 Let us spend some time on the modeling and simulation of distillation processes. After having gone through the basic aspects, now we understand what really is contained in a distillation column, it is nothing but a cascade of flashes ok, cascade of flashes. Of course, we have to make provision for the complex column which I mentioned to you for side draws, multiple feeds so on and so forth. Now, there are two categories in which one can classify the models essentially. The first one I have called them as the shortcut methods, even in simulators they are known as shortcut techniques or shortcut methods and the second is the rigorous methods. There are various ways shortcut methods have been written and what I am going to share with you is those shortcut methods which became popular over the years and have found place in industrial simulators. By enlarge the procedures are same in all the simulators, some minor variations from one simulator to other simulator may be seen. So, one very popular shortcut method which has been around for more than several decades now 5, 6 decades is the method of Fensky, Underwood and Gilliland and when I talk about the shortcut methods the focus by default is multi-component distillation because if it is binary distillation you have very good graphical procedures. So, we are not interested in binary systems here, we are interested in multi-component system that is where simulation plays significant role. So, what is this Fensky Underwood Gilliland method, what does it do? Fensky is I mean due to Fensky you can construct what is known as a Fensky column, a Fensky column is that column in which no product is drawn either at the top or at the bottom. Now, if no product is drawn at the top or at the bottom which means if it is a column at its steady state then there cannot be any feed alright. So, you can start a column and you can bring it to the steady state and then go on increasing the reflux, go on increasing the reflux, go on decreasing the withdrawal of the distillate and go on reducing the withdrawal of the bottoms and go on cutting on the feed also. So, I will have a distillation column in which internal circulation is occurring, vapor is rising, liquid is coming down, heat is being added in the reboiler, heat is being removed from the condenser, but whatever is drawn at the top is reversed it is put back into the system and whatever reaches the bottom is totally vaporized. So, which means that it will be a total refluxed column, a total refluxed column. So, that will be known as or that will be called as the Fensky column. What does Fensky column give us? The Fensky column tells us that if the reflux was total which means reflux ratio is infinity, you are not withdrawing anything, reflux ratio is L over D, D is 0. So, the reflux ratio is infinity, it is a total reflux column and therefore, the corresponding number of stages for the desired separation. So, we do analyze the top and bottom product and try to find out what is its composition. So, if the composition required is 99 percent of the light key at the top, 99 percent of the heavy key at the bottom in terms of pureties, so that is the kind of composition you have in mind. So, Fensky will tell you what is the minimum number of stages which are required to carry out that separation that is the Fensky column. It can be constructed, it can be operated, it can be done in the laboratory. So, and if you write down the equations, you can find out everything about Fensky column. So, based on this concept that a total reflux column will give you minimum number of stages, Fensky derived expression in terms of the compositions of the light key and the heavy key which essentially define the degree of separation in a multi-component distillation and that is related to the minimum number of trays. Of course, from thermodynamics viewpoint you need to know the relative volatility. So, one assumption which Fensky's formula makes that the relative volatility remains constant across the column from top to bottom. Now, when I talk about relative volatility, the focus will be on the key component, the concept of key component is that let us say if I have propane, butane, pentane and it is a multi-component system, it is a ternary system and I need to recover most of my propane at the top and let us say most of the butane and onwards should go to the bottom. If that is the objective then propane is my light key and butane is my heavy key. If butane is heavy key then pentane automatically will go at the bottom because that is heavier than the heavy key. So, in this case the two keys will be adjacent, the light key and the heavy key. Light key is that key which is desired at the top in the distillate, heavy key is that key which is desired in the bottom's product. I can also have a situation where propane is desired at the top as much as possible, propane is desired at the bottom and I do not care where butane goes, butane can get distributed in between. So, then butane will be called an intermediate key or middle key, the light key will be propane, the heavy key will be pentane. So, it is up to us to define what is the key. So, once the keys are known you can calculate average relative volatility. Why average because the temperature from top of the column to bottom of the column is going to change, the compositions are going to change and in thermodynamic session yesterday we had seen that k value if you recall was a function of temperature, pressure, the liquid composition and the vapor composition. So, if the compositions are different pressures are going to be different because there is a pressure drop across the column, temperatures are going to be different because top of the column is colder than the bottom of the column and therefore, you have to work with some average value. So, we know the values will not be same, so we average them out and there are various ways to average them out. The recommended procedure normally is that the calculate alpha at the top, calculate alpha at the bottom and take the geometric mean. Sometimes the feed condition also is given equal importance and therefore, alpha at the top multiplied by alpha at the bottom multiplied by alpha at the feed and take the cube root. So, it is totally depends upon how much weightage you want to give to which section of the distillation column. So, that is the Fenske column. I have not written the equation here, it is available in any good book on multi-component distillation. The second component of this shortcut method is underwood. By the way, these are three procedures established by these three researchers independently. And so, this is a collection of three procedures which is typically called Fenske-Underwood Gilliland method and it should not be understood that these three people together worked and they came out with a method, no that is not true. They have worked at different times. Underwood again is a theoretically sound procedure which tells you that what is the minimum reflux required for a given separation. So, Fenske on one side is telling us total reflux, what is the minimum number of stages. Underwood allows you to calculate, it is an iterative process, it allows you to calculate what is the minimum reflux which is required for given separation. Now, if you recall in binary systems, McCabe-Thiel diagram when we construct, so we have an operating line for rectification zone and we have an operating line for the stripping zone. And the point at which these two lines intersect that is where your feed line passes through is that right and that is the pinch point that is where the driving forces are minimum. In multi-component systems, there are normally two pinches, one above the feed and one below the feed because you have two keys. So, it is not a single pinch like in binary system, you have single pinch. In multi-component, you have two pinch points, one occurs just above the pinch, one just below the pinch. So, Underwood equation enables us to calculate what is the minimum reflux which is required for the given separation which means it should hit the pinch point, it should hit the pinch point. Gilliland correlation which was originally presented by Gilliland in the form of a graph, a logarithmic plot rager. Later on it has been converted into a correlation, so we call this as Gilliland correlation is a relationship among minimum reflux, minimum number of trays, actual reflux and actual number of trays. So, there are four parameters, minimum reflux, minimum number of trays, actual reflux and actual number of trays. Out of these four, Fensky allows me to calculate minimum number of trays, Underwood allows me to calculate minimum reflux. So, which means I have a correlation which now relates to parameters, the actual number of trays and actual reflux. So, this is where the design starts now. If you decide the reflux ratio at which you want to design the column then you can calculate the number of trays or you decide how many trays you want to put in the column, obviously those number of trays should be more than the minimum required then you can calculate at what reflux the column should operate. So, Gilliland correlation will allow you to calculate actual reflux for given number of trays or actual number of trays for given reflux, it is a correlation. Now, based on the experience you will find in literature that the recommended refluxes for simple separations, most of these thumb rules are for simple separations. So, actual refluxes are kept between 1.2, 1.25 to 1.5 times the R minimum. And the guideline on number of stages that number of stages normally lie between 2 times N minimum to 3 times N minimum, so the choice is yours, how you want to configure the column. So, we are not doing calculations for any internal traffic, we are assuming relative volatility can be averaged out, we use Fenske method for minimum number of trays, underwood for minimum reflux, Gilliland for relationship between actual reflux and actual number of trays and the column is configured. There are related techniques to establish the location of feed tray, one method uses the Fenske itself between feed and the top product, there is another method which is popularly known as the Kirkbride equation which enables you to calculate the feed location. So, short cut distillation design using this method, now win underwood Gilliland, win is a slightly modified version of Fenske and I have written win here because you people are doing experiments with Aspen plus and Aspen plus uses win underwood Gilliland, win is not too different from Fenske, slight modifications have been done to the Fenske's approach. So, whether we are using win underwood Gilliland or Fenske underwood Gilliland conceptually we are doing the same thing. So, what does it do this short cut, it determines minimum reflux ratio, it determines minimum number of stages and either actual reflux ratio or actual number of stages, one of the two as I mentioned to you. And it is applicable normally to columns with one feed and two product streams, simple columns. There is another short cut method which is fairly popular, but not as popular as the Fenske underwood Gilliland and this is due to admister, so popularly known as the admisters method. Administer method, if I contrast with Fenske underwood Gilliland, Fenske underwood Gilliland is a design method, it enables us to configure the column. For example, we said that if I want to operate the column at 1.5 times r minimum, I can calculate number of trays, then I can configure the column. Administer method on the other hand is a rating method, it is a performance evaluation method, it is a simulation method. The number of trays should be known, column should be configured and that it enables you to calculate the performance of the column. So, it determines separation based on reflux ratio, number of stages that is very important, number of stages should be known and distillate to feed ratio, D by F ratio should be known. Again it is applicable normally to columns with one feed and two products and this method is also available in Aspen plus. As I mentioned that the distillation columns can have fairly complex configurations, so there are short cut methods which have been developed over the years and are available in commercial simulators which enable calculations to be done or columns to be configured for complex calculations like petroleum fractionation units. So, these are also short cut methods and they again use average values of relative volatilities but it is segmented, it is put in different segments of the column because the column is not divided only in two segments, the conventional column is divided only in two segments, the rectification zone and the stripping zone but when you have a refinery column depending upon the number of side draws, you have various segments, maybe from NAFTA to kerosene one segment, kerosene to diesel another segment, diesel to gas oil another segment. So, for each segment you can apply short cut methods, so this will be a multi product short cut calculation. So, determines product composition and flow, number of stages per section or segment as I am saying and heat duty using fractionation indices. These columns or these procedures are normally used for generating approximate designs for crude columns, vacuum columns, FCC fractionators so on and so forth. So, they find application primarily in refining industry. So, I have described to you two or three different types of short cut methods. Now, they are good to get the design started, something is now available on the paper but short cut methods are available are suitable only for preliminary design studies. What do we mean by preliminary design study? A preliminary design study is primarily for feasibility or maybe for preliminary cost estimation and preliminary cost estimation as you know is that cost estimation in which the costs are expected to be in error up to plus minus 30 percent, typical value is plus minus 30 percent. You want to improve on 30 percent, you want to come down to 20 percent the short cut methods are no good. Why? Because they ignore the detailed internal traffic, they approximate the thermodynamic properties. For example, we have said average relative volatility, relative volatility changes with temperature, changes with pressure, changes with composition. So if you really want to design a real world distillation column, you need to do better than this and go for rigorous methods. But short cut methods have their own importance because they take very small effort or very little effort to have a design. So something is there, I mean you know that column looks like this approximately there will be so many trace. But what is happening inside and then you have to calculate the diameters, for diameter calculation you require the internal traffic. You need to know the vapor loads, you need to know the liquid loads. So you need to go through rigorous modeling. So for final design, now when I say final we are going to improve from the preliminary design. We want to bring the cost estimates to less than plus minus 20 percent or maybe preferably to less than plus minus 10 percent. So designs have to be improved. So for final design of a multi-component distillation column, rigorous determination of temperatures, pressures, stream flow rates at each stage is required. And therefore, you require a mathematical model which will enable you to do this particular calculation. So problem is going to get a very large dimension problem. What does this imply? This implies that we need to take into account all the known processes which are occurring within the distillation column from chemical engineering viewpoint. So we need to balance the material, so material balances in terms of total balance, in terms of component balances we have to write. We have to write the energy balance on every tray. Every tray has to be kept in energy balance. We may work with equilibrium assumption and later on relax that also. So if we work with equilibrium assumption then for every IS species equilibrium is to be written between the vapor composition and the liquid composition, so y is equal to kx. So ks will come into picture and it is not only 1 k or 2 k they will be as many as number of species. So if you have a 10 component system there are going to be 10 ks on one tray and if I have 100 trays then I have that many ks, I have 1000 ks to worry about. So equilibrium relations have to be brought into picture. If you want to further make it more rigorous we have to worry about the hydrodynamic aspects and bring into picture the pressure drops. We also have to bring into account the efficiency into picture because trays are not going to be 100 percent efficient. And therefore the efficiency is to be known, efficiency is something which will depend upon the extent of heat and mass transfer which you get on the tray or on the on a given section of your packing. So the rigor is totally controlled by us. We can decide how much more we want to go deep into this calculation. Typically now what happens is though it is desirable to have very rigorous models where we can have momentum balance, the heat or the energy balance, the material balance, the thermodynamic calculations everything we can throw into, but the problem dimensionality keeps on increasing and we do not want to lose the physics of the process. So we would like to be rigorous at the same time we would like to be rigorous enough so that it serves our purpose and not get into the complications of too many variables which mess up our whole numerical system. So the rigorous methods which we are going to talk about in this particular lecture the focus is only on the study state. Study state models are good for design purposes, for simulation purposes. They will not be very useful for control systems of course some applications can be seen for control systems also, but there you require dynamic analysis. So we are going to work with study state models and we are going to work with models where material balance will be completely characterized, energy balance or which will reduce to enthalpy balance finally and I will show you how that will be completely characterized. We will pay attention, sufficient attention to the thermodynamic support which is required in terms of equilibrium constants, enthalpies and so on and so forth. And this itself will result in a very large non-linear algebraic equation set and therefore we will then require robust numerical technique to find the solution to this kind of set of non-linear algebraic equations. We may not take into account the hydrodynamics at this stage because if you throw in the momentum balance and take into account the hydrodynamics which means in a way you are taking into account the internals also in picture. Why internals can be kept away at this juncture? Because we are assuming that it is an equilibrium stage process, how is the equilibrium reached that we are not focusing attention on whether it is a tray or whether it is part of the packing. It can be done, I do not rule out it can be done, but our process will become very much dependent on the internals so we have to characterize the internal that is one thing the model will not remain very general, model will become very specific and at the same time it will become more complex. So, typically in simulators what is done is that equilibrium is assumed, calculations are carried out once you generate internal traffic then you go to the internals and try to find out what is the kind of pressure drop which is expected or what is the kind of efficiency which you are going to get out of a tray. And if required corrections are significant you can always make those corrections on the pressure profile and do another simulation. So, in our analysis we are going to assume that the pressure is a loaded variable, pressure is a specified variable we are not going to calculate the pressures because we are not going to do any momentum calculations. Now there are various ways these equations can be solved and as I said that we require reliable and efficient numerical strategies. There are two issues just like in any numerical technique which we demand. One is that the procedure should be such that it goes through the execution very fast and it should give correct solution. So, speed is of concern and the accuracy of results of course we would like to have. So, we should have a good convergence behavior that is the purpose one of the purposes we have in mind. The so the convergence behavior should be good. The second is that it should be robust. By robustness we mean that even if user has loaded approximate values because multi-component distillation is a very involved process and if I have a numerical set of equations and if I have to get the iteration started I need to have all sort of starting values, starting guesses. The procedure may not be competent enough to load all the guesses because the number of variables is too large and therefore, the procedure should be such that even if the guesses are poorly described or rough values are given it should be able to come to the light track or right path pretty soon which means degree of convergence or the probability of convergence should be very high, very seldom it should fail. So, convergence procedure has two aspects to it one which is the speed of convergence something to do with the time and so that it gives you the right answers in a reasonable amount of time. So, that is the speed of convergence and the other is the probability of convergence. We are looking for both. We are looking for a good fast method which has sufficient speed and we also are looking for a method where the probability of convergence is fairly high. So, what we are going to do is we are going to construct a general stage. The stage is labeled here as jth stage, stage J. This stage is such that it has provision for a feed, the flow rate is fj, zij means the composition of ith species it is a mole fraction at the jth tray. So, j is the tray index and i is the species index, component index. It has an enthalpy of hf, its temperature is Tf and its pressure is Pf and this pressure can be any pressure higher than the column pressure expected to be higher than the column pressure it typically will be higher. So, we put a valve here so that the feed is throttled and brought to the pressure of the stage. So, this pressure is the feed pressure it has nothing to do with the column pressure or the stage pressure. Now, this is a typical stage internal stage so it is receiving liquid from the stage above. You will find that in literature sometimes the numbering is done from the top of the column. So, condenser may be number 1, re-boiler may be n, sometimes numbering is done from the bottom. So, re-boiler is number 1 and condenser can be n, it really does not matter. In this particular case the convention we have used here is that numbering is done from the top. So, if that is the convention then stage j is receiving liquid from stage j minus 1 and it is supplying liquid stream to stage j plus 1. This is the standard convention that if it is stage j then liquid leaving the stage will be labeled j and vapor leaving the stage will also will be labeled j. This vapor and this liquid they are in equilibrium, they are in equilibrium. Now, if they are in equilibrium which means the temperature should be same the thermal equilibrium, the pressure should be same the mechanical equilibrium and the other condition which is not stated here which we will have to write in our equations that the fugacity of ith component in the vapor phase will be equal to the fugacity of ith component in the liquid phase and that is valid for all i's for all the components. Vapor is coming from the tray below which is j plus 1 and this vapor has to be at a pressure which is higher than the pressure of the stage. Why? Because the stage has a hold up, due to hold up there is a static head, liquid has static head and therefore, this vapor requires certain additional pressure additional force so that it can bubble through. It has to go through that height which is there in the form of the wear height wear height on the tray. So, therefore, that pressure additional pressure that can be shown through this valve. So, this is only a modeling aspect, there is no valve in a distillation column, there is no head like this in a distillation column right. This is just a modeling aspect, so what we are trying to show is on the liquid side there is a static head because of hold up so there will be a pressure loss, a pressure gain if you are coming on this side and on the vapor side there will be a pressure loss and that loss can be shown in the form of throttle because throttling simply means that everything is happening without any heat effects, adiabatic, Joule-Thamson, Joule-Thamson expansion. So, this is what this represents. So, it is going to be an equilibrium stage. On this stage we are making provision for heat transfer and the amount of quantum of heat transfer is Q, convention we have chosen is that if it is positive it is if it is plus then it is from stage, if it is negative it is to stage you can revert it. Everything is the sign in the equation will also have to be reverted accordingly it is just a convention does not matter. So, there is provision for heat transfer. We have made provision for drawing a vapor side stream assuming that this product may be required. We have also made provision for withdrawing a liquid side stream that product may be required. So, we have heat transfer I am sorry we have heat transfer, we have provision for the liquid head, we have provision for the vapor pressure drop, we have provision for feed to be higher at a different pressure, vapor side stream and liquid side stream. This we will call as a generalized stage. This is a generalized stage on which we would like to work. What are the variables of interest? The variables of interest will be I would like to know the composition of this vapor stream completely. So, that is contained in the variables y ij. I would like to know what is its enthalpy for energy balance. I would like to know what is its temperature and obviously, I would like to know pressure but as I said that in this particular modeling we are going to assume that pressure is a loaded variable. So, we are going to define or we are going to specify the pressure. So, pj will be specified, this will be a specified variable, tj will be a computed variable, hvj also will be a computed variable, yij also will be a computed variable. Similarly, on this side liquid composition will be computed, enthalpy will be computed, temperature will be computed, this pressure will be pre specified and the same argument holds for these variables. On the feed side everything is defined, the feed is defined, its flow rate is defined, the composition is defined, the enthalpy is defined, the temperature is defined, pressure is defined. So, we are going to assume that feeds are all the feeds, when I say all I simply mean because every j at stage has a feed and if I have n number of stages I have n feeds. So, all the feeds are fully characterized, all the properties of the feed are known. So, they are fully characterized. So, I am going to construct this cascade and finally, the vapor which is coming out the y will constitute the distillate composition, y will constitute the distillate composition, the temperature tj from the final stage tn or t1 rather that will be the distillate temperature, p will be preloaded. So, I am not saying anything about it, this will be the enthalpy of the distillate and on this side the temperature will be the bottoms temperature, the enthalpy of the bottom stream and this will be the composition of the bottoms. What will Q do? Q will tell us what are the heat loads on every tray and numbering goes from 1 to n. So, the first Q on the top will take care of condenser load and the last Q at the nth stage will take care of the reboiler load. So, this is the dimensionality. So, we are we then have to now write equations which will characterize the system and enable us to calculate all liquid compositions, all vapor compositions on every tray I am saying, the temperatures, the temperature profile, the vapor profile and the liquid profile, total vapor flow rate, total liquid flow rate and the temperature on every stream. And the energy balance will enable us to calculate the heat loads, the condenser load and the reboiler load.