Multiple steady states due

Potentiometric techniques have been used to study autonomous reaction rate oscillations over catalysts and carbon monoxide oxidation on platinum has received a considerable amount of attention43,48,58 Possible explanations for reaction rate oscillations over platinum for carbon monoxide oxidation include, (i) strong dependence of activation energy or heat of adsorption on coverage, (ii) surface temperature oscillations, (iii) shift between multiple steady states due to adsorption or desorption of inert species, (iv) periodic oxidation or reduction of the surface. The work of Sales, Turner and Maple has indicated that the most... 

The key problems in a polymerization CSTR are the determination and characterization of micro- and macromixing, and the possibility of multiple steady states due to the exothermic nature of the reactions. Recent studies of CSTRs for bulk or solution free-radical polymerization indicate the possibility of multiple steady states due to the large heat evolution and difficult heat transfer that are characteristic of the reactors. Furthermore, even in simple solution polymerization (for example, in methyl methacrylate polymerization in ethyl acetate solvent), autocatalytic kinetics can lead to runaway conditions even with perfect temperature control for certain combinations of solvent concentration and reactor residence time. In practice, the heat evolution can be an additional source of autocatalytic behavior. 

Besides these two regimes, another regime, with a temporally periodic change of the chemical composition (chemical oscillation or self-oscillation), may also be observed. A famous example of this phenomenon is the Belousov-Zhabotinsky reaction. Another example of complex kinetic behavior in open chemical systems is the occurrence of multiple steady states due to the fact that for some components of the reaction mixture the rate of consumption and rate of production can be balanced at more than one point. This type of behavior has become the subject of detailed theoretical and computational analyses (Marin and Yablonsky, 2011 Yablonskii et al., 1991). Bespite the fact that there are many experimental data concerning such complex behavior, the steady-state regime with characteristics that are constant in time still is the most observed phenomenon. 

J. (1973) Chemical oscillations and multiple steady states due to variable boundary permeability. J. Theoret. Biol., 41, 503-521. 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

From this expression, one can show that, for reasonable parameter values, there can be no multiple steady states for very large Ts. This will also be the case, independent of Ts, if r is larger than both dA and dB. However, if Er is less than either desorption activation energy, then it is possible to reduce T below the critical value required for multiple steady states by decreasing the surface temperature. The actual values of the reactant partial pressure and surface temperature for which multiple steady states occur can vary by orders of magnitude due to the sensitivity of (46) to the parameters (and the fact that preexponential factors can vary by orders of magnitude between different systems). 

Nonlinear feedback would produce multiple steady states and support oscillatory behavior. The increase of the analytical Cu(II) ion concentration leads to a mass decrease due to enhanced dissolution as shown by the... 

It is demonstrated that volatility of the liquid reactant has a detrimental effect on the enhancement of gas absorption. It is also shown that failing to account for effects due to liquid evaporation in the modeling of gas-liquid CSTRs can lead to significantly different predictions of the number and region of multiple steady states. 

Chemical reactors are inherently nonlinear in character. This is primarily due to the exponential relationship between reaction rate and temperature but can also stem from nonlinear rate expressions such as Eqs. (4.10) and (4.11). One implication of this nonlinearity for control is the change in process gain with operating conditions. A control loop tuned for one set of conditions can easily go unstable at another operating point. Related to this phenomenon is the possibility of open-loop instability and multiple steady states that can exist when there is material and/or thermal recycle in the reactor. It is essential for the control engineer to understand the implications of nonlinearities and what can be done about them from a control standpoint as well as from a process design standpoint. 

Despite the similar reaction mechanism, a completely different type of behavior was found for the TAME process [71-73]. This is due to the fact that the rate of reaction is one order of magnitude slower for TAME synthesis compared to MTBE synthesis. The behavior of the TAME process is illustrated in Fig. 10.14. In contrast to the MTBE process the TAME column is operated in the kinetic regime of the chemical reaction at a pressure of 2 bar. Under these conditions large parameter ranges with multiple steady states occur. The more detailed analysis by Mohl et al. [73] reveals that steady state multiplicity of the TAME process is caused by self-inhibition of the chemical reaction by the reactant methanol, which is adsorbed preferably on the catalyst surface. Steady state multiplicity is therefore caused by the nonlinear concentration dependence of the chemical reaction rate. Consequently, a similar type of behavior can be observed for an isothermal CSTR. This effect is further in-... 

The influence of the die resistance and screw rotation rate on the conversion is shown in Fig. 8.7. A reduction of the die resistance results in a decrease in residence time, and therefore a decrease in conversion can be expected. Experimentally, a very sharp decrease in conversion was observed at a certain die resistance. The sharp decrease in conversion indicates a transition from one regime with a high to another with a low residence time. This transition is due to the occurrence of multiple steady states and will be dealt with in more detail in Chapter 13. After the drop in conversion, the die pressure was almost zero, which indicates that hardly any fully-filled length was present anymore. An increased screw speed resulted also in a decreased residence time and therefore in a decreased conversion. 

Nevertheless, the inclusion of axial dispersion may be interesting from the point of view of the numerical methods used to solve the conservation equations or in studies regarding the appearance of multiple steady-state solutions [141, 142], Petersen [81] presented an analysis for a ID reactor in terms of the dispersion factor E, which is the ratio between the length of a plug-flow reactor (no dispersion) and the one for a reactor with dispersion yielding the same conversion (Lm). Due to the coordinate transformations employed, F is a function of oP = kDAefu (isothermal first-order reaction). Asymptotic solutions for the dispersion ratio were obtained and are given by... 

The results of simulations of TWC model with microkinetics and diffusion resistance within the washcoat enable the interpretation of the dynamics of surface coverages and overal reactions and can serve for the improvement of the washcoat design. It has been found that not only multiple steady states (hysteresis) but also various types of periodic and complex spatiotemporal concentration patterns can exist in the monolith. Thorough analysis of bifurcations and transitions among existing patterns is numerically demanding task due to dimension of the problem. 

It is stressed again that the occurrence of multiple steady states is due to the feedback of heat. Tubular reactors that are not coupled with a heat exchanger generally do not exhibit this feature — except in very particular situations, as will be shown later. This does not mean that perturbations in the operating... 

In general, CPRF is the better mode of operation for the RFBR when multiple steady states are possible. In addition to expediting the approach to steady state, the establishment of the CPRF profile early in the bed is an advantage in cases where the reaction zone gradually creeps toward the bed exit due to catalyst deactivation. When only unique steady states are possible and/or for parallel and series reactions, the preferred flow mode may depend on other considerations. 

The same phenomenon is seen in emulsion polymerization. The phenomenon of multiple steady states arises in anulsion polymerization for much the same reason it appears in solution polymerization The autocatalytic nature of the polymerization (due to the gel effect) combined with the mass balance results in the possibihty of steady-state multipUcity. 

In the general case of nonisothermal shrinking-core systems controlled both by chemical reaction and diffusion, the thermal effect of the reaction may bring about multiple steady states and instability due to sudden transition of rate-controlling steps during the reaction. The problem of thermal instability in noncatalytic gas-solid reactions was first pointed out by Cannon and Denbigh [37] and has been discussed by Shen and Smith [23] and Wen and coworkers [38, 39]. 

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