In this section a systematic approach is proposed to design the control structures for these three types of reactive distillation flowsheets. Because all five reactive distillation systems (Table 7.5) have almost equal molar feedflows (neat flowsheet), the stoichiometric balance has to be maintained.3 Here we adjust the feed ratio to prevent accumulation of unreacted reactants attributable to stoichiometric imbalance. The next issue is, how many product compositions or inferred product purities should be controlled? For the esterification reactions with A + B , C + D with a neat flowsheet, controlling one-end product purity implied a similar purity level on the other end, provided the product flowrates are equal. Thus, a single-end composition (or temperature control) is preferred. This leads to 2 x 2 multivariable control, as opposed to a 3 x 3 multiple-input-multiple-output system. The next problem is consideration of robustness. Because of input multiplicities and potential sign reversals, the uncertainty associated with a linear model can be significant. Instead of inverting all four transfer function models (for a 2 x 2 system) for controller design, a decentralized control is used in which only information on diagonal elements is employed for initial design. In other words, the uncertainty associated with the process is so large that we want to minimize the exercise of model inversion. Thus, decentralized control is preferred. In summary, the following principles are recommended:
1. Maintain the stoichiometric balance using the feed ratio.
2. Control only one-end composition (or temperature).
3. Use decentralized control to maintain robust stability.
This leads to the following design procedure for temperature control of reactive distillation systems:
1. With the feed ratio as one manipulated variable, select a second manipulated variable. Typically, the other manipulated input is the heat input or the reflux ratio.
2. Use the nonsquare relative gain (NRG)8 to select temperature control trays. The larger row sums of the NRG indicate a potential temperature control tray. Note that the temperatures with sign reversal (Fig. 13.3) cannot be used as controlled variables.
3. Use the relative gain array (RGA)9 for variable pairing once the inputs and outputs are determined.
4. Perform a sequential relay-feedback test to find the ultimate gain (Ku) and ultimate period (Pu).
5. Use Tyreus-Luyben tuning to set the tuning constants for the PI controllers. A simple version is Kc = Ku/3 and tj = 2Pu.
In step 2, the row sum of the NRG is used instead of the more familiar SVD. The reason for this is that the row sum of the ith row is equal to the square of the 2-norm of the ith row of the U matrix from the SVD (i.e., G = U S VT and ith row sum = ||eiUk2, where ei is the ith unit vector). Thus, the two approaches are the same.
Sensitivity analyses were performed on these five esterification reactive distillation systems. In order to find the steady-state gains of the tray temperature in the linear region, extremely small step changes (+ 0.1%) are made in the manipulated variables.
For the MeAc system (Fig. 13.1a), a step increase in the heat input (Qr) leads to a temperature increase in the lower section of the column and a relative small temperature decrease toward the top of the column as shown in Figure 13.4. This is uncharacteristic because, for conventional distillation, we generally observe temperature increases throughout the column. For the change in the second manipulated variable (ratio of acid feed flowrates to alcohol feedrate, Facid/Falcohol), an increase in the flowrate of the
8J. W. Chang and C. C. Yu, The relative gain for non-square multivariable systems, Chem. Eng. Sci. 45, 13091323 (1990).
9E. H. Bristol, On a new measure of interaction for multivariable process control. IEEE Trans. Automat. Control AC11, 133-134 (1966).
heavy reactant (Facid) results in a temperature rise in the lower section of the column, followed by a decrease toward the middle section of the column (Fig. 13.4). The reason for the temperature increase in the lower section is the excess of the heavy reactant (HAc). Because of the lowered conversion, less water (the second highest boiler) is formed, which subsequently leads to a lower tray temperature in the midsection of the column.
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