Chemical reactors multiple steady states

The Belousov-Zhabotinskii reaction has also caught the attention of chemical engineers who, in the study of chemical reactor design, have been interested for some time now in chemical instabilities, multiple steady state behaviour and sustained oscillations (Schmitz, 1974). 

Hlavicek V, Hofmann H. Modeling of chemical reactors—XVII Steady state axial heat and mass transfer in tubular reactors. Numerical investigation of multiplicity. Chemical Engineering Science 1970 25 187-199. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Plug flow reactors with recycle exhibit some of the characteristics of mixed reactors, including the possibility of multiple steady states. This topic is explored in other books (Perlmutter, Stability of Chemical Reactors, Chapter 9, 1972). 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

Forced oscillations of chemical reactors with multiple steady states (with G.A. Cordonier and L.D. Schmidt). Chem. Eng. Sci. 45,1659— 1675 (1990). 

The starting point of a number of theoretical studies of packed catalytic reactors, where an exothermic reaction is carried out, is an analysis of heat and mass transfer in a single porous catalyst since such system is obviously more conductive to reasonable, analytical or numerical treatment. As can be expected the mutual interaction of transport effects and chemical kinetics may give rise to multiple steady states and oscillatory behavior as well. Research on multiplicity in catalysis has been strongly influenced by the classic paper by Weisz and Hicks (5) predicting occurrence of multiple steady states caused by intrapellet heat and mass intrusions alone. The literature abounds with theoretical analysis of various aspects of this phenomenon however, there is a dearth of reported experiments in this area. Later the possiblity of oscillatory activity has been reported (6). 

In a chemical packed-bed reactor in which a highly exothermic reaction is taking place conditions may be encountered under which, for a given set of input conditions (feed rate, temperature, concentration), the exit conversion is either high or low. To the experimental study of conditions connected with the existence of multiple steady states in packed catalytic reactors has not been paid as much attention in the past as to the study of... 

The area of reactor design has been widely studied, and there are many excellent textbooks that cover this subject. Most of the emphasis in these books is on steady-state operation. Dynamics are also considered, but mostly from the mathematical standpoint (openloop instability, multiple steady states, and bifurcation analysis). The subject of developing effective stable closedloop control systems for chemical reactors is treated only very lightly in these textbooks. The important practical issues involved in providing reactor control systems that achieve safe, economic, and consistent operation of these complex units are seldom understood by both students and practicing chemical engineers. 

Chemical reactors are inherently nonlinear in character. This is primarily due to the exponential relationship between reaction rate and temperature but can also stem from nonlinear rate expressions such as Eqs. (4.10) and (4.11). One implication of this nonlinearity for control is the change in process gain with operating conditions. A control loop tuned for one set of conditions can easily go unstable at another operating point. Related to this phenomenon is the possibility of open-loop instability and multiple steady states that can exist when there is material and/or thermal recycle in the reactor. It is essential for the control engineer to understand the implications of nonlinearities and what can be done about them from a control standpoint as well as from a process design standpoint. 

Here we have dealt with the control of chemical reactors. We covered some of the fundamentals about kinetics, reactor types, reactor models, and open-loop behavior. In particular we have shown that reactors with recycle or backmixing can exhibit multiple steady states, some of which are unstable. Nonlinearities in reactor systems also frequently give rise to open-loop parametric sensitivity. 

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). 

In section 2.2.7, multiple steady states in a heterogeneous chemical reaction (one dependent variable) and a jacketed stirred tank reactor (two dependent variables) were analyzed. Both stable and unstable steady states were obtained. The transient behavior of the system was found to depend on the initial conditions. The methodology and Maple programs presented in this chapter should be valid for any system of IVPs with multiple steady states. In section 2.2.8, phase plane behavior of a jacketed stirred tank reactor was analyzed. The program provided should be of use for analyzing phase plane behavior of different chemical systems. A total of ten different examples were presented in this chapter. 

In section 3.2.7, boundary value problems were solved as initial value problems. This methodology is especially useful for predicting the performances in chemical reactors. Maple s stop condition was used in this section to obtain t] vs. O curves. This is very useful because, it is generally easier to solve an initial value problem than a boundary value problem. This technique was then used in section 3.2.8 to predict multiple steady states in a catalyst pellet in section 3.2.8. This methodology is extremely useful for predicting the hysteresis curves in multiple steady state problems. 

In addition to the multiple steady states and sustained oscillations, chemical reactors are a rich source of other types of behavior including deterministic chaos [18, 17, 23, 14], The chemical engineer should be aware of the complex behavior that is possible with simple nonlinear CSTR models, especially if confronted with apparently complex operating data. 

Therefore, a systemic approach is necessary in designing chemical reactors. In this chapter, we will demonstrate that a minimum reactor volume is required in a recycle system for flexible operation. Moreover, the nominal design should be placed sufficiently far away from the maximum sensitivity region of the relation conversion-reactor volume. It is worthy to note that multiple steady states are possible in the case of several recycles of reactants. The recycle of heat gives an even more complicated behaviour. Moreover, the stability and performance of the reactor system depends also on the control structures of other units. Hence, reactor design and feasibility of the control structures at the plant level should be examined simultaneously. 

The nonlinearity of chemical processes received considerable attention in the chemical engineering literature. A large number of articles deal with stand-alone chemical reactors, as for example continuously stirred tank reactor (CSTR), tubular reactor with axial dispersion, and packed-bed reactor. The steady state and dynamic behaviour of these systems includes state multiplicity, isolated solutions, instability, sustained oscillations, and exotic phenomena as strange attractors and chaos. In all cases, the main source of nonlinearity is the positive feedback due to the recycle of heat, coupled with the dependence of the reaction rate versus temperature. 

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. 

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