## Summary Of Methods

6.1.1 Slope Criterion

Satisfaction of the slope criterion consists in selecting the tray where there are large changes in temperature from tray to tray.

The temperature profile at design conditions is plotted, and the "slope" of the profile is examined to find the tray where this slope is greatest. Large changes in temperature from tray to tray indicate a region where compositions of important components are changing. Maintaining a tray temperature at this location should hold the composition profile in the column and prevent light components from dropping out the bottom and heavy components from escaping out the top.

### 6.1.2 Sensitivity Criterion

Satisfaction of the sensitivity criterion consists in finding the tray where there is the largest change in temperature for a change in the manipulated variable.

A very small change (0.1% of the design value) is made in one of the manipulated variables (e.g., reflux flowrate). The resulting changes in the tray temperatures are examined to see which tray has the largest change in temperature. The procedure is repeated for the other manipulated variable (e.g., reboiler heat input). Dividing the change in the tray temperature by the change in the manipulated variable gives the openloop steady-state gain between temperature on that tray and each manipulated variable. The tray with the largest temperature change is the most "sensitive" and is selected to be controlled. A large gain indicates that the temperature on that tray can be effectively controlled by the corresponding manipulated variable. A small gain indicates that valve saturation can easily occur and the operability region could be limited.

6.1.3 Singular Value Decomposition Criterion

Singular value decomposition (SVD) of the steady-state gain matrix is thoroughly treated by Moore.1

The steady-state gains between all the tray temperatures and the two manipulated variables are calculated as described in the previous section. A gain matrix K is formed, which has NT rows (the number of trays) and two columns (the number of manipulated variables). This matrix is decomposed using standard SVD programs (e.g., the svd function in Matlab) into three matrices: K = UsVT. The two U vectors are plotted against the tray number. The tray or trays with the largest magnitudes of U indicate locations in the column that can be most effectively controlled. The s matrix is a 2 x 2 diagonal matrix whose elements are the singular values. The ratio of the larger to the smaller is the condition number, which can be used to assess the feasibility of dual-temperature control. A large condition number (or small minimum singular value) indicates a system that is difficult to control. The controller is the inverse of the plant gain matrix, and a singular value of zero means that the matrix is "singular" and cannot be inverted.

### 6.1.4 Invariant Temperature Criterion

With both the distillate and bottoms purities fixed, we change the feed composition over the expected range of values. We select the tray where the temperature does not change as feed composition changes.

The difficulty with this method is that there may be no constant-temperature tray for all feed compositions changes. This is particularly true in multicomponent systems, where the amounts of the nonkey components can vary and significantly affect tray temperatures, especially near the two ends of the column.

### 6.1.5 Minimum Product Variability Criterion

Satisfaction of the minimum product variability criterion consists in selecting the tray that produces the smallest changes in product purities when its temperature is held constant in the face of feed composition disturbances.

Several candidate tray locations are selected. The temperature on one specific tray is fixed, and a second control degree of freedom is fixed such as reflux ratio or reflux flow-rate. Then the feed composition is changed over the expected range of values, and the resulting product compositions are calculated. The procedure is repeated for several

1C. F. Moore, Selection of controlled and manipulated variables, in Practical Distillation Control, Van Nostrand-Reinhold, 1992, Chapter 8.

control tray locations. The tray is selected that produces the smallest changes in product purities when its temperature is held constant in the face of feed composition disturbances.

### 6.1.6 Summary

In the sections above, we have described the five most frequently used criteria. Sometimes these criteria recommend the same control tray location. In other cases, they recommend different control tray locations. In the next sections we apply these criteria to several typical industrial distillation systems to assess their relative effectiveness.

6.2 BINARY PROPANE/ISOBUTANE SYSTEM

The first separation system examined is a binary mixture of propane and isobutane. The feed flowrate is 1 kmol/s, and the design feed composition is 40 mol% propane. We use the conventional notation that the composition of the feed is z, the composition of the distillate is xD, and the composition of the bottoms is xB (all in mole fraction propane). Column pressure is set at 13.5 atm so that cooling water can be used in the condenser (reflux drum temperature is 315 K). The column has 37 stages and is fed on stage 16, using Aspen notation of numbering stages from the reflux drum on down the column.

Distillate purity is specified to be 98 mol% propane. Bottoms impurity is specified to be 2 mol% propane. The reflux ratio required to achieve these purities is 1.08.

### 6.2.1 Slope Criterion

The upper graph in Figure 6.1 gives the temperature profile at design conditions. The lower graph shows the differences between the temperatures on adjacent trays. The location of the tray with the largest slope is stage 8. There is another tray (stage 29) that has a slope that is almost as large. We will compare the use of both of these later in this section.

### 6.2.2 Sensitivity Criterion

The upper graph in Figure 6.2 gives the openloop gains between tray temperatures and the two manipulated variables reflux R and reboiler heat input QR. The solid lines show reflux flowrate changes, and the dashed lines represent reboiler heat input changes. Very small increases from the steady-state values (+0.1%) of the two inputs are used. As expected, the gains between the tray temperatures and reflux are negative, while they are positive for heat input.

These curves show that stage 8 is sensitive to changes in reflux and both stages 8 and 29 are sensitive to changes in heat input. Therefore stage 8 can be controlled using either reflux or heat input, while stage 29 can be controlled by only heat input.

It should be remembered that these are steady-state results and tell us nothing about dynamics. Temperatures on all trays in the column are quickly affected by changes in heat input, so pairing heat input with any tray temperature is dynamically feasible.

Figure 6.1 Propane/isobutane temperature profile and slope.

However, a change in reflux flowrate takes a significant time to affect temperatures on trays near the bottom of the column because of liquid hydraulic lags (3-6 s per tray). Therefore poor control can be expected when reflux is paired with a tray temperature significantly down from the top of the column.