Interaction among various oscillating inputs may result in significant modifications in the behavior of a system under forced periodic control.Forced oscillations in the input flow rate and input concentrations of an exothermic continuous stirred tank reactor enable the stabilized operation of the CSTR in the unstable steady state region. Reactor temperature oscillations under forced periodic control are similar to the oscillations resulting from proportional-integral feedback or nonlinear (push-pull) feedback control, and under some operating conditions the oscillation amplitude is significantly lower. Theoretical and experimental studies illustrate the effects of forcing frequency and phase shift on reactor behavior. Konstantinos Rigopoulos Xianshu Shu Ali CinarDepartments of Chemical and Electrical Engineering Illinois Institute of Technology Chicago, IL 60616 IntroductionForced periodic operation of chemical reactors has been used effectively for conversion improvement, selectivity, and yield enhancement in complex reactions and the stabilization of reactor operation. References to various review papers, different analysis techniques developed, and applications reported are given by Cinar et al. (1987a). Recently there has been a revival of interest in periodically forced systems. Promising experimental results on yield improvement in catalytic reactions have been reported (Cutlip, 1979, Jain et al., 1982Wilson and Rinker, 1982;Barshad and Gulari, 1985,1986;Silveston et al., 1986). A technique based on the Carleman linearization of nonlinear plant equations and the maximization of a time-averaged performance measure has been proposed (Lyberatos and Svoronos, 1987) for determining the optimal periodic operation and has been applied to an isothermal reactor model for the selection of the best forcing frequency, amplitude, and waveform. Common features of periodically forced reacting systems, particularly properties of spontaneously oscillating systems that are forced periodically, have been considered (Kevrekidis et al., 1986) and an algorithm using stroboscopic representation has been presented for the numerical computation and stability analysis of invariant tori. In stabilizing reactor operation by periodic forcing, the vibrational control approach has provided successful theoretical and experimental results (Cinar et al., 1987a, b).Vibrational control (Meerkov, 1982) is a method for modification of the dynamic properties of a system by introducing fast,

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Vibrational control is an open-loop control strategy implemented by fast zero-average oscillations in the input variables of a system. It has been previously shown that vibrational control of an exothermic reaction in a CSTR, introduced via oscillations in the input flow rate, modifies the dynamic behavior of the reactor and enables its stabilized operation in the unstable steady-state region. In the present paper it is shown that simultaneous vibrations in input flow rate and concentration lead, in addition, to a reduction of the amplitude of oscillations in reactor temperature and/or increase in reactant conversion. It is also shown that the phase shift between the vibrations of the two input variables affects the behavior of the system as well as the amplitude, the frequency and the form of forcing functions.

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