Multiplicity steady state

Up to scale, this is the dependence of overall reaction rate on concentration Cb in the assumption of constant temperature and concentrations c 2 and Cab- All figures in this chapter illustrate certain qualitative features of kinetic behavior, i.e. rate-limitation, vicinity of equilibrium, steady-state multiplicity, etc. Parameter values are selected to illustrate these qualitative features. Certainly these features could be illustrated with "realistic" kinetic parameters. 

Figure 6 Approximations of the thermodynamic branch steady-state multiplicity case (see Figure 1). Solid line is the first-term hypergeometric approximation. Circles correspond to the higher-order hypergeometric approximation (m = 3). Dashed line is the first-order approximation in the vicinity of thermodynamic equilibrium. Dash-dots correspond to the second-order approximation in the vicinity of thermodynamic equilibrium.

Figure 9 shows the convergence domain as well as steady-state multiplicity domain on fi, h plane. We can see that steady-state multiplicity is not generally... 

Figure 9 A convergence domain and steady-state multiplicity domain.

Figure 13 Dependencies from Figure 10 at f = 4 case of the steady-state multiplicity.

The conditions in terms of N4 for the occurrence of the steady state multiplicity can be summarized as follows. According to (4.3.17b), for co 1, there exists a low current branch with current saturation as long as the following condition holds ... 

Balakotaiah, V. and Luss, D., 1982b, Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter-systems. Chem. Engng ScL 37,433-445. 

Steady state multiplicity in surface reactions with coverage dependent parameters (with... 

So far there is a number of experimentally oriented papers dealing with steady-state multiplicity and stability, however, the majority of them are devoted to homogeneous systems. The important papers dealing with heterogeneous systems are discussed below. 

Fig. 13. The effect of the reactor volume V on the stability of steady states. Multiple steady states 45). (Copyright by Pergamon Press. Reprinted with permission.)...

Ciric AR, Miao P. Steady-state multiplicities in an ethylene glycol reactive distillation column. Ind Eng Chem Res 1994 33 2738-2748. 

For a standpipe system, different flow patterns of steady-state flows may exist, depending upon the ranges of operational parameters of the system. This phenomenon is commonly known as the steady-state multiplicity. We introduce the general concepts of the steady-state multiplicity based on the standpipe system in 8.3.3, in which the solids in the feed hopper are kept in a steady moving bed motion. 

For a system with a fixed y, the steady-state multiplicity may be revealed by the curve of superficial velocity of solids Up versus Ap, as illustrated in Fig. 8.17, where a hysteresis loop is formed between two branch (upper and lower) curves [Chen et al., 1984], The upper branch corresponds to Regimes 2, 3, or 4, whereas the lower one corresponds to Regime 1. At Ap less than (Ap)i, the system is operated on the upper branch of the curves, while it is operated on the lower branch of the curves at Ap larger than (Ap)2- Between these two... 

Follstad BD, Balcarcel RR, Stephanopoulos G, Wang DIC (1999), Metabolic flux analysis of hybridoma continuous culture steady state multiplicity, Biotechnol. Bioeng. 63 675-683. 

Figure 3.6 shows codimension-1 singularities in a parameter plane spanned by the activation energy of the anodic reaction yA and the activation energy of the electrolyte conductivity yK, when the total cell current I is used as the bifurcation parameter. Hysteresis and isola varieties are found. The nonlinear behavior becomes increasingly complicated for increasing values of yK. Steady-state multiplicities even exist for rather small values of yK, but vanish if the electrolyte s conductivity is temperature-independent, i.e., yK = 0. This confirms that the multiple steady states are caused not by the electrochemical reaction alone, but by the combination of reaction and varying electrical conductivity. 

J. Moxley, S. Tulyani, J. Benziger, Steady-state multiplicity in the autohumidification polymer electrolyte membrane fuel cell. Chem. Engng. Sci., 2003, 58, 4705-4708. 

Steady-State Multiplicity and Stability A simple example of steady-state multiplicity is due to the interaction between kinetics and heat transport in an adiabatic CSTR. For a first-order reaction at steady state, Eq. (19-13) gives... 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

Some specific aspects in the modeling of gas-liquid continuous-stirred tank reactors are considered. The influence of volatility of the liquid reactant on the enhancement of gas absorption is analyzed for irreversible second-order reactions. The impact of liquid evaporation on the behavior of a nonadiabatic gas-liquid CSTR where steady-state multiplicity occurs is also examined. 

The occurrence of steady-state multiplicity in gas-liquid CSTRs has been demonstrated in experimental (9) and theoretical investigations (cf., 10). The irreversible second-order reaction system, in particular, has been treated extensively in several theoretical studies (10-15). These studies are however based on neglecting energy and material losses which result from evaporation of the liquid. 

Example 12.10 Multiple steady states Multiple steady states and dissipative structures may play an important role in nerve excitations. Consider the following simple set of reactions ... 

Packed-bed reactors are discussed qualitatively, particularly with respect to their models. Features of the two basic types of models, the pseudohomogene-ous and the heterogeneous models, are outlined. Additional issues — such as catalyst deactivation steady state multiplicity, stability, and complex transients and parametric sensitivity — which assume importance in specific reaction systems are also briefly discussed. 

Steady State Multiplicity, Stability, and Complex Transients. This subject is too large to do any real justice here. Ever since the pioneering works of Liljenroth (41), van Heerden (42), and Amundson (43) with continuous-flow stirred tank reactors, showing that multiple steady states — among them, some stable to perturbations, while others unstable — can arise, this topic has... 

Jensen and Ray (50) have recently tabulated some 25 experimental studies which have demonstrated steady state multiplicity and instabilities in fixed-bed reactors many of these (cf., 29, 51, 52) have noted the importance of using a heterogeneous model in matching experimental results with theoretical predictions. Using a pseudohomogeneous model, Jensen and Ray (50) also present a detailed classification of steady state and dynamic behavior (including bifurcation to periodic solutions) that is possible in tubular reactors. 

A feature related to steady state multiplicity and stability is that of "pattern formation", which has its origins in the biological literature. Considering an assemblage of cells containing one catalyst pellet each, Schmitz (47, 53) has shown how non-uniform steady states - giving rise to a pattern - can arise, if communication between the pellets is sufficiently small. This possibility has obvious implications to packed-bed reactors. 

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