## Design Degrees Of Freedom

The design of a distillation column involves many parameters: product compositions, product flowrates, operating pressure, total number of trays, feed tray location, reflux ratio, reboiler heat input, condenser heat removal, column diameter, and column height. Not all of these variables are independent, so a "degrees of freedom" analysis is useful in pinning down exactly how many independent variables can (and must) be specified to completely define the system.

A rigorous degrees-of-freedom analysis involves counting the number of variables in the system and subtracting the number of equations that describe the system. For a multi-component, multistage column this can involve hundreds, if not thousands, of variables and equations. Any error in counting is grossly amplified because we are taking the difference between two very large numbers. A simple intuitive approach is used below.

The normal situation in distillation design is that the feed conditions are given: flowrate F [moles per hour (mol/h)], composition zj (mole fraction component j), temperature TF and pressure PF. The desired compositions of the product streams are also typically known. We consider a two-product column, so the normal specifications are to set the heavy-key impurity in the distillate xD HK and the light-key impurity in the bottoms xBLK.

Distillation Design and Control Using Aspen™ Simulation, By William L. Luyben Copyright © 2006 John Wiley & Sons, Inc.

The design problem is to establish the operating pressure P, the total number of trays Nt, and the feed tray location NF that produces the desired product purities. All other parameters are then fixed. Therefore, the number of design degrees of freedom is 5: xD,HK, -%,lk, P, Nt, and Nf. So if the desired product purities and the pressure are given, there are 2 degrees of freedom.

Just to emphasis this point, the five variables that could be specified might be the distillate flowrate D, reflux ratio RR = R/D, P, NT, and NF. In this case the product compositions cannot be specified but depend on the distillate flowrate and reflux ratio selected.

The steps in the design procedure will be illustrated in subsequent chapters. Our purpose in this chapter is to discuss some of the ways to establish reasonable values of some of the parameters such as the number of stages or the reflux ratio.

### 2.2 BINARY McCABE-THIELE METHOD

The McCabe-Thiele method is a graphical approach that shows very nicely in pictorial form the effects of VLE, reflux ratio, and number of trays. It is limited to binary systems, but the effects of parameters can be extended to multicomponent systems. The basic effects can be summarized:

1. The easier the separation, the fewer trays required and the lower the required reflux ratio (lower energy consumption).

2. The higher the desired product purities, the more trays required. But the required reflux ratio does not increase significantly as product purities increase.

3. There is an engineering tradeoff between the number of trays and the reflux ratio. An infinite number of columns can be designed that produce exactly the same products but have different heights, different diameters, and different energy consumptions. Selecting the optimum column involves issues of both steady-state economics and dynamic controllability.

4. Minimum values of the number of trays (Nmin) and of the reflux ratio (RRmin) are required for a given separation.

All of these items can be visually demonstrated using the McCabe-Thiele method.

The distillation column considered is shown in Figure 2.1 with the various flows and composition indicated. We assume that the feed molar flowrate F and composition z are given. If the product compositions are specified, the molar flowrates of the two products D and B can be immediately calculated from the overall total molar balance and the overall component balance on the light component:

For the moment let us assume that the pressure has been specified, so the VLE is fixed. Let us also assume that the reflux ratio has been specified, so the reflux flowrate can be calculated R = (RR)(D). The "equimolal overflow" assumption is usually made in the McCabe-Thiele method. The liquid and vapor flowrates are assumed to be constant in a given section of the column. For example, the liquid flowrate in the rectifying section LR is equal to the reflux flowrate R. From an overall balance around the top of the column, the vapor flowrate in the rectifying section VR is equal to the reflux plus the distillate (VR = R + D).

This method uses an xy diagram whose coordinates are the mole fraction of the light component in the liquid x and the mole fraction of the light component in the vapor phase y. The VLE curve is plotted for the selected pressure. The 45° line is plotted. The specified product compositions xD and xB are located on the 45° line, as shown in Figure 2.2. Figure 2.2 McCabe-Thiele method: locate products and VLE. Figure 2.3 McCabe-Thiele method: draw operating lines.

Figure 2.3 McCabe-Thiele method: draw operating lines.

### 2.2.1 Operating Lines

Next the "rectifying operating line" (ROL) is drawn. This is a straight line with a slope equal to the ratio of the liquid and vapor flowrates in the rectifying section:

The line intersects the 45° line at the distillate composition xD, so it is easy to construct (see Fig. 2.3). The proof of this construction can be derived by looking at the top of the column, as shown in Figure 2.4. Component Balance:

Intercept on 45° line (x = y) Xint = (Lr/Vr) xint + D xd/Vr

Cut above Tray n

Component Balance:

Intercept on 45° line (x = y) Xint = (Lr/Vr) xint + D xd/Vr

Figure 2.4 ROL construction.

The liquid and vapor flowrates in the stripping section (LS and VS) can be calculated if the thermal condition of the feed is known. Since the temperature, pressure, and composition of the feed are given, the fraction of the feed that is liquid can be calculated from an isothermal flash calculation. This fraction is defined as the variable q. Knowing q, we can calculate the liquid and vapor flowrates in the stripping section. If the feed is saturated liquid, q is 1; if the feed is saturated vapor, q is 0:

The stripping operating line (SOL) can be drawn. It is a straight line with slope Ls/Vs that interects the 45° line at the bottoms composition xB. The proof of this construction can be derived by looking at the bottom of the column, as shown in Figure 2.5. Figure 2.6 shows the two operating lines.

There is a relationship between the intersection point of the two operating lines and feed conditions. As shown in Figure 2.7, a straight line can be drawn from the location of the feed composition z on the 45° line to this intersection point. As we will prove below, the slope of this line is only a function of the thermal condition of the feed, which is defined in the parameter q. The slope is — q/(1 — q), which makes the construction of the McCabe-Thiele diagram very simple:

1. Locate the three compositions on the 45° line (z, xD, xB).

2. Draw the ROL from the xD point with a slope of RR/(1 + RR). Figure 2.5 SOL construction. Figure 2.6 Operating lines.

3. Draw the q line from the z point with a slope of — q/(1 — q).

4. Draw the SOL from the xB point to the intersection of the q line and the ROL.

The equations of the rectifying and stripping operating lines are given below in terms of the point of intersection of the two lines at yint and xint:

q-Line VLE

q-Line VLE Figure 2.7 q line.

Subtracting the two equations gives

The last term on the right is just Fz. Using the definition of q leads to

Substituting these relationships into the previous equation gives

This is the equation of a straight line with slope —q/(1 — q). The q line is vertical for saturated liquid feed (q = 1), and it is horizontal for saturated vapor feed (q = 0). On the 45° line, xint is equal to yint. We can define this as X45:

Thus the q line intersects the 45° line at the feed composition z.

### 2.2.3 Stepping off Trays

The number of trays is determined by moving vertically from the xB point on the 45° line to the VLE line. This is the composition of the vapor yB leaving the partial reboiler. Then we move horizontally over to the SOL. This step represents the partial reboiler. The value of x on the SOL is the composition of liquid x1 leaving tray 1 (if we are numbering from the bottom of the column up). This stepping is repeated, moving vertically to y1 and horizontally to x2. Stepping continues until we cross the intersection of the operating lines. This is the feed tray. Then the horizontal line is extended to the ROL. Continuing to step until the xD value is crossed gives the total number of trays. A numerical example is given below.

### 2.2.4 Effect of Parameters

We know enough now about the McCabe-Thiele diagram to make several observations, which can be applied to any distillation system, not just a binary separation:

1. The farther the VLE curve is from the 45° line, the smaller the slope of the rectifying operation line. This means a smaller reflux ratio and therefore lower energy consumption. A "fat" VLE curve corresponds to large relative volatilities and an easy separation.

2. The easier the separation, the fewer trays is takes to make a given separation.

3. The higher the product purities, the more trays it takes to make a given separation.