A fundamental assumption that is often made in the analysis of distillation control structures is perfect (or nearly perfect) inventory control. This reduces the full 5x5 control system to a 2 x 2 system and, thereby, substantially simplifies the controller design. This assumption is usually valid as the pressure and level control loops are normally fast acting and globally stable. Indeed, this assumption is always acceptable if a product stream is used to control the levels in the reflux accumulator and the reboiler sump.
However, if the level controller is paired with an internal flow (e.g. the reflux or boilup rate), the stability is only ensured if the following conditions are satisfied for the reflux accumulator (inequality 10.4) and reboiler sump (inequality 10.5):
Considering the reflux accumulator only, inequality (10.6) holds for all values of the reflux rate so that inequality (10.4) can be extended to produce inequality (10.7) which is a necessary condition for the global stability of a linear (e.g. PI) level controller.
Similarly, inequality (10.8) can be derived as a linear stability condition for the reboiler sump level controller.
The converses of inequalities (10.7) and (10.8) (i.e. positive values of the derivatives for any value of the reflux rate or reboiler duty) describe necessaiy geometric conditions for output multiplicity. Therefore, wherever output multiplicity exists, at least one of the level control loops will not be globally stable if a linear level controller is used with an internal flow. This creates a strong incentive to use an energy balance control configuration (e.g. the LV scheme) for reactive distillation.
Interestingly, there is no evidence of output multiplicity in non-reactive distillation where a material balance control structure (e.g. DV, LB, etc.) is used unless the associated level control is very poor (Jacobsen and Skogestad, 1994). This result appears to extend to reactive distillation although pseudo-multiplicity has been detected with a material balance control configuration. This suggests that a material balance control configuration could be useful to avoid possible transitions between parallel steady states where they exist but such a scheme can not be implemented with linear level controllers. It is also doubtful whether a complex non-linear controller could adequately account for the observed behaviour in order to satisfactorily control the column inventory near unstable operating points (i.e. where unplanned steady state transitions are possible).
This apparent paradoxical situation creates a dilemma for control system design. If an energy balance control structure is used, multiple steady states can occur and there is the possibility of unwanted transitions from desirable to undesirable steady states. However, if a material balance structure is used, only one steady state is possible but satisfactory control of the column inventory is unlikely and this could ultimately lead to equipment failures or the complete shut down of the column. The events that trigger a transition in the first case and déstabilisation in the second case are likely to be similar (e.g. feed rate or composition perturbations, or disturbances to external variables such as the steam pressure or cooling water temperature). The lesser disaster is probably an unwanted transition as it should be possible to recover the original operating state with an appropriate response from the process operator. Techniques for achieving the return to the desirable steady state are discussed in the next section.
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