Flow reactors multiple steady states

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... 

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... 

Piston flow reactors lack any internal mechanisms for memory. There is no axial dispersion of heat or mass. What has happened previously has no effect on what is happening now. Given a set of inlet conditions (flin, 7i , Text), only one output (flout, 7 out)is possible. A PFR cannot exhibit steady-state multiplicity unless there is some form of external feedback. External recycle of mass or heat can provide this feedback and may destabilize the system. Figure 14.7 shows an example of external feedback of heat that can lead to the same multiple steady states possible with a CSTR. Another example is when the vessel walls or packing has significant thermal capacity. In such cases, a second heat balance must be added to supplement Equation 14.16. See Section 10.6 for a comparable result. 

It is interesting to note that this control structure exhibits multiple steady-state solutions. There are two sets of recycle flow rates, reactor temperatures, and reactor compositions that give the same production rates for the same feed rates. Structures that give multiple steady states should be avoided because the operation of the plant may be quite erratic. 

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). 

Multiple steady-state behavior is a classic chemical engineering phenomenon in the analysis of nonisothermal continuous-stirred tank reactors. Inlet temperatures and flow rates of the reactive and cooling fluids represent key design parameters that determine the number of operating points allowed when coupled heat and mass transfer are addressed, and the chemical reaction is exothermic. One steady-state operating point is most common in CSTRs, and two steady states occur most infrequently. Three stationary states are also possible, and their analysis is most interesting because two of them are stable whereas the other operating point is unstable. 

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. 

Interest in flowing systems focuses on (i) the existence of multiple steady-state solutions of the reactor equations and (ii) the stability of such solutions. These are not independent the existence of parametrically sensitive regions of dynamic behaviour can give rise to oscillations in both temperature and concentration, constant in period and amplitude. The anticipation and control of such oscillatory modes of reaction is clearly of no less importance to the successful operation of the reactor than is the prediction of its stability. 

As shown on the CD-ROM that accompanies this book (following the links HYSYS — Chemical Reactors Setting Up Reactors —> CSTR or ASPEN —> Chemical Reactors Kinetic Reactors — CSTRs RCSTR), analysis of this process shows the possibility of multiple steady states. For example, at a water flow rate of 400 kmol/hr, the following steady states are obtained (1) conversion of 83% with an effluent temperature of 62°C, (2) conversion of 45% with an effluent temperature of 44°C, and (3) conversion of 3% with an effluent temperature of 25°C. The intermediate steady state at 45% conversion is unstable, while the other two steady states are stable. Furthermore, a controllability and resiliency (C R) analysis on this process is carried out in Case Study 21.1, where a design involving a single CSTR is compared with one utilizing two CSTRs in series. ... 

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... 

Basic PFR equation Design equations Nonisothermal operation Perfectly mixed flow reactor (MFR) Basic CSTR equation Nonisothermal operation Multiple steady states MSS In a CSTR Adiabatic CSTR... 

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