Again, you have possibilities. You may have stable point. You may have saddle and you may have unstable point, right. X is equal to y. At pure component, X is equal to y always, right. When I say I have pure components, suppose I take A here. I do not have B and C. X is equal to y, right, pure component. So, all, suppose you have terminal system, all three components, you have X equal to y. But apart from these three points in the terminal diagram, A, B, C, there are other points also, where X becomes equal to y, which are these points azeotrop. In the captain method, you have pure points where X is equal to y, right. At this point X is equal to y. At this point again X equal to y. At this point again X equal to y, right. In terminal diagram, at this point X is equal to y. I am not seeing y, but I am sure X is equal to y at pure component. At this point X is equal to y. At this point X is equal to y, okay. And suppose your azeotrop form, you may have certain point inside a triangle. If there is a ternary azeotrop or if there is a binary azeotrop, if there is a binary azeotrop here, here, you have certain points there where X becomes equal to y, right. Okay. How to identify these points? Okay. Let us see. So, this is your equation for RCM. You have derived it already. Okay. You have still, you are taking vapor out, which is composition y. And then this is your equation. Very simple equation, okay. But as I said before, it is a representation of the vapor liquid equilibrium curve because the entire course or the way X will change will depend on VLE, nothing else, right. Okay. Now, can I plot this on the ternary diagram? Okay. Now, in this case, you have a zeta. Okay. This is your dimensionless time. But if I want to plot it on a ternary diagram, there is no zeta there, no. There is no zeta there. I am plotting it for XA and XB, right. In a ternary diagram, it is XA and XB. So, even if I solve these equations, right, if I solve these equations, suppose there are two equations for three component system, what do I get? Zeta, for different zeta values, I get XA, I get XB. And of course, XC would be 1 minus say minus XB, right. So, I am going to get for different values of zeta, I am going to get different values of XA, XB and XC. If I solve this equation, this is ODE, Initial Value Problem, right. And once I have this table, okay, I can plot XA versus XB, right. I use these two columns, XA column and XB column to plot it on the ternary diagram, right. What, how will it behave on ternary diagram? Suppose I start with any particular point, it needs the starting point, Initial Value Problem, it needs the initial value, no. So, I start with some point here, okay. I go on solving this equation. I go on solving this equation. Where will I go? In which direction will I go? Will I go in this direction? Will I go in this direction? Will I go in this direction? Will I go in this direction? Can you tell me? For ideal system, let us not worry about non-ideal initially. Where will I go? Yeah, you will definitely go in this direction. You won't go in this direction. Why? Because I am looking at the composition of still, okay, residue, okay. It will move towards the, either towards the intermediate boiling or towards the least volatile, right. In fact, it is going to follow the same behavior as that of the rectifying section profile. It will initially move towards intermediate boiling and then will move towards, right. And what is the final point? Where will it stop? So, it is gone here. It is going to come down. Where will it stop? At least volatile components. It will stop at point C. It won't stop in between. Then I don't have feed pinch there like what I get in rectifying section. Why do I get feed pinch in rectifying section? Because you have the material balance line equation, okay. Instead of, and that depends on the reflux ratio, right. I don't have reflux ratio here. Just look at the equation, okay. The system is different. The physics is different. But equations are similar, okay. I don't have reflux ratio here. So my feed pinch is nothing but the stable point here, okay, right. It is equivalent to the stable point here. I hope it is clear. So, any point I start with, I am going to go to this point. If I start from this point, I am going to go here. If I start from this point, I am going to go here, right. And now I am going to show some arrows. So, it tells me the direction as well, okay, right. Suppose I am on this binary edge. Forget presence of components C. I start with A and B. I know A is more volatile. What will happen? I will go in which direction, right? Because A is more volatile. This is A. This is B. And this is C. A is more volatile. So, I will move in this direction on binary edge. If I start with binary mixture of A and C, I will move in which direction, right? If I start with binary mixture of B and C, I will move in this direction. Look at the arrows, okay. Let me complete this diagram. If I start solving residue curve map in opposite direction, where will I go? Yeah. There is no other way. There is no other point, okay. All the residue curves will go and meet point A, right. What does it mean? If I start with pure A, will I go in this direction? No, right. If I start with pure A, what will happen? There is no B and C, right. I will get only A. I will remain there itself, okay. So, when I extend these residue curve maps in backward direction towards point A, doesn't mean that all the residue curves are starting from exactly from point A. So, if you just expand this, it's like this. Your point A is here, right? We are going very close to A. That means you have a system or your mixture with A composition 99.99999999, okay. 99.999999 not exactly 100 all right. If it is pure A then there is no question of ternary system it is just a single component system right. But these points are ternary points ok. So, that is the meaning of it right. So, this is your residue curve map this is your RCM ok. This is your residue curve map for ideal system. What if you have azeotron ok. So, ideal system this is a residue curve map ideal system. Now, before we go ahead let us try and understand ok. This point is never realized when we solve residue curve map equation right. So, this point is called as unstable point or unstable node right. When I go on solving the equation a stage comes when the moment is stopped right. This point is called as stable point and depending on where I start suppose I start very close to this binary edge and go on solving the equation I may realize a point right. There the moment is very slow, but then it takes the course from there and it goes to point C. What is this point? This point is called as saddle ok. So, what we learned earlier for rectifying section profile stripping section profile you see the similarity right. But of course now the behavior is independent of the reflux ratio right. The behavior is independent of the reflux ratio unstable point saddle stable point right ok. Now, there is a mathematical condition of course I am not going to get into that, but if you see the right hand side x minus y if you take the Jacobian of this Jacobian matrix and find out its eigenvalues all the eigenvalues negative means the stable point. All the eigenvalues positive means unstable point some of them are positive some of them are negative means the saddle ok. So, you can independently find out whether a given point is saddle stable or unstable just by looking at a Jacobian of x minus y ok right. Let us go ahead. See the emphasis is more on understanding the concept as I said before you need to work on this and then only you will have mastery over it. So, I am trying to tell you that similarity between residue curve map and rectifying section profile and stripping section profile because now I need to move ahead and correlate it with the column behavior. But I am going to use this concept for knowing the column behavior I want to design a column right. I have just looked at the still in which I am boiling the mixture and all that right. One thing you would have noticed now is there is no zeta appearing here I am just plotting x a versus x b ok. So, I am not worried about zeta and looking at the nature of this particular diagram or map I know something about a vapor liquid equilibrium. Suppose I get this type of RCM right I am sure the system is ideal I am sure the system is ideal or non-asiotropic for that matter ok non-asiotropic. If there was some azeotrop then I will realize that point somewhere because my moment will be restricted I do not see any distillation boundary here I am going to give the example later what happens if azeotrop I do not see any boundary here ok. So, the system is ideal the moment I see the residue curve map I get some idea about a system vapor liquid equilibrium. So, that is what I was telling you even if I am not plotting y here even if I am not plotting vapor liquid equilibrium here I get some idea about the vapor liquid equilibrium of the system right. This tells me that there are no boundaries there are no azeotrops. If there is azeotrop we will have a different RCM we will see ok. Before that let us look at the RCM the relationship between RCM and column profiles as I told you in column profiles here reflux ratio in stripping section you have rebound ratio, but then if we make reflux ratio infinity what will happen? What will happen to this? This will become for r equal to infinity this factor r by r plus 1 is 1 right very high value of reflux ratio r by r plus 1 will be 1 and this will be 0 because you have in denominator you have infinity. So, this becomes 0. So, these two equations at infinite reflux ratio we will get reduced to the residue curve both the equations because now they are independent of reflux ratio and rebound ratio right. Of course, this equation in the differential form this equation is a discrete form. So, I will just spend two three minutes to show you a derivation how to correlate these two. But from this slide it is known that the rectifying section profile and stripping section profile under extreme conditions that means at infinite value of reflux ratio will boil down or again large number of stages will boil down a single equation that is of residue curve map right. So, that is the relationship between column profile and RCMs and I am going to use this concept later to synthesize a sequence column sequence for a ternary system or azeotropic system. So, I have rectifying section profile equation y n plus 1 i is equal to r by r plus 1 x n i plus x d i r plus 1. I want to see whether I can convert this equation to residue curve map. Now the condition first condition is r is tends to infinity what do I get n plus 1 i is equal to x n i right because this becomes 0 this becomes 1 at infinite reflux ratio same is true for stripping and do it separately ok. Now this equation and saying it is equivalent to d x i by d zeta is equal to x i minus y i is that true I just instead of this equation I write it as n plus 1 i minus y n plus 1 i is equal to x n plus 1 i tell me whether it is true or not. Can I do this? I have just multiplied it by minus 1 both the sides and then added x n plus 1 i on both the sides right. So, I can convert this I can write this particular equation alright. Now if you have a sufficiently tall column I am just divided by 1 I can divided by 1 not a problem. If I tall column with say 100 stages 200 stages or infinite stages that one is a very small number one is very small number that means for a very tall column 100 stage column I am just looking at one stage I am just looking at one stage that means a differential element of that column right. Can I convert this to delta x divided by delta h I can do that no delta h of course is no unique, but this is delta x that means the change in composition along the height of the column n plus 1 n right divided by delta h very small number for a very tall column right for a very tall column. So, I can write this as a differential delta x by delta h ok. So, what is this now in a very tall column you have x minus y you have x minus y equal to this differential and what do you have here you have x minus y equal to this differential right. Only difference is now instead of delta h you have delta zeta there, but anyway on RCM or in a triangle or ternary diagram plot I do not plot zeta I am only interested in x versus x a versus x b right ok. So, there is a direct relationship between residue curve equation and the column profile as far as this ternary diagram is concerned right I hope it is clear this is basically the transformation of discrete dynamic approach to continuous dynamic approach. So, this is where I will stop you have this profile now relationship between residue curves and operating lines ok. Now look at the residue curve you have residue curve map I have just plotted the operating lines or other trajectories of rectifying and stripping sections. Look at the similarity and look at a difference as well. Now rectifying section profile will go this way compare it with residue curve you have residue curve also going this way right. So, rectifying section profile will follow the residue curve in terms of its direction, but depending on how much is the reflux ratio it will go away from the residue curve because in rectifying section equation you have reflux ratio. So, finite reflux it will not exactly fall on the residue at finite reflux ratio if you start on the same point residue curve will move this way whereas, rectifying section profile will take a different path it depend on your reflux ratio if it is infinite yes then it will follow the residue curve right. And same is true for stripping, but of course stripping now it is going to go in opposite direction. Now stripping section profile start with XB, but it will go in opposite direction as that of the residue curve because in stripping section you start with the least volatile component and go towards a more volatile mixture right whereas, in rectifying you start with more volatile that is XD and go towards less volatile look at what happens to the temperature. In the residue temperature is going to increase with respect to time right it will increase right because you have less volatile component getting accumulated or increasing rather with respect to time. So, the temperature will increase with respect to time in residue curve in rectifying section profile when you go from top to bottom the temperature will increase right rectifying section start with XD lowest temperature at the top the temperature is lowest right when you go down it will increase that means there is a similarity between rectifying section profile and residue curve in terms of temperature the way temperature changes right I hope it is clear that is why rectifying section profile follows residue curve in terms of its direction it will overlap on residue curve only when the ripple ratio is infinity whereas, in stripping section temperature is high and as you go up along the height temperature goes down. So, you are going exactly opposite to residue curve right. So, I have got vital information now if I plot a residue curve it will not only tell me whether there is a formation of azeotrope or not, but it will also tell me the movement of direction for the movement of rectifying section profile and stripping section profile ok. Then next question is how to use this information to design a column for azeotropic systems ok. As far as calculation number of stages is concerned it is going to be same as what we have learned before, but when it comes to feasibility we need to use the information which is obtained from residue curve map. So, it becomes now our tool our main I would say tool or base on which further analysis would be developed for azeotropic and extractive distillation systems ok. So, in the next lecture we will look at azeotropic and extractive distillation.