Up to this point we have concentrated on binary distillation, the McCabe-Thiele method, and the accompanying equations that describe distillation design using CMO models. In this tutorial, we extend the equations from the McCabe-Thiele method and develop a graphical procedure for three component or ternary mixtures. We discuss in detail

1. Ternary or triangular diagrams and distillation lines.
2. Pinch points.
3. Minimum energy use and stripping line distance.

4.1 Ternary Diagrams and Distillation Lines. In the McCabe-Thiele method we can plot mass balance (or operating lines) and equilibrium (or yx) information on a single diagram using one vapor and one liquid composition. Unfortunately, for three component or ternary mixtures, we cannot plot all of the information describing phase equilibrium and mass balance as compactly because we simply have too many variables – two liquid compositions and two vapor compositions. Remember the third composition in each phase can be calculated from x₃ = 1 – x₁ – x₂ or y₃ = 1 – y₁ – y₂. So we must find another way of representing things graphically. Figure 4.1 shows one way of doing this for three component mixtures on what is called a triangular or ternary diagram.

As you can see, figures like Fig. 4.1 can be confusing at first because they contain a lot of direct as well as hidden information. However, with patience, these ternary diagrams are easy to understand. Let's go through the information contained in Fig. 4.1.

1. Any point in the diagram represents a liquid composition expressed in terms of x₁ and x₂. That is, we can look at the horizontal and vertical axes as a typical xy plot like the ones you learned to do in high school – only we use x₁ and x₂ instead of x and y. In this particular case, x₁ = x (chloroform) and x₂ = x (acetone) and remember x₃ = x (benzene) = 1 – x (chloroform) – x (acetone).
2. The points denoted by F, B, and D represent the feed, bottoms, and distillate compositions respectively. The straight line that connects the points B, F and D is the overall mass balance for the distillation column.
3. The primary curve of interest in Fig. 4.1 is called a distillation line and is made up of two parts. It begins at B and contains the small filled squares. Each little filled square represents the liquid composition on a stage in the distillation column. In this example, the black curve that moves right and then up toward the point F is the stripping section. Also, there are many (300) stages in the stripping section. The second part of the distillation line that runs from the point F to D represents the rectifying section of the column.
4. We could also plot y₁ and y₂ as a separate set of curves representing the vapor composition on the stripping and rectifying stages in the column. The vapor in equilibrium with liquid on any stage can be identified by connecting these equilibrium compositions by a straight line called a tie line. Generally vapor phase information is not included in diagrams like Fig. 4.1.

Equations Used to Generate Distillation Lines. The equations needed to generate the two portions of the distillation lines shown in Fig. 4.1 are simple modifications of the equations we used in the McCabe-Thiele method. For the stripping section, we have the mass balance (or stripping) equation

which is just a rearrangement of Eq. 2.6 from Tutorial II. Remember j is a stage index that runs from 1 to N and the stages are numbered from bottom to top. For the rectifying section, the governing mass balance (or rectifying) equation is

It is important to understand that for a ternary mixture we actually have two of equation 4.1 and two of equation 4.2 – one for component 1 (chloroform) and one for component 2 (acetone). It is also important to remember that these equations contain the CMO assumption – just like the McCabe-Thiele equations.

So how do we use Eqs. 4.1 and 4.2 to 'generate' a distillation line? Well, we need to do this on a computer (because it involves a lot of calculations) and we also need to include what we know about bubble point calculations (in order to get y values to use in Eqs. 4.1 and 4.2). Here are the steps you would need to use.

1. Starting at x₁ = x_B, calculate the bubble point temperature and the values of y₁ at the bubble point. The composition y₁ is the composition of the vapor stream leaving the reboiler or stage 1.
2. Pick a value of s and use x_B and y₁ to calculate x₂ (the liquid composition on stage 2) from Eq. 4.1. Remember you need to do this for components 1 and 2.
3. Using x₂ calculate the bubble point temperature and values of y₂ at the bubble point, where y₂ is the vapor composition leaving stage 2.
4. Using the same value of s, x₂, and y₂ calculate x₃ (the liquid composition on stage 3) from Eq. 4.1. Remember you need to do this for components 1 and 2.
5. Repeat steps 3 and 4 for as many stripping stages as required.
6. Calculate the reflux ratio r = (s – q + 1) [x_B – x_F] / [x_F – x_D] – q, which is just a rearrangement of the equation s = (r + q)[x_F – x_D] / [x_B – x_F] + q – 1 from Tutorial III.
7. Using r, x_D, and y (say y_f ) for the last stripping stage, calculate the liquid composition, x_f +1, for the first stage in the rectifying section using Eq. 4.2.
8. Using x_{f +1}, calculate the bubble point temperature and vapor composition, y_{f +1}.
9. Using r, x_{f +1}, and y_{f +1}, calculate x_{f +2} from Eq. 4.2.
10. Repeat steps 8 and 9 for as many rectifying stages as desired.

Practice Exercises. Answer the following:

1. Equations 4.1 and 4.2 along with bubble point calculations can actually be used to generate a McCabe-Thiele plot. See if you can figure out how to do that numerically.
2. Generate distillation lines for a mixture of n-propane, n-butane and n-pentane at 300 psia assuming that the liquid and vapor phase behave according to the following.
   
   

   yi = Ki xi where Ki = exp[ ln(pci / p) + 5.37 (1 + wi) (1 – Tci / T)] and where pci, Tci, and wi are the critical pressure, critical temperature, and acentric factor, respectively.

   
   
   Let the feed, bottoms, and distillate compositions be
   
   

   Component	Feed	Bottoms	Distillate
   n-propane	0.45	0.00001	0.99
   n-butane	0.30	0.54	0.007
   n-pentane	0.25	0.46	0.003

   
   
   
3. Pick three distinct values of s and generate the corresponding distillation lines. Plot these distillation lines on a triangular diagram.

Several Distillation Lines and the Design of Ternary Distillation Columns. The design of any ternary distillation column is, in many ways, similar to the design of binary distillation columns. We are usually given a feed and have some desired separation that we want to make. This desired separation could be driven by economics or by processing constraints. We also generally want to pay attention to energy use. Also, in the McCabe-Thiele method you saw that you can generally find many distillation columns that perform a desired separation and that the differences between these different distillation columns was the number of stages and the reflux (and reboil) ratio. Here again, things are very similar. In the distillation of ternary mixtures, we can usually find many distillation columns that will make the same desired separation. Therefore, we often generate several distillation lines and compare their performance, number of stages and reboil and reflux ratios, to decide which is the design that meets our needs best. Figure 4.2 shows three distillation lines for the separation of a mixture of chloroform, acetone, and benzene by atmospheric distillation. The goal of this separation is to produce relatively pure acetone as the distillate product (i.e. x_D (acetone) ≥ 0.99).

Each of the three distillation columns represented in Fig. 4.2 produces a distillate product that is highly concentrated in acetone. The significant differences between each of these distillation columns are the purity of the overhead product, the number of stages in the rectifying section, and the reflux and boil-up ratios. As you move from left to right, the purity of the distillate product increases slightly, the number of stages in the rectifying section increases, and the boil-up and reflux ratios decrease. These differences can be small or large, depending on the situation.

The table below illustrates that the distillation columns shown in Fig. 4.2 actually show small differences in reflux and boil-up ratio but, larger differences in the number of stages and liquid stage compositions in the rectifying section of the column.

The numbers in this table correspond to the distillation lines in Fig. 4.2 as you move from left to right and notice that the one furthest to the left actually does not meet the desired acetone concentration in the distillate.

Practice Exercises. Repeat the previous practice exercises paying more careful attention to the specifications for the distillate and the bottoms product compositions and determine values of s that meet or exceed these product specifications.

4.2 Pinch Points and Pinch Point Curves. Just as we learned how to determine feed and tangent pinch points in the McCabe-Thiele method, it is possible to represent and find feed and tangent pinch points in distillations of ternary mixtures. There is also a third type of pinch point in three component mixtures, called a saddle pinch point. Figure 4.3 shows both a feed and saddle pinch point for the atmospheric distillation of chloroform, acetone, and benzene.

Each of the three distillation lines in Fig. 4.3 exhibits a feed pinch while the red curve also exhibits a saddle pinch. The saddle pinch point in Fig. 4.3 is the point where the red curve takes an abrupt turn along the hypotenuse of the triangle toward the acetone corner. These pinch points are recognized by the fact that the liquid composition on many adjacent stages changes very, very little. We illustrate this by showing the liquid compositions on two sections of the red curve – around the feed stage and in the neighborhood of the saddle pinch point.

The last 10 stages in the stripping section have the following liquid compositions, which identify the feed pinch point in the stripping section of the column.

Note that the feed pinch point occurs at (x₁, x₂) = (0.129997082392705, 0.187132126994375) and is not equal to the feed composition, which is x_F = (0.11, 0.17)!!!

Stages in the middle of the rectifying section in the neighborhood of the saddle pinch have the following liquid compositions.

By adjusting the boil-up ratio to higher precision, we can actually force many more stages into the neighborhood of the saddle pinch point.

Figure 4.3 also contains a dashed curve showing all potential feed pinch points in the stripping section as a function of the boil-up ratio for a fixed value of the bottoms composition, x_B. This dashed curve is called a stripping pinch point curve and is made up of a number of compositions called pinch points. The stripping pinch point curve is easily computed by setting the number of stripping stages to a high value (say 300) and then simply using Eq. 4.1 and bubble point calculations over and over again. The resulting composition at stage 300 is the pinch point. Any feed pinch point in the stripping section must land on the stripping pinch point curve. There is also a rectifying pinch point curve (not shown in Fig. 4.3) that can be used to identify potential feed pinches in the rectifying section.

Practice Exercises. Using the same practice exercise example of the distillation of propane, butane, and pentane, do the following.

1. Find the stripping pinch point composition for three different values of boil-up ratio.
2. Find the stripping pinch point curve.
3. Plot the stripping pinch point curve along with the distillation lines from the previous practice exercise on a triangular diagram.

Minimum Energy Requirements and Stripping Line Distance. Remember that we used pinch points in the McCabe-Thiele method to determine minimum energy requirements for binary distillations. The great utility of finding pinch point curves is that it provides information that helps us find minimum energy requirements for some ternary distillations. However, there is more to it than that.

Feed Pinch Points. In the McCabe-Thiele method there is only one point at which a feed pinch can occur – the feed composition! So in binary distillation, once you know where the feed line intersects the equilibrium line, you know the feed pinch point composition. From there it is easy to find minimum reflux and boil-up ratios. On the other hand, as we illustrated for the distillation of chloroform, acetone and benzene, feed pinch points do not have to occur at the feed composition – and usually don't. The fact that we have a stripping pinch point curve tells us that there are many, many potential feed pinch points. At this point it may be beneficial for you to study Fig. 4.3 and to pay careful attention to the fact that all three of the distillation lines shown in that figure exhibit a feed pinch! Determining the boil-up ratio that still makes the desired separation and corresponds to minimum energy use requires some additional work.

One way to determine minimum energy use is by trial and error. For example, you could

1. Pick a value of the stripping ratio, s.
2. Calculate a stripping pinch point composition for that value of s. This generates the stripping portion of a distillation line with a pinch.
3. Calculate the corresponding reflux ratio, r.
4. Generate the rectifying section and see if the top stage of the rectifying portion of the distillation line meets the desired distillate composition.
5. Repeat steps 1 through 4 until you find the smallest value of s that gives the desired separation.

However, we have discovered a more straightforward way of doing this automatically. It still involves trial and error but the trial and error can be done systematically using a computer and by searching for something rather simple – the shortest stripping line distance. Figure 4.4 shows the connection between stripping line distance and energy use for the illustrative example of the distillation of chloroform, acetone and benzene.

The table below gives some numerical values that coincide with the three distillation lines shown in Fig. 4.4.

It is also interesting to note from this table that the acetone concentration in the distillate actually increases as boil-up ratio decreases. That seems weird, but it's true. It's an example of what is called reverse separation.

Practice Exercises. For the propane, butane, and pentane separation in this tutorial, determine the following.

1. Minimum boil-up ratio.
2. Minimum reflux-ratio.
3. Minimum reboiler duty.
4. Minimum condenser duty.
5. Is the pinch point corresponding to minimum energy use a feed or saddle pinch point? Why?
6. What is the minimum stripping line distance and does it correspond to the minimum boil-up ratio?

Total Reflux and the Minimum Number of Stages. Here again, as in the McCabe-Thiele method, the minimum number of stages required for a separation can be easily determined by determining distillation lines at total reflux. To do this, you need only set the boil-up ratio to a very large number (e.g. s = 1 x 10²⁰⁰), use Eq. 4.1, and count the number of stages. However, you also need to pay careful attention to determine if the desired separation has been made.

Practice Exercise. Determine the minimum number of stages for the propane, butane, and pentane separator. Can the desired separation be made?

Summary Exercises. Consider the distillation of methanol(1), ethanol(2) and n-propanol(3) at atmospheric pressure. Assume that the liquid and vapor phases are ideal and therefore obey Raoult's law. Let the feed consist of 200 lbmol/h of saturated liquid with the composition x_F = (0.33, 0.33, 0.34). It is desirable to remove methanol as the distillate product and a mixture of ethanol and n-propanol in the bottoms. Let x_D = (0.99, 0.007, 0.003) and x_B = (0.001, 0.493, 0.506). Answer the following:

1. What are the distillate and bottoms flow rates?
2. Determine the minimum number of stages needed to make the desired separation.
3. Determine the minimum boil-up ratio, corresponding reflux ratio, minimum reboiler duty, and condenser duty using the concept of shortest stripping line distance that satisfies x_D (methanol) ≥ 0.99.
4. What is the pinch point composition corresponding to minimum boil-up ratio?
5. Is this a feed or a saddle pinch. Why?
6. If it is a feed pinch point, is the feed pinch point composition equal to the feed composition? Why or why not?
7. Show that your solution to question 2 corresponds to minimum energy by generating distillation lines for various values of s – for both s > s_min and s < s_min.
8. Explain the difference between the distillation lines generated in question 7 in terms of whether or not they meet the distillate composition specification for methanol.
9. If s = 1.2 _min is a reasonable design criterion for an actual distillation column, determine the actual number of stages in this methanol recovery column. How does the actual number of stages compare to the minimum? Does this make sense?
10. How many stages are there in the stripping section and rectifying section of the design in question 9? What is the feed stage?
11. What is the actual amount of energy used in the reboiler?
12. What are the corresponding reflux ratio and condenser duty?