Bifurcation behavior

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

After the bifurcation behavior is examined, the role of flame-wall thermal interactions in NOj is studied. First, adiabatic operation is considered. Next, the roles of wall quenching and heat exchange in emissions are discussed. Two parameters are studied the inlet fuel composition and the hydrod3mamic strain rate. Results for the stagnation microreactor are contrasted with the PSR to understand the difference between laminar and turbulent flows. 

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. 

Bifurcation behavior in homogeneous-heterogeneous combustion II. Computations for stagnation-point flow (with X. Song, W.R. Williams, and L.D. Schmidt). Comb. Rome, 292-311 (1991). 

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... 

It is important to introduce the reader at an early stage to simple examples of nonlinear models. We will first present cases with bifurcation behavior as the more general case, followed by special cases without bifurcation. Note that this is deliberately the reverse of the opposite and more common approach. We take this path because it sets the important precedent of studying chemical and biological engineering systems first in light of their much more prevalent multiple steady states rather than from the rarer occurrence of a unique steady state. 

To gain further and broader insights into the bifurcation behavior of nonadiabatic, nonisothermal CSTR systems, we again use the level-set method for nonalgebraic surfaces such as z = /(K,., y). This particular surface is defined via equation (3.14) as follows for a given constant value of yc with the bifurcation parameter Kc ... 

Equation (3.17) allows a different interpretation of the underlying system s bifurcation behavior by taking Kc and yc as fixed and letting a vary, for example. We now study the bifurcation behavior of nonadiabatic and nonisothermal CSTR systems via their level-zero curves for the associated transcendental surface z = g(a,y). The surface is defined as before, except that here we treat Kc and yc as constants and vary a and y in the 3D surface equation... 

Thus far we have explored the bifurcation behavior of equation (3.14) with respect to Kc via equation (3.17) in Figures 3.14 through 3.16, and with respect to a via (3.19) in Figures 3.17 and 3.18. Since different Kc and a values can lead to bifurcation behavior for the same nonadiabatic, nonisothermal CSTR system, it is of interest and advantageous to be able to plot the joint bifurcation region for the parameters Kc and a as well. 

We are made aware of the lively change in bifurcation behavior here just a third digit change in a can cause absolutely different bifurcation behavior of the associated CSTR system, as witnessed by Figures 3.20 to 3.22. 

The current section has covered numerical techniques and MATLAB codes for investigating the static bifurcation behavior of nonadiabatic lumped systems. 

To solve equation (4.86) for y and K > 0 we use the graphical level set method that we have introduced in Chapter 3 for the adiabatic and nonadiabatic CSTRs and draw the surface z = F(y, K), as well as the y versus K curve of solutions to equation (4.86) in order to exhibit and study the bifurcation behavior of the underlying system. 

The static bifurcation behavior and its practical implications have been investigated. We have also formulated the unsteady-state dynamic model and we have used it to study the dynamic behavior of the system by solving the associated IVP numerically. Both the controlled and the uncontrolled cases have been investigated. Two particular reactions have been studied, one with three steady states, and one with five steady states. 

Combined Static and Dynamic Bifurcation Behavior of Industrial FCC Units... 

We have studied the dynamic behavior of FCC units in Section 7.2.3. Here we explain the dynamic bifurcation behavior of FCC type IV units. The dynamic model that we use will be more general than the earlier one. Specifically, we will relax the assumption of negligible mass capacity of gas oil and gasoline in the dense catalyst phase. This relaxation is based upon considering the catalyst chemisorption capacities of the components. 

In this fermentation process, sustained oscillations have been reported frequently in experimental fermentors and several mathematical models have been proposed. Our approach in this section shows the rich static and dynamic bifurcation behavior of fermentation systems by solving and analyzing the corresponding nonlinear mathematical models. The results of this section show that these oscillations can be complex leading to chaotic behavior and that the periodic and chaotic attractors of the system can be exploited for increasing the yield and productivity of ethanol. The readers are advised to investigate the system further. 

We will use the model to explore the complex static/dynamic bifurcation behavior of this system in the two-dimensional D — Cso parameter space and show the implications of bifurcation phenomena on substrate conversion and ethanol yield and productivity. The system parameters for the specific fermentation unit under consideration are given below. 

Figure 7.25 compares the experimental results for this data, drawn in a solid curve, with the simulated results in dashed form. Further details of the static and dynamic bifurcation behavior of this system are shown in Figure 7.27. 

Later development in singularity theory, especially the pioneering work of Golubitsky and Schaeffer [19], has provided a powerful tool for analyzing the bifurcation behavior of chemically reactive systems. These techniques have been used extensively, elegantly and successfully by Luss and his co-workers [6-11] to uncover a large number of possible types of bifurcation. They were also able to apply the technique successfully to complex reaction networks as well as to distributed parameter systems. 

Many laboratory experiments have been successfully designed to confirm the existence of bifurcation behavior in chemically reactive systems [25-29], as well as in enzyme systems [18]-... 

Multiplicity or bifurcation behavior was found to occur in many other systems such as distillation [30], absorption with chemical reaction [31], polymerization of olefins in fluidized beds [32], char combustion [33, 34], the heating of wires [35] and in a number of processes used for manufacturing and processing electronic components [36, 37]. 

The work of Uppal et al. [20, 21] and that of Ray [56] on the bifurcation behavior of continuous stirred tank reactors. 

Fig. 4.31. Potential singular point surfaces and stable node bifurcation behavior of reactive membrane separation at different mass transfer conditions B + C< > A Keq = 5 ccba = 5.0, acA = 3.0.

Figure 4.33 illustrates the PSPS and bifurcation behavior of a simple batch reactive distillation process. Qualitatively, the surface of potential singular points is shaped in the form of a hyperbola due to the boiling sequence of the involved components. Along the left-hand part of the PSPS, the stable node branch and the saddle point branch 1 coming from the water vertex, meet each other at the kinetic tangent pinch point x = (0.0246, 0.7462) at the critical Damkohler number Da = 0.414. The right-hand part of the PSPS is the saddle point branch 2, which runs from pure THF to the binary azeotrope between THF and water. 

Because the boiling temperature of 1,4-BD is much higher than of the two reaction products and the reaction is irreversible, the bifurcation behavior is only affected by the mass transfer coefficient ratio Kwater/KTHF, if kbd is not extremely high or low. There exists a critical value of Kwater/KTHF = 2.1, above which the stable node branch approaches the THF-vertex. 

Fig. 4.33. Potential singular point surface and bifurcation behavior for reactive distillation 1,4-BD —> THF + Water p = 5 atm.

It should be noted that if n is of the order 2, this is a cusp manifold (which is one which will show bifurcation behavior of the form discussed by Shore and Comins ). The analysis of this syst n presented by Ferrini et al. shows that this is indeed a system with multiple equilibrium states. Having this form for the potential, there exists a Fokker-Planck equation for the... 

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