Closed Loop Transitions

Steady state transitions are clearly possible for a range of realistic disturbances when open-loop operation is considered. Are transitions still possible if closed-loop control is used? Output multiplicity implies that there is more than one set of output variables which satisfy a single set of inputs, if each set of outputs are separate and distinct, perfect control of any output variable will confine the column to a single steady state for any known set of inputs. However, if the parallel output sets have common elements, control of one of those common elements might not be sufficient to prevent a transition to a parallel steady state. Therefore, the choice of appropriate controlled variable(s) within the control scheme is paramount.

The selection of controlled and manipulated variables is discussed extensively in Chapters 9 and 10. However, four variable pairings are considered here to demonstrate the importance of making effective choices where steady state transitions must be avoided. Figure 8.25 shows the relationships between the reboiler temperature, Tb, and the temperature on stage 12 (i.e. top of the stripping section), T12, and the reboiler duty. Figure 8.26 examines the relationships between Tb and T12 and the bottoms draw rate. The points designated A, A', B and B' correspond to the same operating conditions in Figures 8.25 and 8.26.

Figure 8.25 - Relationships Between Reboiler Duty and Column Temperatures

Bottoms Draw Rate (m3/hr)

Figure 8.26 - Relationships Between Bottoms Rate and Column Temperatures

Bottoms Draw Rate (m3/hr)

Figure 8.26 - Relationships Between Bottoms Rate and Column Temperatures

The upper part of Figure 8.25 indicates the variation in Tb with the reboiler duty. The important aspect of this plot is that most values of Tb correspond to two separate values of the reboiler duty, QR. In contrast, the lower part of Figure 8.25 shows that each value of T,2 corresponds to one (and only one) value of the reboiler duty. Therefore, the perfect control of T|2 will confine the column to a single operating point (i.e. a single steady state) but control of Tb will not. Figure 8.26 shows that either Tb or T,2 could be used in conjunction with the bottoms draw rate without risking an unwanted steady state transition. In fact, open-loop operation is sufficient to avoid a transition if the bottoms rate and the reflux rate are the variables which are fixed since only one steady state exists for all values of the bottoms rate. However, this analysis conceals that a linear controller for the reboiler sump level would be unstable as the sign of the gain is indeterminate between A and A' (or B and B'). This can be inferred from Figures 8.25 and 8.26 together: some values of the bottoms rate correspond to multiple values of the reboiler duty.

This analysis suggests that a linear controller using the pairing T12-QR is acceptable but the other three c ombinations are not. The T12-QR controller will prevent unwanted steady state transitions and simplify the associated inventory control system. However, a linear controller will be unstable for 124.5°C < T,2 < 125.5°C as the process gain is negative in this interval and positive elsewhere. The controller may also be inefficient for T12 > 125.5°C (i.e. the low conversion steady state) because of tuning as the process gain is approximately 50 times lower than for Tl2 < 124.5°C (i.e. the high conversion steady state and normal operating region).

A similar logic can be applied to more complex, multivariate controllers to determine if closed-loop control will necessarily prevent a steady state transition. Regardless of the type of controller, the key consideration is whether the controlled variables are common to multiple steady states. Two-point control would make a transition almost impossible since there must be two common elements in each output set (i.e. the values of both controlled variables).

In practice, reasonably tight control (but not necessarily perfect control) of almost any temperature within the column should be sufficient to prevent a steady state transition. This ensues from the differences between the temperature profile of the different steady states, particularly in the stripping section where the controlled variable is likely to be located. However, although tight control of any temperature is adequate, this might only be realisable for some temperatures. For example, the process gain, Tb-QR> is both positive and negative in the normal operating region (i.e. the high conversion steady state) so that a linear controller for this temperature could not be globally stable.

This result provides further encouragement that once the high conversion steady state is reached, the column operation should be stable, especially if closed-loop control is used. However, the possibility of a start-up sequence ending at an undesirable steady state still remains as start-up is almost always conducted in open-loop.

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