Accumulation with state feedback

The general equation for an accumulator with state feedback is ... 

Fig. 2.12. Stability diagram for the allosteric model with positive feedback, as a function of the nonexclusive binding coefficient c and of the allosteric constant L, for (a) = 10 and (b) e = 10". System (2.7) admits a unique steady state. This state is stable in domain I and unstable in domain II in the latter domain, the system evolves towards a limit cycle. For parameter values corresponding to domain III no steady state is reached as the substrate can only accumulate in the course of time (Goldbeter Lefever, 1972).

Chemical conversion is an effective way to counteract the accumulation of impurities due to positive feedback. Also, changing the connectivity of units may be used to modify the effect of interactions, for example by preventing an excessive increase in recycles due to snowball effects. Effective plantwide control structures may imply controlled and manipulated variables belonging to different but dynamically neighbouring units. The methodology to evaluate the dynamic inventory of impurities consists of a combination of steady state and dynamic flowsheeting with controllability analysis. This is used to assess the best flowsheet alternative and propose subsequent design modifications of units. Case Study 3 in Chapter 17 will present this problem in more detail. 

Figure 17.19b presents a direct comparison of all alternatives with respect of impurity L. It may be seen that the effect depends on the recycle structure deflation for the base-case and inflation in alternatives A and C, while in the alternative B the steady state effect is almost negligible. The explanation can be found in the negative feedback effect of reactor Rl, where L is destroyed in heavies. In alternatives, by suppressing the back flow from S4 to Rl, the converted L diminishes and the accumulation becomes positive. The effect is maximum in alternative C and minimum in the alternative B, corresponding to differences in the path of the recycle loops. 

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