Multiple steady states isothermal

The same complex kinetics gives multiple steady states isothermally in thin or thick porous electrocatalysts (418). A simple, graphical orthogonal collocation method (422) can show the existence of multiple solutions for concentration within a certain potential range (418). If ohmic losses in the pores cause a potential change within the electrode structure, multiplicity can also arise with respect to potential, even with simpler rate expressions... 

Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbohc rate equations and processes with porous catalysts also can have such behavior. 

FIG. 23-17 Multiple steady states of CSTRs, stable and unstable, adiabatic except the last item, (a) First-order reaction, A and C stable, B unstable, A is no good for a reactor, the dashed line is of a reversible reaction, (h) One, two, or three steady states depending on the combination Cj, Ty). (c) The reactions A B C, with five steady states, points 1, 3, and 5 stable, (d) Isothermal operation with the rate equation = 0 /(1 -I- C y = (C o Cy/t. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

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. 

Particular forms of the rate equation can give rise to multiple steady states even under isothermal conditions, as in problem P4.10.09 where... 

Finally, we note that Dudukovic has shown that, for isothermal reactors, the degree of micromixing can significantly influence the appearance of multiple steady-state operating conditions, this being particularly likely for kinetic mechanisms which display maximum reaction rates at intermediate concentration levels [38, 39]. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

Thus we see that for nonisothermal reactors this 1/r versus Cao Ca curve is not always an increasing function of conversion as it was for isothermal reactors even with positive-order kinetics. Since the 1/r curve can have a rninimum for the nonisothermal reactor, we confirm the possibility that the CSTR requires a smaller volume than the PFTR for positive-order kinetics. This is hue even before the multiple steady-state possibilities are accounted for, which we will discuss in the next chapter. This is evident from our 1 /r plot for the PFTR and CSTR and will occur whenever r has a sufficiently large maximum that the area under the rectangle is less than the area under the curve of 1/r versus Cao Ca-... 

In this and the previous chapters we considered the effects of nonisothermal operation on reactor behavior. The effects of nonisothermal operation can be dramatic, especially for exothermic reactions, often leading to reactor volumes many times smaller than if isothermal and often leading to the possibility of multiple steady states. Further, in nonisothermal operation, the CSTR can require a smaller volume for a given conversion than a PFTR. In this section we summarize some of these characteristics and modes of operation. For endothermic reactions, nonisothermal operation cools the reactor, and this reduces the rate, so that these reactors are inherently stable. The modes of operation can be classified as follows ... 

Emulsion Polymerization in a CSTR. Emulsion polymerization is usually carried out isothermally in batch or continuous stirred tank reactors. Temperature control is much easier than for bulk or solution polymerization because the small (. 5 Jim) polymer particles, which are the locus of reaction, are suspended in a continuous aqueous medium as shown in Figure 5. This complex, multiphase reactor also shows multiple steady states under isothermal conditions. Gerrens and coworkers at BASF seem to be the first to report these phenomena both computationally and experimentally. Figure 6 (taken from ref. (253)) plots the autocatalytic behavior of the reaction rate for styrene polymerization vs. monomer conversion in the reactor. The intersection... 

Samer also demonstrates the existence of multiple steady states in isothermal miniemulsion polymerization in a CSTR. This is not surprising, since multipH-city is a function of gel or Trommsdorf effect, and not of nucleation mechanism. 

Matsuura and Kato (3 ) predict the possibility of multiple steady states under isothermal conditions because of the gel effect, and Gerrens et al (20) experimentally verified these predictions with styrene emulsion polymerization. 

Multiple steady states as discussed in the previous subsection are related to the nonisothermicity of the CSTR. However, even in the isothermal case, a CSTR is known to be able to exhibit multiple steady states, periodic orbits, and chaotic behavior for sufficiently complex reaction network structures (see, e.g.. Gray and Scott, 1990). When the number of reactions is very large, the problem becomes a formidable one. In a series of papers (Feinberg 1987, 1988, and the literature quoted therein), Feinberg and his coworkers have developed a procedure for CSTRs that can be applied to systems with arbitrarily large numbers of reactants and reactions. The procedure is based on the deficiency concept discussed in Appendix C. 

There are many (otherwise unremarkable) instances of reaction networks where the Schlosser and Feinberg method gives no answer. Multiple steady states are not excluded but are not guaranteed to be possible either. In actual fact, multiplicity of steady states in isothermal homogeneous CSTRs has seldom been observed. This suggests, as Schlosser and Feinberg conclude, the existence of as yet unknown theorems which, like theorem 4.1, deny the capacity for multiple steady states but which have an even wider range. ... 

Feinberg, M., Chemical reaction network structure and the stability of complex isothermal reactors. II, Multiple steady states for networks of deficiency one. Chem. Eng. Sci. 43, 1 (1988). 

Rumschitzki, D., and Feinberg, M., Multiple steady states in complex isothermal CFSTR s. II, Homogeneous reactors. Chem. Eng. Set. 43,329 (1988). 

Example 3.2.2. Series Solutions for Non-isothermal Catalyst Pellet - Multiple Steady States... 

Lowe, A. and Bub, G., Multiple Steady States for Isothermal Catalytic Gas-Solid Reactions with a Positive Reaction Order, Chem. Eng. Sci., 31, 175-178 (1976). 

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