Multiple Steady-State Operation

In addition to the performance variations with reactant concentration and gas hourly space velocity (GHSV), there can be multiple steady states observed. Generally, a reactor in which a single, exothermic reaction is occurring will operate in one of two stable steady states. Additionally, an unstable steady-state solution to the mass 

Additional experiments have been conducted that varied A from 0.7 to 3.0, holding all other feed conditions constant.8 Dual steady-state operation was observed for the entire range tested providing more evidence that the water amount in the feed composition is probably the dominant factor. This type of experiment was repeated for a different CO feed concentration and nearly the same behavior was recorded. This dual steady-state nature persists for a wide range of lambda, which points to the fact that dual steady state is more influenced by feed composition and more specifically by water amount. Since water has high heat capacity, this may temper the heat rise and delay the onset of H2 oxidation reaction. 

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For the parameters used in the program, the system is characterised by multiple steady-state operating conditions. For example, four of them are as follows... 

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]. 

Chen, R., McAvoy, T.J., and Zafiriou, E. (2004). Plantwide control system design extension to multiple-forcing and multiple-steady-state operation. Ind. Eng. Chem. Res., 43, 3685-3694. 

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

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. 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

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. 

The global design equations for packed beds—e.g.. Equations (10.1), (10.9), (10.39), and (10.40)—all have a similar limitation to that of the axial dispersion model treated in Chapter 9. They all assume steady-state operation. Adding an accumulation term, da/dt accounts for the change in the gas-phase inventory of component A but not for the surface inventory of A in the adsorbed form. The adsorbed inventory can be a large multiple of the gas-phase inventory. 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

Which of the possible multiple states actually occurs in steady-state operation ... 

It is suspected that this nonlinear rate form, which has a maximum value, may cause certain regions of unstable operation with multiple steady states. How should the operation be conducted to ensure unique steady conditions ... 

Whenever multiple steady states in a reactor are possible, we must be very concerned that we are operating on the desired steady-state branch. This requires a proper startup procedure to attain the desired steady state and suitable operation limits to make sure that we never exhibit a sufficiently large transient to cause the system to fall off the desired conversion branch. We will consider transients in the CSTR in the next section. 

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 ... 

In order to assist convergence in this circuit, the UIC statement was included in the. TRAN simulation. This statement helps in circuits where a steady-state operating point may not exist, multiple stable operating points exist, or it is difficult for SPICE to determine the correct operating point. 

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

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. 

The model is solved numerically. Figure 9.12 presents the reaction temperature and conversion versus heat-transfer capacity of the reactor, UA. From Figure 9.12, we find that we need to provide UA = 95 kW/K heat-transfer capacity, if we want to reach the operating point T = 268 K, xA = 0.4. Figure 9.12 also shows that the combined FEHE-reactor system does not show multiple steady states, which should always be a concern when designing heat-integrated reactors [16]. 

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

Whether or not multiple steady states will appear, and how large the deviation of the effectiveness factors between both stable operating points will be, is determined by the values of the Prater and Arrhenius numbers. Effectiveness factors above unity generally occur when p > 0 (exothermal reactions). However, for the usual range of the Arrhenius number (y = 10-30), multiple steady states are possible only at larger Prater numbers (see Fig 13). For further details on multiple steady states, the interested reader may consult the monograph by Aris [6] or the works of Luss [69, 70]. 

To give an example, Fig. 20 shows a diagram for E = 10 and various selected values of Kp fi. The dashed lines indicate the range over which multiple steady states of t/(ip) occur. Here, by means of numerical methods it is not possible to determine a unique solution of the effectiveness factor of the pellet for given conditions at the external pellet surface [91]. Which operating point will be observed in a real situation again depends upon the direction from which the stationary state is approached [91]. 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

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