In general, separation by distillation can be accomplished in two ways – by adding energy in the form of heat or by adding stages to a distillation column. However, there are limits to what can be done. For example, we do not usually build distillation columns that are in excess of 300 ft. Thus for something like a C3 splitter, which has almost 200 stages, height is a real consideration. There is also generally a minimum amount of energy that is required to accomplish any separation.

Introductory Concepts. In this tutorial, we discuss concepts such as

1. Pinch points.
2. Minimum reflux and boil-up ratios.
3. Maximum number of stages.
4. Total reflux and minimum number of stages.

all of which relate to energy requirements in distillation

3.1 Pinch Points. The easiest way to understand energy consideration in distillation is graphically using a McCabe-Thiele plot. Figure 3.1 shows the McCabe-Thiele plot for the C2 splitter from Tutorial II. For the given reflux ratio of r = 3.9, there 16 stages are required to produce product compositions x_B = 0.05 and x_D = 0.99.

Trading Stages for Energy. To understand the idea of trading stages for energy, we reduce the reflux ratio from r = 3.9 to r = 3.0. Using the same saturated liquid feed and desired product compositions, we re-calculate the number of stages required and this number turns out to be 20. See Fig. 3.2. However, note by reducing the reflux ratio, we have also reduced the stripping ratio s from s = 4.5 to s = 3.75 because

However, reducing the reflux and boil-up ratios also reduces the cooling and heating requirements for the condenser and reboiler respectively since r = L/D and s = V/B. To understand why, note that if F, x_F , x_D, and x_B are fixed (which they are), then B and D are fixed by total mass balance for the distillation column. Reducing r means L, the amount of reflux is reduced, and reducing s means the amount of vapor boil-up, V, is also reduced. Smaller amounts of reflux and vapor boil-up require smaller amounts of energy since

Quick Exercise. Compare the values of Q_R and Q_C for the C2 splitter for reflux ratios of r = 3.9 and r = 3.0.

Minimum Energy Requirements. Now there is a limit to which you can trade stages for energy and this can be illustrated very easily using a McCabe-Thiele diagram. Using the C2 splitter as an illustration, we draw the line connecting the distillate composition, x_D, to the intersection of the feed line with the equilibrium line. Now if we step off stages, an infinite number of rectifying stages are required to reach the feed composition. This is called a feed pinch point. Since the slope of this rectifying line is r/(r+1), the value of r that actually gives the slope corresponding to an infinite number of stages is called the minimum reflux, r_min. Also note that if we choose a value of r < r_min, then we will never reach the feed stage and the desired separation is not possible.

If we now connect the feed pinch point to the bottoms composition, x_B, we also see that there are an infinite number of stripping stages required to go from x_F to x_B or vice versa. Here the slope of the stripping line is (s+1)/s and the value of s that actually gives this stripping line corresponding to an infinite number of stages is called the minimum boil-up ratio, s_min. We can also calculate s_min from Eq. 3.1 and the value of r_min. All of this is illustrated in Fig. 3.3.

Quick Exercises. Answer the following.

1. Calculate the values of r_min, s_min, Q_R, and Q_C for the feed pinch shown in Fig. 3.3.
2. Compare these values to the values for r = 3.9 and r = 3.0.
3. Show that the feed pinch does, in fact, give the minimum energy requirements.

Practice Exercises. For a C3 splitter at 200 psia with a feed consisting of a 50-50 mol % mixture of saturated vapor and a flow rate of 100 lbmol/h, x_D = 0.99 and x_B = 0.01, answer the following.

1. Calculate
   
   1. Minimum reflux ratio.
   2. Minimum reboil ratio.
   3. The heat duty of the reboiler.
   4. The cooling requirements for the condenser.
2. Show that the values obtained in 1 are less than those obtained in your solution from Tutorial II.

Total Reflux and the Minimum Number of Stages. An infinite number of stages results in the smallest amount of energy required for a desired separation. Conversely, if we withdraw no distillate product, we have a situation called total reflux and total reflux corresponds to the fewest number of stages required to make a desired separation. To see this, note that withdrawing no distillate product is equivalent to saying that D = 0. However, since r = L/D, r must be infinite and the slope of the rectifying line is r/(r+1) = 1. Thus, at total reflux the rectifying line corresponds to the 45-degree line on a McCabe-Thiele plot. In a similar manner, if we withdraw no bottoms product, we have a situation called total reboil and here s = V/B is infinite if B = 0. Thus the slope of the stripping line is (s+1)/s = 1 and the stripping line corresponds to the 45-degree line at total reboil.

Stepping off stages on the McCabe-Thiele plot, we easily see that this will give the fewest number of stages to go from x_D to x_B. See Fig. 3.4.

Total reflux provides a lower limit on the number of stages when considering the design of a distillation column. It is also used when first starting up a distillation column – so it is a practical part of distillation operation.

Practice Exercise. Once again, consider the C3 splitter and assume that all conditions specified in the previous practice exercise hold. Answer the following

1. Calculate the minimum number of stages required for the desired separation.
2. Calculate the heating and cooling requirements.

Tangent Pinch Points. In binary distillation, there is another type of pinch point that can occur. It's called a tangent pinch point. To illustrate a tangent pinch and its effect on minimum energy requirements, consider the distillation of acetone and water at atmospheric pressure and the yx diagram shown in Fig. 3.5. For this illustration we have specified x_F = 0.2, x_D = 0.96, and x_B = 0.01.

The first thing to notice about the yx diagram is that there is a dip or inflection in the equilibrium curve. Raoult's law does not predict this behavior – so you should understand that the way we have calculated K-values and the equilibrium for acetone and water is by using a more complicated expression for the K-values. While you need not be concerned with these details it is important that you realize that it is this inflection that gives rise to the tangent pinch point shown in Fig. 3.5, which occurs at a composition of x = 0.89475. The reason this is called a tangent pinch point is because the rectifying line is tangent to the equilibrium curve at the pinch point.

From the perspective of operating this distillation column, the presence of the tangent pinch point means that the minimum reflux ratio is much higher than expected. By this we mean that minimum reflux ratio predicted by a feed pinch point would not be sufficient to make the desired separation.

Practice Exercises. For the acetone-water distillation at atmospheric pressure, calculate

1. The boil-up ratio predicted by feed pinch.
2. The corresponding reflux ratio from Eq. 3.1.
3. The distillate composition that results from this value of reflux ratio.
4. Compare the results in part 3 to the desired distillate composition.

Summary Exercises. To get some practice with the material learned in this tutorial, consider the distillation of acetone and water at atmospheric pressure. Let F = 100 lbmol/h of saturated liquid, x_F = 0.2, x_D = 0.98, and x_B = 0.01. Answer the following by calculating

1. The minimum number of stages required for the desired separation.
2. The minimum reflux ratio.
3. The minimum reboil ratio.
4. The minimum heat requirements.
5. The minimum cooling requirements.
6. What is the largest value of x_D that can be reached from a feed pinch?
7. What is the minimum reflux ratio for this feed pinch point?
8. How do the reflux ratios for the tangent and feed pinch points compare?
9. Why must the largest value of r be selected in order to make the desired separation with x_D = 0.98?
10. If a reasonable choice of actual reflux ratio is 1.2r_min, what is
1. The actual number of stages required for the desired separation?
2. The feed stage?
3. The actual boil-up ratio?
4. The vapor flow up the column?
5. The liquid flow down the column?
6. The heat requirements for the reboiler?
7. The cooling reqquirements for the condenser?
11. Repeat parts 1 through 10 for the atmospheric distillation of methanol and water where x_F = 0.4, x_D = 0.99, and x_B = 0.01. To do this you will need the yx data given below.

Equilibrium Data for Methanol-Water at Atmospheric Pressure

Self-Study Exercises. Many distillation columns have much more complicated structure than just one feed and two product streams. There are columns with side-stream products, multiple feeds, pump-arounds, divided wall columns, and so on. The following self-study exercises are intended to give you some experience with multiple feed columns. Therefore, consider a C2 splitter that is fed with two feed streams. One feed is a saturated liquid with a feed rate of 100 lbmol/h and feed composition of x_F = 0.3; the other feed that is saturated vapor with a fed rate of 50 lbmol/h and has a composition of x_F = 0.7. The purpose of this distillation column is to produce a distillate with composition x_D = 0.999 and a bottoms stream of composition x_B = 0.001. Answer the following.

1. What is the minimum number of stages required for this separation?
2. What is the minimum reflux ratio?
3. What is the minimum boil-up ratio?
4. What are the minimum heat and condenser duties?
5. If 1.2r_min is a reasonable design condition for operation, then how many stages are needed to perform the desired separation?
6. What are the feed tray locations?
7. What is the heat duty of the reboiler?
8. What is the condenser duty?
9. Write the equation for the operating line in each of the three sections of this column.
10. Explain why each pair of adjacent operating lines must meet at the appropriate feed composition.
11. Suppose both feed streams were fed to the middle tray in the column. What effect would this have on the energy requirements for this column?