Multiple steady states plug flow

Plug flow reactors with recycle exhibit some of the characteristics of CSTRs, including the possibility of multiple steady states. This topic is explored by Penmutter Stah dity of (%emical Reactors, Prentice-Hall, 1972). 

Just as we approached reactor control in Chap. 4, we will start by exploring the open-loop effects of thermal feedback. Consider Fig. 5.19, which shows an adiabatic plug-flow reactor with an FEHE system. We have also included two manipulated variables that wall later turn out to be useful to control the reactor. One of these manipulated variables is the heat load to the furnace and the other is the bypass around the preheater. It is clear that the reactor feed temperature is affected by the bypass valve position and the furnace heat load but also by the reactor exit temperature through the heat exchanger. This creates the possibility for multiple steady states. We can visualize the different... 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

Three important (complicating) possibilities were not considered in the treatment of reactors presented in earlier chapters (1) the residence time of the reactant molecules need not always be fully defined in terms of plug flow or fully mixed flow (2) the equations describing certain situations can have more than one solution, leading to multiple steady states and (3) there could be periods of unsteady-state operation with detrimental effects on performance, that is, transients could develop in a reactor. 

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... 

Industrial fixed-bed catalytic reactors have a wide range of different configurations. The configuration of the reactor itself may give rise to multiplicity of the steady states when other sources alone are not sufficient to produce the phenomenon. Most well known is the case of catalytic reactors where the gas phase is in plug flow and all diffusional resistances are negligible, while the reaction is exothermic and is countercurrently cooled. One typical example for this is the TVA type ammonia converter [38-40]. 

Inverse response creates control difficulties. Assume, for example, that we wish to control the exit temperature of an adiabatic plug-flow reactor by manipulating the inlet temperature as shown in Fig. 4.13. From a steady-state viewpoint this is a perfectly reasonable thing to consider, since there are no issues of output multiplicity or open-loop instability, assuming the fluid is in perfect plug flow and there is no... 

Plug-flow membrane reactors arc not faced with potentially unstable states since no back mixing of mass or heat is involved. However, when the product is recycled, heat is exchanged between the product and the feed streams or dispersive backmixing exists, multiple steady slates can occur and membrane reactor stability needs to be considered. 

Falling Off the Steady State 623 Nonisothermal Multiple Reactions 625 Unsteady Operation of Plug-Flow Reactors... 

The phenomenon of parametric sensitivity is not directly related to the multiplicity phenomenon, for parametric sensitivity is found using plug flow pseudo-homogeneous models which, by definition, do not show multiplicity of the steady states. A good review of the phenomenon is given by Rajadhyaksha and Palekar (1984). More recent research on this problem include the extensive work of Westerterp and co-workers (e.g. Westerterp and Ptasinky, 1984a,b Westerterp el al., 1984 Westerterp and Overtoom, 1985). 

Nonlinear phenomena can be induced by material recycles. For examples, the plug-flow reactor (PFR) with recycle of a fraction of reactor s effluent can exhibit state multiplicity, sustained oscillations around a unique steady state and chaotic behaviour. If the material recycle has the temperature equal to the reactor effluent the true cause of the nonlinear phenomena is the material feedback. 

A few words on the stability of steady states of polymerization. This question arises immediately as soon as the multiplicity of steady-state conditions spears. It is well known that three solutions are possible in the flow of reactants. The general theory of thermal instability of reactors has been developed in detail in Refe. [16-20,30,31], and the theory of kinetic instability caused by peculiarities of the kinetic schenK (self-acceleration. gel-effect, etc. in Refs. [37-40]). The instability of steady states of poly-nKiization plug reactors of a hydrodynamic nature is more interesting for the present paper. It can be assumed that the state corresponding to the negative slopes of the P(Q) curve are unstable if P = const is maintained [30, 33, 34]. At Q = const, all states are stable and realizable. The analysis of this problem in zero-dimensional formulation [41], for a reactor determined by only one value of T, p, q and a complex variable hydrodynamic resistance has shown that the slope of the curve is not an exhaustive stability criterion. 

Aris et al. have primarily analyzed whether the steady-state multiplicity features in a CSTR arising from a cubic rate law also can arise for a series of successive bimolecular reactions [26]. Aris et al. have showed that the steady-state equations for a CSTR with bimolecular reactions scheme reduces to that with a cubic reaction scheme when two parameters e(=k,Cg/k j) and K(=kjC /k j) arising in system equations for the bimolecular reactions tend to zero. Aris et al. have shown that the general multiplicity feature of the CSTR for bimolecular reactions is similar to that of the molecular reactions only at smaller value of e and K. The behavior is considerably different at larger values of e and K. Chidambaram has evaluated the effect of these two parameters (e and K) on the periodic operation of an isothermal plug flow reactor [18]. 

This multiplicity of steady states is caused by the highly nonlinear nature of the heat generation expression and by the internal thermal feedback associated with complete mixing. Note that the phenomenon is not limited to completely mixed reactors but can also occur with plug flow reactors with external recycle, as discussed by van Heerden [1953] and, a long time before, by Liljenroth [1918] (see Chapter 11). 

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