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Abstract:
Industrial process plants are primarily made up of two basic units, i.e. reactors and separators. The reactants are fed to the reactor unit and partially converted to products. The mixture of reactants and products is then transferred to a separator unit. In the separator the unreacted material is separated from the product and recycled back to the reactor (Figs. 1 and 2). However, recycles may introduce complex dynamics involving instability and sensitivity to disturbances. So far a nonlinear exothermic first order reaction [1] and an autocatalytic isothermal reaction [2] have been studied in a continuous flow stirred tank reactor (CSTR) coupled to an isothermal separator represented by a heated flash corresponding to a one-step distillation unit. Recently these studies have been extended to the investigation of an isothermal first order reaction [3]. Although this model system is very simple we found surprisingly complex dynamic phenomena, such as multiplicity (static instability) and autonomous periodic oscillations (dynamic instability). It was shown that the dynamic behavior depends on the flow and the flash control strategy. Especially for the case of fixed reactor inlet flow and constant heating power of the flash we found the most complex behavior involving multiplicity and oscillations. Model system: In the present paper we are focussing on the case of fixed CSTR inlet and constant heating of the flash mentioned above. The system is further simplified by skipping the reaction completely so that the reactor can now be regarded as a simple mixing unit. The whole set-up can therefore be interpreted as a single step distillation unit with recycle. The mixture is assumed to be binary, ideal and homogeneous. All flow rates are based on molar units. The controllers are assumed to be perfect. For recycling we have investigated two different modes i) vapor phase (Fig. 1) and ii) liquid phase (Fig. 2) recycling. In a second step we have introduced a delay time in the recycle which is caused by a transportation lag. Due to the simplicity of the model system an analytical approach is possible. The results are confirmed by model calculations, such like integration and continuation of stationary and oscillatory states, which were performed using the software package DIVA [4].
Results: In the case of vapor recycle we prove static as well as dynamic stability for all operating conditions. The results are confirmed by model calculations. For the case of liquid recycle static stability can be proven. However, dynamic stability can be guaranteed only for DhA > DhB and Tfl < TR, where DhA and DhB are the evaporation enthalpies and Tfl and TR are the temperatures of the flash and the reactor unit, respectively. Here A is assumed to be the heavy boiling component. For the reversed case where A is the light boiling component dynamic stability can be proven only for DhA < DhB and Tfl < TR. For all other cases dynamic instabilities can occur. Examples of stable and unstable periodic oscillations have been found in our model calculations. The oscillations can emerge smoothly at supercritical Hopf bifurcation points or rapidly at subcritical Hopf points. Dynamic instabilities and Hopf bifurcation points can be found for the binary mixtures of water and acetic acid as well as for ethanol and acetic acid. Taking a set of parameters where stable periodic oscillations occur we introduced delay times in the recycle of the liquid phase. For increasing delays the period of the oscillations increases and decreases in a saw-tooth like manner. The individual saw-tooths are overlapping so that different oscillatory states are obtained for increasing and decreasing delay time. Therefore, two or more different stable oscillatory states are coexisting for the same parameter set. Further analysis shows that the coexisting stable oscillatory states are separated by unstable periodic orbits (multirhythmicity). This behavior is analogous to studies of delayed feed back in the isothermal nonlinear minimal bromate oscillator.
