Observation of Multiple Steady States

We next examine the variations of these solutions with respect to parameters of the system, Tq and V. We consider the situations where we vary these parameters slowly and examine how the steady states change. 

Note that in this experimenf in which we varied Tq, we somehow never reached the middle intersection but jumped from the lower steady state to the upper one and back again. We said previously that this middle state is unstable, and we see that it is never attained in a slow variation of Tq. 

It is seen that multiple intersections exist for a limited range of r. For z 1.3 min the reaction goes nearly to completion, while for r 0.8 min the conversion is very small. Between these TS the system exhibits two steady states as x (or i ) is varied. 

The previous examples showed that we can attain multiple steady states in a CSTR by varying the feed temperature T. Many other parameters in the reactor such as v and Cao can also be varied, and these can give multiple steady states. 

Whenever multiple steady states in a reactor are possible, we must be very concerned that we are operating on the desired steady-state branch. This requires a proper startup procedure to attain the desired steady state and suitable operation limits to make sure that we never exhibit a sufficiently large transient to cause the system to fall off the desired conversion branch. We will consider transients in the CSTR in the next section. 

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A. Experimental Observation of Multiple Steady-State Phenomena... 

So far, there have been published only a few papers devoted to experimental investigation of multiplicity and oscillatory activity of a single catalyst particle. Observations of multiple steady states and/or oscillations for a single catalyst particle are reported in Table IV. Evidently three types of strong exothermic reactions have been investigated ... 

When a catalytic reaction takes place on a nonporous metal surface the coupling between the exothermic chemical reaction and the transport effects may also give rise to multiple steady states. Apparently, in the realm of chemical reaction engineering the first experimental observation of multiple steady states was done just for the catalytic wire problem [see Tamman (29), Davies (30), and Buben (5/)]. Catalytic gauzes consisting of wire screens or layers of metal pills (e.g., the silver crystals) are used for a number of industrially important catalytic reactions as, e.g., synthesis of... 

Experimental Observations of Multiple Steady States in Tubular Catalytic Adiabatic Reactors... 

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

In vitro experiments carried out in a reconstituted system containing PFK show that such a minimal system is capable of exhibiting transitions between multiple steady states or sustained oscillations (Eschrich, Schellenberger Hofmann, 1980,1983). So far, however, experiments on a reconstituted system have led primarily to the observation of multiple steady states (Hofmann, Eschrich Schellenberger, 1985 Eschrich et al, 1990). 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

Experimentally, in chemical systems a variety of dynamic states can be observed resulting from thermodynamic and kinetic conditions as defined as a prerequisite for the evolution of dissipative structures. Today we distinguish the following states (1) the maintenance of multiple steady states with transitions from one to another, (2) rotation on a limit cycle... 

In this paragraph, based on experimental observations, we are going to review a number of observed facts which may help to elucidate the lows governing the occurrence of multiple steady states in tubular packed catalytic reactors. 

Carbon Monoxide Oxidation. Analysis of the carbon monoxide oxidation in the boundary layer of a char particle shows the possibility for the existence of multiple steady states (54-58). The importance of these at AFBC conditions is uncertain. From the theory one can also calculate that CO will bum near the surface of a particle for large particles but will react outside the boundary layer for small particles, in qualitative agreement with experimental observations. Quantitative agreement with theory would not be expected, since the theoretical calculations, are based on the use of global kinetics for CO oxidation. Hydroxyl radicals are the principal oxidant for carbon monoxide and it can be shown (73) that their concentration is lowered by radical recombination on surfaces within a fluidized bed. It is therefore expected that the CO oxidation rates in the dense phase of fluidized beds will be suppressed to levels considerably below those in the bubble phase. This expectation is supported by studies of combustion of propane in fluidized beds, where it was observed that ignition and combustion took place primarily in the bubble phase (74). More attention needs to be given to the effect of bed solids on gas phase reactions occuring in fluidized reactors. 

The research of Roy Jackson combines theory and experiment in a distinctive fashion. First, the theory incorporates, in a simple manner, inertial collisions through relations based on kinetic theory, contact friction via the classical treatment of Coulomb, and, in some cases, momentum exchange with the gas. The critical feature is a conservation equation for the pseudo-thermal temperature, the microscopic variable characterizing the state of the particle phase. Second, each of the basic flows relevant to processes or laboratory tests, such as plane shear, chutes, standpipes, hoppers, and transport lines, is addressed and the flow regimes and multiple steady states arising from the nonlinearities (Fig. 6) are explored in detail. Third, the experiments are scaled to explore appropriate ranges of parameter space and observe the multiple steady states (Fig. 7). One of the more striking results is the... 

Before we leave this topic, it would be wise to note the results of some recent research on heterogeneously catalysed gas reactions. Here finite rates of adsorption and desorption had to be introduced into the reaction scheme in order to explain the occurrence of multiple steady states and oscillatory phenomena. This observed exotic behaviour could be reproduced by solving a set of coupled equations for the rates of adsorption/desorption, the rate of the surface reaction, and the mass balance relations [22, 23], Adsorption steps (ii) and (iv) may therefore need to be invoked for any heterogeneously catalysed solution reactions that are found to exhibit similar dynamic behaviour. 

The question of multiple steady states is not simply an academic problem, since at certain operating conditions a small perturbation can shift the rate, output, and selectivity into a new unfavorable regime. Potential and current oscillations observed experimentally show that such regimes are within the usual operating conditions of electrocatalysts. 

Jaisinghani and Ray (40) also predicted the existence of three steady states for the free-radical polymerization of methyl methacrylate under autothermal operation. As their analysis could only locate unstable limit cycles, they concluded that stable oscillations for this system were unlikely. However, they speculated that other monomer-initiator combinations could exhibit more interesting dynamic phenomena. Since at that time there had been no evidence of experimental work for this class of problems, their theoretical analysis provided the foundation for future experimental work aimed at validating the predicted phenomena. Later studies include the investigations of Balaraman et al. (43) for the continuous bulk copolymerization of styrene and acrylonitrile, and Kuchanov et al. (44) who demonstrated the existence of sustained oscillations for bulk copolymerization under non-isothermal conditions. Hamer, Akramov and Ray (45) were first to predict stable limit cycles for non-isothermal solution homopolymerization and copolymerization in a CSTR. Parameter space plots and dynamic simulations were presented for methyl methacrylate and vinyl acetate homopolymerization, as well as for their copolymerization. The copolymerization system exhibited a new bifurcation diagram observed for the first time where three Hopf bifurcations were located, leading to stable and unstable periodic branches over a small parameter range. Schmidt, Clinch and Ray (46) provided the first experimental evidence of multiple steady states for non-isothermal solution polymerization. Their... 

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. 

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. 

Shown in Figure 4.6 is this relationship obtained by Carberry and Kulkarini (1973). Similar behavior as in Figure 4.5 can be observed. However, multiple steady states do not exist. It is obvious that the external effectiveness factor by itself is mainly of academic interest except for those unusual cases where the catalyst has only an external surface. 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

Tamman (29) observed in 1920 that for electrically heated catalytic wire multiple steady states occur for a certain value of electrical current. Similar experimental observations were done by Buben (31) and Davies (30) and recently by Rader (35), Barelko (36), and Cardoso and Luss (33). The Luss results involving oxidation of butane and carbon monoxide on a platinum wire will be discussed in detail. 

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