 Now, let us look at top composition, feed composition and bottom composition. There is certain rule there. They should follow certain rule which is called as a lever rule. What is that? Look at these equations. I have a material balance equation for every species. This equation should be satisfied. This is for species 1 or A. This is for species 2 or B. See distillate bottom feed, right? Distillate bottom feed and then this is nothing but overall balance. Now, you have these three equations. I play with these three equations. I convert them this particular form. What is this form? It tells me, tells me that when you plot these compositions x d, x f and x b on a ternary diagram, the slope of this line is equal to the slope of this line. This is the slope of this line. This is the slope of this line. If the slopes are equal, then these three points are co-linear. These three points are co-linear. It is lever rule. So, in nutshell, distillation column you have feed coming in, distillate going out, bottom going out. These three compositions, if I plot them on a ternary diagram, right, they should be on the straight line. Always remember this rule. When you are going to construct the diagrams later, sometimes quite likely that you will forget this rule and then you will end up getting a wrong sequence, right? This is very important that my x d, x f and x b are on the same line, right? This is one possibility where my x d is here, right? My x f is here, x b has to be here, right? Can we have x b here? Now, if you just go by boiling points, I say, my top composition should be the most volatile component, right? That means, somewhere here. My bottom composition would be the least volatile component that is here, is it correct? Where will we go then? In a ternary system, in a ternary system, if you have a continuous distillation, I cannot say that I will get at the top pure A and I will get at the bottom pure C at a time. If, is it possible? It is not possible because you have an intermediate component in the feed. It has to go somewhere, right? So, at a time I can separate only one component in pure form like what I showed you just now, right? So, you get either pure A or you get either pure C. Now, let us look at these profiles or trajectories, how they behave. Again, we will go back to the capital method. You have the equilibrium curve, right? And you start with x d, like in a triangle for a ternary system, I am going to start with this x d. So, here in binary, I am just looking at a similarity. I am trying to work out an analogy between these two. This is your x d here for a ternary system. In binary system, this is your x d. Now, when I solve the equation for the rectifying section line, I get this, right? So, which are the composition I am going to get? This is the first composition, right? The equilibrium stage composition. Then the second would be this, right? Third would be this, okay? And then, right? A point will come where my movement is restricted. I am not able to move. What is this point? This point is the intersection of rectifying section line and the vapor-equilibrium curve, okay? If I just go on doing the calculations, right? I will reach a stage where the further movement is not possible. I realize a pinch, okay? A fixed point, right? That is where the vapor-equilibrium curve and the material balance line, they will intersect. The further movement is not possible, further change in composition is not possible, right? No driving force, okay? You are on the equilibrium line, right? The same behavior will be observed here, okay? But since I am not seeing the vapor-equilibrium curve, it is not possible to visualize the vapor-equilibrium curve. I am not seeing the vapor compositions. I will realize it in a different way. Let us see, okay? You go on doing these calculations. I plot these points, okay? I am plotting XA versus XB. So, they are corresponding Y values, okay? Right? I go on doing this. A stage will come where you will not be able to move, okay? What does it mean? That means you have got something similar to this. That is where all these equations of vapor-equilibrium curve and the material balance line are satisfied, okay? You are on the equilibrium composition, equilibrium surface, I would say, right? So, further movement is restricted. This point is a pinch, typically called as feed pinch, right? Okay? We do not need to really define it in the case of my capital method, but it has a significance when you go to multi-component system, right? So, this is nothing but a feed pinch. You have one feed pinch for rectifying section. Similarly, you will get another feed pinch for stripping sections. Like this, you have XB, right? And you may have me draw a line, right? This is one point, another point, another point and for the further movement is restricted. So, you will just hit the vapor-equilibrium curve. Similarly, if you do these calculations here, you go this way, right? And stop there. You will not be able to move ahead. You can do this calculation on your own. You will realize that, okay? What does it mean? It means that now further movement is not possible because you are on the equilibrium surface. As I said before, no driving force for mass transfer, right? So, you get a feed pinch, right? So, you have feed pinch for rectifying section. You have feed pinch for stripping section. Is that clear? It is quite similar to what we see in my capital, okay? Basically, it is just an interplay of vapor-equilibrium curve and material balance line. We get intersection here. That is why the further movement is restricted. Similarly, I am not able to see the intersection here because of the dimensionality of the system. But then, definitely all those equations are satisfied, okay? Right? And that is the reason further movement is not possible. Now, which are all these equations? Now, in this case, which are the equations? For this particular point, I have y is equal to alpha x upon 1 plus alpha minus 1x, right? For a rather, okay? More volatile component. And which is the other equation? Intersection of vapor-equilibrium curve and material balance line, right? So, which is that other equation? y a equal to r upon r plus 1 x a plus x d by r plus 1. Why I am not writing n there? Because n has no meaning, right? And even if I change n, it does not matter, right? n is going to infinity and I am getting this particular point. So, if you want to get this point without actually solving the equation step by step, without doing this exercise, if you want to get this point, I can solve these two equations together, right? Simultaneously. If I solve this equation, I will locate this point. I do not need to do this exercise, right? What about this point? This point again, you have the vapor-equilibrium equation, right? But now, you have two equations like this, for a and b, right? And then, material balance equation, again, two equations. So, binary system, you need to solve these two equations. But for terminal system, you need to solve four equations simultaneously, right? So, by solving all those equations, you will get this point, right? Because so many equations, then these are nonlinear algebraic equations. You will get multiple roots. You will get multiple roots. One of those roots is this, field pinch, right? So, you have something similar to what happens in my capital method and that is what I have shown here. See this, you have a profile this way and you see the points are very close here and further movement is difficult or rather impossible, right? Now, this profile, you see the equation, it depends on the reflux ratio, right? The rectifying section equation, if you see, it is r by r plus 1, right? What happens in my capital? If I change the reflux ratio, if I change the reflux ratio, yeah, the field pinch shifts, right? Same thing I am going to see here. See, the field pinch has shifted. It has gone ahead. If I increase the reflux ratio, if I increase the reflux ratio, the field pinch has gone ahead. My composition profile is traveling maximum space in the triangle. Now, can you imagine what if you have reflux ratio infinity? What will happen? Start with this. Where will it go? Where will you see the field pinch? This pinch is moving in this direction. What is the maximum possibility? What is the extreme case? Again, think of my capital, okay? If you increase reflux ratio, go to infinity. PoC. PoC, right? So, this point, this field pinch will go here, right? And same is true for stripping section exactly in the opposite direction. If I go increasing the reflux ratio, rebound ratio will also increase and my field pinch for stripping section will move and for infinite value of reflux or rebound ratio, that pinch will go to this particular point, right? That is what I have written here. At field pinch, the profiles do not move, okay? Profiles means the trajectories. This point is equivalent to point of intersection of operating line and equilibrium curve in the captain method, right? And at infinite reflux ratio, the field pinch of rectifying section is nothing but C, the least volatile component. And for stripping section is nothing but point A, which is the most volatile component, right? Right now, I am just looking at the rectifying section profile behavior and the behavior of the stripping section profile. We will correlate it with the column performance later, okay? I am just trying to devise a method, okay, to find out the minimum reflux ratio, okay? That is what we do in Macapthal method, right? So, in order to find out the minimum reflux ratio, I need to know the behavior of these profiles, okay? So, that behavior we are studying now, okay? Alright? So, this is the way the rectifying section profile will behave. This is the way stripping section profile will behave, right? And you have something called as field pinch. So, we have defined field pinch, okay? We know how to get this particular point. So, as I said before, can we determine the field pinch without solving the column trajectory? That means without doing that step by step calculation, I can do that, right? I can get the field pinch without actually solving these equations by dealing with this equation and the vapor liquid equilibrium equation, right? Okay? So, this equation is for rectifying section profile. This equation is for stripping section profile. Now, I want to get a field pinch for the rectifying section profile. I have this equation as I told you before for a Turner system, how many equations you have? You have two such equations for rectifying section. Then you have the vapor liquid equilibrium equation. I just solve them simultaneously and get the solution, right? I get a solution, okay? So, that is your field pinch. But as I told you, I am going to talk about it later as well, that you are not going to get one solution because there are many equations, they are highly non-linear. So, you get multiple roots, okay? And this is one of them. The field pinch is one of them, right? Similarly, stripping section, you are going to get the field pinch for the stripping section by solving this equation. When I say solve, that means direct solution. I am not going to do step by step calculations starting with x d and all. This equation should be solved simultaneously with vapor liquid equilibrium equation. Like what I told you before for, so these equations are solved together simultaneously to get this point directly instead of doing these calculations. Right? Similarly, here I am going to solve these equations. Now, there are many equations, okay? For binary system, you have only two equations. For quaternary system, you have four equations, right? Sorry, quaternary system, you have four equations. Quaternary, how many? Yeah, six equations, okay? So, you can get these points directly, okay? Feed pinches. Now, what is minimum reflux ratio? Can I get minimum reflux ratio? Look at this. This is your stripping section. This is your rectifying section. Don't forget x d, x b and x f. I am not showing x f here. They should be on the same line, right? Because now I am talking about a column, right? The overall material balance should be satisfied. So, this is my x d. This is my x b and there will be some x f on the line joining these two, right? Okay? Suppose I plot these profiles. I plot these profiles and if your reflux ratio is less than the minimum reflux ratio, okay? Then will the profiles intersect? Profiles won't intersect, right? If the reflux ratio, see, in order to solve this, in order to get this profile, I need a reflux ratio. I assume some value, okay? I assume some reflux ratio, right? And get this profile and corresponding value of free ball ratio, I plot this profile, okay? I see this particular behavior. What is this behavior? There is no intersection, okay? It means that your reflux ratio is less than minimum reflux ratio. Profiles are not intersecting. I am coming from top, I am coming from bottom, okay? But there is no intersection. So, it is not feasible, right? So, in this case, there is no intersection. So, you agree that if the reflux ratio is less than the minimum reflux ratio, the two profiles that I plotted, they won't intersect, okay? Operation is not feasible, right? If the reflux ratio is greater than minimum reflux ratio, I can see the profiles are intersecting, right? And there is an intermediate stage where the value of reflux ratio is such that, now this is very important, the value of reflux ratio is such that pinch, the feed pinch of one of the sections, either stripping section or rectifying section falls exactly on the other section. Now in this case, what is happening? In this case, the pinch of stripping section is falling on the rectifying section. There is a possibility that the pinch of rectifying section may fall exactly on the stripping section, okay? Getting both the pinches falling on each other is a very specific case, we will come to that later, very specific case, okay? Right? What happens in my capital? What is the minimum reflux ratio condition? Number of stages equal to, at minimum reflux ratio, number of stages equal to infinity, right? See, that's what we see. At this point, you have almost infinite stages, but these two operating lines are connected to each other in a feasible composition space. If they intersect outside this, no meaning. They have to intersect in this region. This is the region, okay? Feasible region. I can't go out of this. I can't go out of this. I can't go here, right? So, the profiles they intersect or trajectories they intersect here, there is no meaning. They should intersect inside. And what's the limitation? What's the point where they can just intersect and give infinite stages? Yeah, this is that particular point, right? And I get many, for that, I get infinite stages, right? I get infinite stages, right? And we have similar behavior here for ternary system. Now, in this case, since it's a binary system, you have the feed pinch of stripping section falling on feed pinch of rectifying section. But now, since we are dealing with ternary system, okay? There's a slight difference, okay? There's a slight difference that a feed pinch of any of the sections should lie on the other profile, okay? Right? So, what is happening here? Feed pinch of the stripping section is falling on the rectifying section. How many stages you have here in this particular situation? How many stages you need? Infinity. Infinity. Why? Because you have feed pinch here and at the feed pinch, you have no moment possible. So, stages, number of stages is almost infinity, right? Okay? So, you have minimum reflux ratio there. There's this particular arm of the rectifying section which has no significance because in the column at infinite reflux ratio, if you start from bottom, this is the way you will go, okay? Right? It will come up to this point and then here, right? So, these compositions, you won't realize them in the column at all, okay? Right? So, reflux ratio equal to minimum reflux ratio. So, there are three different possibilities at reflux ratio less than minimum reflux ratio. You see this particular behavior. There is no intersection at all. The feed pinches of both the sections are far away from each other or I would say the feed pinch of stripping section is away from the rectifying section. In this case, the feed pinches are such that the intersection is possible where the reflux ratio is greater than minimum reflux ratio. So, you have a situation where this exact falling of the feed pinch of stripping section on the rectifying section, right? Now, the next question is who will fall on the other profile whether it is a feed pinch of stripping section or feed pinch of rectifying section. In this case, what I have shown is the feed pinch of stripping section is falling on rectifying section. Sometimes you may have other way around, okay? Right? You may have feed pinch of rectifying section falling on stripping section, right? When does that happen? Look at this. The first case, the first case where you have as I showed you before the feed pinch of stripping section falling on rectifying section, right? In this case, the top composition is almost pure A, okay? Look at this column. Top composition is though I have said A here, it is close to A, okay? Right? Not necessarily pure A, okay? When I have pure A, I call this as sharp split, okay? But it is not necessary that it should be pure A. It would be close to A, right? So, when you are close to A, you see a direct split. Something that I showed you before, you have mixture of ABC, you are separating pure A from the top, BC goes down and this BC goes to another column and you are separation of B and C, right? Okay? So, that is the sequence that I am talking about here, okay? And this particular split is called as direct split, okay? This particular split is called as direct split. So, what happens in direct split? You have the feed pinch of stripping section falling on rectifying section, right? Okay? Then the next is the other way round where the feed pinch of stripping section falling on, sorry, the feed pinch of rectifying section falling on the stripping section where your bottom product is almost pure C. So, you are close to C, you see these two positions and of course, this would be slightly above depending on where your feed is, right? I cannot define these two independently, right? These three should be on the same line, this, this and a feed. Now, what do you mean? Is this a feed? Is this a feed? Intersection, will it give you the feed, feed composition? Is it same as a feed composition? See these two profiles, they intersect. That intersection is it same as a feed composition? Need not be, right? That is correct. So, let us see, okay? Now, when I am talking about the intersection, all these points are satisfying the equilibrium constraint, right? Phase equilibrium, okay? This is your feed plate, okay? This is your column, this is your feed plate, your feed going in, right? And you have this, right? And when I plot these points on a triangular diagram, all these points are what? They are the compositions of the streams leaving any stage. Now, when there is an intersection, there is an intersection, okay? I am going from stripping section, I am going from stripping section and then I am switching over to rectifying section, right? From at this point, I am switching over to rectifying section, right? Okay? And I am looking at these points. Now, these points are not, there is no connection, direct connection, okay? There is no feed point in this, right? Feed point is independent. So, when I say, I when I talk about this point, this is nothing but the composition of the stream leaving the feed plate, right? Composition of the stream leaving a feed plate and not a feed composition. So, feed composition would be different and composition of the stream leaving that stage would be different, okay? So, do not make an impression that this intersection is a feed composition, okay? It is a composition of the stream leaving the feed stage and not a feed composition, okay? Feed composition can be anywhere. Now, in this case, the feed composition would be on the line joining these two points. Why? This is XB, this is XD and feed composition would be on the line somewhere and not necessarily the intersection, right? I hope it is clear, okay? Now, there is something else I am going to define which has spatial significance in ternary systems or a multi-component system. There is something called as saddle, okay? There is something called as saddle. You realize that when you actually plot these profiles, okay? We do not realize that in binary systems, okay? I told you feed pinch. How do you get feed pinch? If you want to get feed pinch, you have to solve these equations simultaneously. Which equations? I will repeat, okay? You have rectifying section Y, A, sorry, sorry, okay? Rectifying section, okay? Component A, okay? Component B, then equilibrium equations, phase equilibrium equations. YA is equal to alpha A XA divided by 1, sorry, sigma alpha I XI, right? Multi-component system, okay? Then for YB alpha B XB, okay? And then summation equations, right? Sigma YI is equal to 1, sigma XI is equal to 1. Right? So, all these equations for a ternary system, if I solve them together, okay? I get feed pinch, right? I get feed pinch, clear? Right? I get feed pinch if I solve all these equations together. But then look at the nature of these equations. They are linear equations or non-linear equations? What am I solving them for? I am solving them for YA XA YB XB YC XC, right? Six equations, six unknowns. So, if you look at these variables in the equations, they are, these equations are non-linear equations, okay? These equations are non-linear equations. Look at these equations. This is non-linear equation. What do you mean by non-linear equation? Y is equal to MX plus C, linear equation. Y is equal to X square plus C is non-linear equation, right? So, somewhere you have non-linearity, okay? Then we call it as non-linear equation and it is a set of non-linear algebraic equations. And if you solve these equations together, okay? It is quite likely that you will get multiple roots. You will get multiple roots. A quadratic equation has two roots, right? So, here also you are going to get multiple roots. One of the roots, as I told you before, is the feed pinch, right? Is the feed pinch where the profile is not able to move, okay? This is an intersection. But there are other roots also possible, okay? Right? The feed pinch you get in the ternary diagram in the inside the triangle, okay? Whereas, see, the feed pinch you get somewhere here, right? But there is one point on the binary edge also, okay? On the binary edge where all these equations are satisfied and which is called as saddle, okay? This point is called as saddle. We do not realize that or we do not get such point in binary system in macapthil, okay? Because the dimensionality is less there, okay? It is a two-dimensional system. Whereas, for ternary system, since we are increasing number of state variables, okay, you get multiple roots and there is something called as saddle which is realized. Now, how does this saddle behave? And where do we realize this saddle? Do you, every time come across saddle? Answer is no. For example, if you solve the equation for the stripping section profile with a point on the binary edge, this is your xB. This is your xB. You start solving equation for the stripping section profile, okay? You move in the upward direction like what I told you before, okay? You will hit a point. Not exactly, you will realize that point. We will go very close to that point and you see there are so many points here, okay? The moment is very slow. The moment is very slow. You will realize, okay? I am not able to, like you may think that, okay? Now, I have got a feed pinch. It is not correct. You go on solving that equation, right? And after some time, we will go away from that point and then we will follow this particular track. And then you will hit the feed pinch, okay? So, this particular point or rather this particular behavior is realized because you are very close to the saddle. Your saddle is sitting somewhere here and since I am starting with a composition on the binary edge, okay? Very close to the binary edge. That means the composition of A, the most volatile component in XB or rather in bottom is very less. Say 10 raise to minus 13, okay? I cannot make it 0. In that case, it becomes a binary system, right? But if I make that composition very, very small and I start solving the equation, I will go very close to this saddle and then we will change the course thereafter and you will see the profile like this. So, that is how I will realize this saddle on the composition space, okay? In the composition profile. Actually, as I said here, composition profile does not pass through the saddle, but it approaches saddle very closely and changes its course thereafter, okay? Something you might have learned stability analysis, right? Some points are stable, some roots are stable. What does it mean? When I go on solving the equation, I reach that point and the movement is not possible. Some points are unstable, that means you do not realize them, okay? And some points are saddle, okay? You go very close to them and then you change your course thereafter. And if you go into the mathematics of that, then you have to work out the Eigen values and of the Jacobian related equations, okay? We are not going to get into that. We are not going to do stability analysis, okay? We are just going to see the behavior because this point has a special significance and I am going to use this behavior to calculate minimum reflux ratio, okay? So, what we have learned is there is something called a saddle point in ternary system, okay? So, saddle on the rectify, in the rectifying session lies on the binary age, okay? Of course, for small xdc values, right? That means here, small xdc value and it is other way around for small xba value, okay? Your saddle lies on the binary age again, okay? This particular binary age. So, you have one saddle here, you have one saddle here, right? And this position of the saddle can be used to calculate minimum reflux ratio, right? I have already told you that this is the condition for minimum reflux ratio, right? Okay? You actually draw profiles, okay? And get to a reflux ratio where they exactly fall on each other or one feed pinch falls on the other profile, okay? So, that is your minimum reflux ratio. But then, is this the complete procedure? Can I use it for four component system? Yeah, I can use it. There also I can do visualization. But five component, six component, it is not possible, right? Because this triangle is going to help me only for ternary system, right? If you have quaternary system, I may have 3D space, I can still visualize with difficulty, little bit difficulty, but I can visualize, right? For five component or multi-components, it is not possible, right? Okay? So, with this visualization, I can get minimum reflux ratio for a ternary system. See, that is my purpose, okay? Getting minimum reflux ratio somehow, right? But then for multi-component systems, it is not possible. So, now I am going to convert this concept to some mathematical form, okay? So, that I can use it, I can extend this concept to multi-component system. And that is why I need this presence of saddle. I need to identify the presence of saddle, okay? So, what do I do for any given system? Okay? Now, this is a certain algorithm, a systematic methodology, okay? I just get this saddle point, right? Okay? I get this saddle point for rectifying section profile. I know the feed pinch for the stripping section profile. I know the feed pinch for the rectifying section profile. Now, suppose I draw a triangle. Now, these are the three points, okay? Okay? These are the three points. If I draw this triangle, okay? The area of this triangle, area of this triangle would be 0 if you have the minimum reflux ratio, right? These three points will be co-linear. These three points will be co-linear, okay? What is this point? This is your feed pinch of the stripping section. This is your feed pinch for the rectifying section. And this is your saddle, okay? Look at this particular diagram. If you have reflux ratio, less than minimum reflux ratio, there will be certain space and this triangle, this triangle has certain area, okay? Right? But if this point exactly falls on this profile, okay? In that case, these three points would be co-linear and the area of the triangle would be 0, okay? Right? That is the way the profiles would behave, okay? And then if your reflux ratio is greater than minimum reflux ratio, this point will move here. Again, the triangle will have certain area, okay? Now, the area of triangle is proportional to the determinant. What is this determinant? You have two vectors. This is the point. This is E1 and E2. You just get a determinant, okay? That determinant will tell you the area, not exactly the area, but it is proportional to the area. And if you make this determinant equal to 0, that means the area is 0, right? So, if you plot determinant versus reflux ratio, you will get a plot like this. So, you will have positive area if the reflux ratio is less than the minimum reflux ratio, right? You will have negative area. Now, negative area is a bit regular, okay? But then determinant would be negative rather, not area because you are in this particular region, okay? It depends on where you put this E1 and E2. If you say E2, E1, then it will be positive anyway. Area is always positive, but determinant will change the sign depending on how you define it. So, it will be negative, right? So, it will cross this 0 and when the area becomes 0 or determinant becomes 0, your reflux ratio is equal to the minimum reflux ratio. So, that is the general procedure. Now, suppose I have four component system, five component system, six component system, how do I extend it? I just change it. Suppose you have four component system, instead of area, what will you have? You will have volume, okay? Because there will be another vector. There will be two feed pinches, okay? Or two saddles rather, two saddles, okay? I will have another point and I will have volume here, right? So, you will have determinant E1, E2, E3. There are three vectors, right? And suppose you have five component system, you will have four vectors, five vectors. But the calculation of determinant is not difficult, right? Okay? Here it is 2 by 2 determinant. Then it will be 3 by 3. Then it will be 4 by 4 depending on number of components you have. You just calculate the magnitude of determinant and plot it against reflux ratio and you will get a minimum reflux ratio. So, I have extended the concept to a multi-component system, right? I think we can take a break at this moment, okay? And later on we will see how to do the design of distillation column based on the minimum reflux ratio.