Oscillatory behavior—

For the MTBE process also oscillatory behavior was reported in the literature. Potential sources for such an oscillatory behavior are either unwanted periodic forcing (e. g., by badly tuned controllers), fluid dynamic instabilities, or instabilities of the concentration dynamics. 

Fluid dynamic instabilities were reported for the MTBE process by Sundmacher and Hoffmann [104]. The cycle times for fluid dynamic oscillations are typically in 

Although many oscillating reactions have been studied experimentally, as many or perhaps even more models describing the oscillatory behavior have been discussed. At present there is no universal mechanism that explains oscillations in all heterogeneously catalyzed reactions. Every reaction has to be thoroughly investigated to discover its specific oscillation mechanism. There are, nevertheless, certain classes of models under which several oscillating systems can be considered. 

We therefore choose a different way to present the various models and their interrelationships (Fig. 5). This procedure was selected in part because our aim was to relate the models discussed to additional experimental evidence beyond mere kinetic studies. We thus distinguish on a first level between general mathematical models and reaction models for particular catalytic reactions. These general mathematical models will be discussed first and they will later be referred to in connection with the impact they have had on the formulation of the more specific experiment-related models. 

The specific models will be further subdivided into isothermal and non-isothermal models. This distinction is justified because mathematical modeling of a nonisothermal system involves a heat balance in addition to coverage equations (or reactor mass balances), and therefore introduces strong Arrhenius-type nonlinearities into the coverage equations. Nonisothermal processes are much more dependent on the reactor type and the form of the catalyst (supported, wire, foil, or single crystal). Thus these heat balance equations that describe them must take into account the type of catalyst and 

The isothermal models can be divided into elementary-step models wherein interaction of the elementary steps (adsorption, desorption, and reaction) can produce oscillations without additional mechanisms, and models with additional, non-Langmuir-Hinshelwood steps, such as phase transitions and oxidation-reduction cycles. The latter models are usually supported by experimental evidence obtained by the methods discussed in Section III. 

In general it should be noted that isothermal models are not strictly isothermal, but would typically display oscillations under nonisothermal conditions as well. In the case of supported catalysts at atmospheric pressure, oscillations are probably never purely isothermal. Usually, however, thermal effects tend to amplify instabilities, so that a model that predicts oscillations for the isothermal case will most probably predict oscillations under nonisothermal conditions. 

---

Perhaps the most fascinating detail is the surface reconstruction that occurs with CO adsorption (see Refs. 311 and 312 for more general discussions of chemisorption-induced reconstructions of metal surfaces). As shown in Fig. XVI-8, for example, the Pt(lOO) bare surface reconstructs itself to a hexagonal pattern, but on CO adsorption this reconstruction is lifted [306] CO adsorption on Pd( 110) reconstructs the surface to a missing-row pattern [309]. These reconstructions are reversible and as a result, oscillatory behavior can be observed. Returning to the Pt(lOO) case, as CO is adsorbed patches of the simple 1 x 1 structure (the structure of an undistorted (100) face) form. Oxygen adsorbs on any bare 1 x 1 spots, reacts with adjacent CO to remove it as CO2, and at a certain point, the surface reverts to toe hexagonal stmcture. The presumed sequence of events is shown in Fig. XVIII-28. 

Matrix QMC procedures, similar to configuration interaction treatments, have been devised in an attempt to calculate many states concurrently. These methods are not yet well developed, as evidenced by oscillatory behavior in the excited-state energies. 

General Second-Order Element Figure 8-3 illustrates the fact that closed loop systems often exhibit oscillatory behavior. A general... 

Rajamani and Herbst (loc. cit.) compared control of an experimental pilot-mill circuit using feedback and optimal control. Feedback control resulted in oscillatory behavior. Optimal control settled rapidly to the final value, although there was more noise in the results. A more complete model should give even better results. 

Recently, Vigil and Willmore [67] have reported mean field and lattice gas studies of the oscillatory dynamics of a variant of the ZGB model. In this example oscillations are also introduced, allowing the reversible adsorption of inert species. Furthermore, Sander and Ghaisas [69] have very recently reported simulations for the oxidation of CO on Pt in the presence of two forms of oxygen, namely chemisorbed atomic O and oxidized metal surface. These species, which are expected to be present for reaction under atmospheric pressure, are relevant for the onset of oscillatory behavior [69]. 

It is well known that the catalytic oxidation of CO on certain Pt surfaces exhibits oscillatory behavior, within a restricted range of pressures and temperatures, which are coupled with adsorbate-induced surface phase transitions [16,17]. In fact, in their clean states the reconstructed surfaces of some crystallographic planes, e.g. Pt(lOO) and Pt(llO), are... 

The HS model exhibits a rich variety of spatio-temporal patterns. During the oscillatory behavior, if the simulation starts with an empty grid in the hexagonal phase the only possible event is CO adsorption. Consequently, when a certain CO coverage is reached, the surface starts to convert into the 1 X 1 phase. Oxygen cannot adsorb yet, due to the lack of empty sites. 

A lattice gas model with adsorbate-induced surface reconstructions has also very recently been proposed by Kusovkov et al. [73]. This model also exhibits a rich oscillatory behavior. 

Very recently, considerable effort has been devoted to the simulation of the oscillatory behavior which has been observed experimentally in various surface reactions. So far, the most studied reaction is the catalytic oxidation of carbon monoxide, where it is well known that oscillations are coupled to reversible reconstructions of the surface via structure-sensitive sticking coefficients of the reactants. A careful evaluation of the simulation results is necessary in order to ensure that oscillations remain in the thermodynamic limit. The roles of surface diffusion of the reactants versus direct adsorption from the gas phase, at the onset of selforganization and synchronized behavior, is a topic which merits further investigation. 

An entirely different approach to the correlation problem is taken in the plasma model (Bohm and Pines 1953, Pines 1954, 1955), in which the electrons in a metal are approximated by a free-electron gas moving in a uniform positive background. According to classical discharge theory, such a plasma is characterized by an oscillatory behavior having a frequency... 

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

C.G. Vayenas, and J. Michaels, On the Stability Limit of Surface Platinum Oxide and its role in the oscillatory behavior of Platinum Catalyzed Oxidations, Surf. Sci. 120, L405-L408 (1982). 

An important class of cycles with non-linear behavior is represented by situations when coupling occurs between cycles of different elements. The behavior of coupled systems of this type has been studied in detail by Prigogine (1967) and others. In these systems, multiple equilibria are sometimes possible and oscillatory behavior can occur. There have been suggestions that atmospheric systems of chemical species, coupled by chemical reactions, could exhibit multiple equilibria under realistic ranges of concentration (Fox et ai, 1982 White, 1984). However, no such situations have been confirmed by measurements. 

Microbial kinetics can be quite complex. Multiple steady states are always possible, and oscillatory behavior is common, particularly when there are two or more microbial species in competition. The term chemostat can be quite misleading for a system that oscillates in the absence of a control system. 

VHiereas the previous case revealed temperature and conversion profiles propagating with almost constant velocity ("constant-pattern profiles"), the next case shows oscillatory behavior of the filtration combustion process for parameters a = 1.0, p = 0.08, y = 0.05, 6 = 1.0, (A) = 100.0, L =50.0 and 8 = -10.0. Figure 3a... 

Ionization potential of metal clusters is one of the factors affected by cluster size [33]. This study represents the most extensive effort so far to determine the size dependence of IP. The measurements on these clusters showed a decreasing IP with size with apparent oscillatory trend. Even-size particles had a relatively larger IP compared to their odd-size counterparts. The data show oscillatory behavior for small Na clusters with a loss of this oscillation for the larger Na clusters. The IP decreases with cluster size, but even at Nai4 the value 3.5 eV is far from... 

At time t=212 h the continuous feeding was initiated at 5 L/d corresponding to a dilution rate of 0.45 d . Soon after continuous feeding started, a sharp increase in the viability was observed as a result of physically removing dead cells that had accumulated in the bioreactor. The viable cell density also increased as a result of the initiation of direct feeding. At time t 550 h a steady state appeared to have been reached as judged by the stability of the viable cell density and viability for a period of at least 4 days. Linardos et al. (1992) used the steady state measurements to analyze the dialyzed chemostat. Our objective here is to use the techniques developed in Chapter 7 to determine the specific monoclonal antibody production rate in the period 212 to 570 h where an oscillatory behavior of the MAb titer is observed and examine whether it differs from the value computed during the start-up phase. 

We have not encountered examples with a second order equation, especially one that exhibits oscillatory behavior. One reason is that processing equipment tends to be self-regulating. An oscillatory behavior is most often a result of implementing a controller, and we shall see that in the control chapters. For now, this section provides several important definitions. 

---