Bifurcation reactor

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

However, because of the strong nonllnearltles In the reactor fiow problem, continuation procedures must be used to obtain a good Initial guess for the Newton Iteration. A simple first order continuation scheme falls at fiow transition points (bifurcations) where the Jacobian, G, becomes singular. To circumvent this problem an arclength continuation scheme discussed by Keller (26.27) and Chan (2S.) Is used which leads to the Infiated system ... 

Since the solution changes stability at the turning points, these are Important to the understanding of the overall reactor behavior. In principle, the Infiated system may be used to determine the bifurcation points and switch solution branches by Increasing the arclength parameter, s, and... 

Reactor 11 [R 11] Bifurcation-distributive Chip Micro Mixer... 

Reactor type Bifurcation-distributive chip micro mixer ... 

F. Teymour and W.H. Ray. The dynamic behavior of continuous solution polymerization reactors-IV. Dynamic stabihty and bifurcation analysis of an experimental reactor. Chem. Eng. Sci., 44(9) 1967-1982, 1989. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

Eq.(50) shows the variation of the equilibrium dimensionless temperature as a function of the maximum value of the dimensionless coolant flow rate X6max- Plotting XQmax versus X3e a bifurcation curve can be obtained, from which it is possible to determine the value of xsmax which gives a different behavior of the reactor in steady state. It is interesting to note that Eq.(50) is equal to Eq.(47) when we make the substitutions of Eq.(49) into Eq.(47). 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

Figure 26.56 is the corresponding plot for 12% inlet H2 in air. In this case, there is an extinction at about 1000 K for both reactors. The qualitative features are similar to that of the PSR discussed above for 28% H2 in air. For such fuel-lean mixtures, the flame is attached to the surface. As a result, the thermal coupling between the surface and the gas phase is strong, and reduction in surface temperature affects the entire thermal boundary layer resulting in significant reduction of NOj,. These results indicate that the bifurcation behavior, in terms of extinction, determines the role of flame-wall thermal interactions in emissions. 

Kalamatianos, S., and D. G. Vlachos. 1995. Bifurcation behavior of premixed hydrogen/ air mixtures in a continuous stirred tank reactor. Combustion Science Technology 109(l-6) 347-71. 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

Fig. 1.14. Four more of the possible stationary-state bifurcation diagrams for chemical systems (see also Fig. 1.2) in flow reactors (a) isola (b) mushroom (c) isola + hysteresis loop ...

Guckenheimer, J. (1986). Multiple bifurcation problem for chemical reactors. Physica, D20, 1-20. 

When the catalyst decays we have a two-variable system and hence there is the potential for Hopf bifurcations and sustained oscillations. In our flow reactor, we have the possibility of oscillation about one stationary state... 

The interest in periodically forced systems extends beyond performance considerations for a single reactor. Stability of structures and control characteristics of chemical plants are determined by their responses to oscillating loads. Epidemics and harvests are governed by the cycle of seasons. Bifurcation and stability analysis of periodically forced systems is especially important in the... 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

Chang, H. C., 1983, The domain model in heterogeneous catalysis. Chem. Engng ScL 38,535-546. Cohen, D. S. and Neu, J. C., 1979, Interacting oscillatory chemical reactors. In Bifurcation Theory and Applications in Scientific Disciplines. N.Y. Acad. Sci., New York. 

Lamba, P. Hudson, J. L. 1987 Experiments on bifurcations to chaos in a forced chemical reactor. Chem. Engng Sci. 42,1-8. 

One of the well-studied systems that illustrates this successive-bifurcation behavior is the Belousov-Zhabotinski reaction. Let me briefly show you the results of some experiments done at the University of Texas at Austin,8 referring for further details to the discussion by J. S. Turner in this volume. The experimental setup of the continuously stirred reactor... 

Third, the line can become unstable during laser writing, and instead of a single line, a periodic pattern of discrete deposits is obtained (233-235). This pattern is analogous to bifurcations in other spatially distributed systems, such as catalytic fixed-bed reactors, and can be analyzed in the same manner (235). 

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