CSTR dynamics

The widest spectrum of dynamic behavior is observed in the CSTR. As we have seen, the use of a CSTR or CSTR train for polymerization reactions may be justified in some cases by kinetic considerations. However, before implementing CSTR polymerization, the engineer should be aware of the unique dynamics associated reactions in a CSTR which are exothermic and/or autocatalytic, or involve nucleation phenomena. 

CSTR polymerization reactors can also be subject to oscillatory behavior [12]. A nonisothermal CSTR free radical solution polymerization can exhibit damped oscillatory approach to a steady state, unstable (growing) oscillations upon disturbance, and stable (limit cycle) oscillations in which the system never reaches steady state, and never goes unstable, but continues to oscillate with a fixed period and amplitude. However, these phenomena are more commonly observed in emulsion polymerization. 

---

J.B. Planeaux and K.F. Jensen. Bifurcation phenomena in CSTR dynamics A system with extraneous thermal capacitance. Chem. Eng. Sci., 41 (6) 1497-1523, 1986. 

CSTR (dynamic and steady state). If the outflow of the CSTR is regulated so that the CSTR has constant volume, then we can conclude from Equation 4.55 that Q = Q/. 

For the most part we have labored over the analysis of steady-state problems, although there have been some important side trips into the unsteady state. Principal among these were the analysis of CSTR startup, visits to fixed-bed and CSTR dynamics arising from catalyst deactivation, and some discussion on adsorption variations. The purpose of this chapter is to pursue some of these topics in more detail the range of interests here is rather broad, but all can be linked through a common concern with fixed-bed dynamics. 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

In cases where a large reactor operates similarly to a CSTR, fluid dynamics sometimes can be estabflshed in a smaller reactor by external recycle of product. For example, the extent of soflds back-mixing and Hquid recirculation increases with reactor diameter in a gas—Hquid—soflds reactor. Consequently, if gas and Hquid velocities are maintained constant when scaling and the same space velocities are used, then the smaller pilot unit should be of the same overall height. The net result is that the large-diameter reactor is well mixed and no temperature gradients occur even with a highly exothermic reaction. 

The dynamic behavior of nonisothermal CSTRs is extremely complex and has received considerable academic study. Systems exist that have only a metastable state and no stable steady states. Included in this class are some chemical oscillators that operate in a reproducible limit cycle about their metastable... 

There are many variations on this theme. Fed-batch and continuous emulsion polymerizations are common. Continuous polymerization in a CSTR is dynamically unstable when free emulsifier is present. Oscillations with periods of several hours will result, but these can be avoided by feeding the CSTR with seed particles made in a batch or tubular reactor. 

FIGURE 14.1 Dynamic response of a CSTR to changes in inlet concentration of a component reacting with first-order kinetics. 

Dynamic differential equation balances were written to calculate the molar concentration of each species in the reactor. These equations consist of inflow, outflow, accumulation, and reaction terms for a CSTR. If there are no outflow terms, the equations reduce to semibatch... 

This, like the other dynamic balances for the CSTR, follows the full generalised form, of Sec. 1.2.5, giving... 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

CSTR WITH EXOTHERMIC REACTION AND JACKET COOLING Dynamic solution for phase-plane plots Located steady-states with THERMPLO and use same parameters. 

Fig. 6. Dynamical phase diagram of the ascorbic acid/copper(II)/oxygen system in a CSTR in the kf — [Cu2+]0 plane. Fixed reactor concentrations [H2Asc]0 = 5.0x10 4M [H2SO4]0 = 6.0 x 10-5 M [Na2SO4]0 = 0.04M. Symbols O, steady state , oscillations , bistability. The asterisk ( ) marks the Takens-Bogdanov point. Strizhak, P. E. Basylchuk, A. B. Demjanchyk, I. Fecher, F. Shcneider, F. W. Munster, A. F. Phys. Chem. Chem. Phys. 2000, 2, 4721. Reproduced by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

We now proceed to demonstrate the application of the NDDR technique using a simulated CSTR with a first-order, exothermic reaction. The example was taken from Liebman et al. (1992). The dynamic model is given by... 

SJi. The initial startup of an adiabatic, gas-phase packed tubular reactor makes a good example of how a distributed system can be lumped into a series of CSTRs in order to study the dynamic response. The reactor is a cylindrical vessel (3 feet ID by 20 feet long) packed with a metal packing. The packing occupies 5 percent of the total volume, provides 50 ft of area per of total volume, weighs 400 ib yft and has a heat capacity of 0.1 Btu/lb °F. The heat transfer coefficient between the packing and the gas is 10 Btu/h It "F. 

Sometimes useful information and insight can be obtained about the dynamics of a system from just the steadystate equations of the system. Van Heerden Ind. Eng. Chem. Vol. 45, 1953, p. 1242) proposed the application of the following steadystate analysis to a continuous perfectly mixed chemical reactor. Consider a nonisothermal CSTR described by the two nonlinear ODEs... 

Example 2. Let us now test the robust linear regulator under model mismatch. For this application let us consider again the CSTR of Example 1. Here, we assumed that the CSTR model dynamics is given by... 

J. Hamer, T. Akramov, and W. Ray. The dynamic behavior of continuous polymerization reactors II, nonisothermal solution homopolymerization and copolymerization in a CSTR. Chem. Eng. Sci., 36 1897-1914, 1981. 

In this section an innovating approach based on the use of a battery of interval observers functioning in parallel is presented. Such an approach allows us to detect a violation of the assumptions related to the unknown inputs (substrates concentrations at the input of the process). In order to illustrate the principle of this approach, consider again a mono-biomass, mono-substrate bioprocess within a CSTR, described by equations (3) to which the dynamics of one of the products, represented by P, has been added ... 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

From the study presented in this chapter, it has been demonstrated that a CSTR in which an exothermic first order irreversible reaction takes place, can work with steady-state, self-oscillating or chaotic dynamic. By using dimensionless variables, and taking into account an external periodic disturbance in the inlet stream temperature and coolant flow rate, it has been shown that chaotic dynamic may appear. This behavior has been analyzed from the Lyapunov exponents and the power spectrum. 

The non-linear dynamics of the reactor with two PI controllers that manipulates the outlet stream flow rate and the coolant flow rate are also presented. The more interesting result, from the non-linear d mamic point of view, is the possibility to obtain chaotic behavior without any external periodic forcing. The results for the CSTR show that the non-linearities and the control valve saturation, which manipulates the coolant flow rate, are the cause of this abnormal behavior. By simulation, a homoclinic of Shilnikov t3rpe has been found at the equilibrium point. In this case, chaotic behavior appears at and around the parameter values from which the previously cited orbit is generated. 

In regard dynamics and control scopes, the contributions address analysis of open and closed-loop systems, fault detection and the dynamical behavior of controlled processes. Concerning control design, the contributors have exploited fuzzy and neuro-fuzzy techniques for control design and fault detection. Moreover, robust approaches to dynamical output feedback from geometric control are also included. In addition, the contributors have also enclosed results concerning the dynamics of controlled processes, such as the study of homoclinic orbits in controlled CSTR and the experimental evidence of how feedback interconnection in a recycling bioreactor can induce unpredictable (possibly chaotic) oscillations. 

Gray, P. and Scott, S. K. (1985). Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem., 89, 22-32. Lin, K. F. (1981). Multiplicity, stability and dynamics for isothermal autocatalytic reactions in CSTR. Chem. Eng. Sci., 36, 1447-52. 

Two dynamic alternatives to the static approach have been used in HO calibration and measurement. In the CSTR (continuously stirred tank reactor) approach, air containing the tracer or tracers flows into the reactor to balance the bulk flow out to the HO measuring devices, and the contents are stirred by a fan or other means. The HO chemical tracer is measured in the inlet flow to obtain [T]() and in the outlet flow to obtain [T], Mass balance requires... 

The second dynamic approach is the atmospheric pressure flow tube, in which an organized two-dimensional flow field replaces the bulk mixing of the CSTR, as has been used by Davis and co-workers (77) for in-flight calibration of their EAPF system. It might be difficult to adapt this method to generate known HO for a instrument intercomparison, however. 

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). 

---