Reaction rates bifurcation points

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

We have stressed that both the real and imaginary parts depend on the parameter n because we are imagining experiments where the reactant concentration will be varied whilst k is held constant. If we were doing the experiments another way so that n was held fixed and the dimensionless reaction rate constant varied in the vicinity of the Hopf bifurcation point we would then wish to consider v(/c) and a>(/c). 

When the dimensionless reaction rate constant lies in the range given by eqn (10.77), the well-stirred system has two Hopf bifurcation points /i 2. Over the range of reactant concentration... 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

FIGURE 3 Three-dimensional view of the steady-state reaction rate surface when -y, = 0.001, and y-i = 0.002. The inset shows a isola for a section of constant at where H and H are Hopf bifurcations and T and T are turning points. 

VRI, the actual reaction path bifurcates into two products before reaching TS7. The rate is determined by TS6, whereas the product ratio is controlled by the shape of the PES near the VRI point and TS7. Thus, a question arises how the product selection occurs when a subtle perturbation, such as isotopic substitution, is introduced and two symmetrical products become asymmetric ... 

In contrast to classical chemical reactors, a fuel cell provides the possibility to control the reaction rate directly from outside by setting the cell current, because the local cell current density and the local reaction rate are related by a constant factor. This operation of a fuel cell at constant cell current is more important than the potentiostatic operation from a technical point of view, as fuel cells typically are characterized by current-voltage plots. Because the integral Eq. (15) has to be included in the analysis, the investigation of the galvanostatic operation is more difficult and requires numerical methods. In the following, numerical bifurcation... 

The problem is reduced to finding the phase trajectories of the equation system (104) at the (g, 0)-plane at different y values (dimensionless reaction rate) and values of p (relationship of the rates of relaxation g and heat removal at T = Tq). Dependence of the solution on x and in the physically justified ranges of their variation (tj > I at q qi ij< 1) turns out to be relatively weak. The authors of ref 234 applied the well-known method of analysis of specific trajectories changing at the bifurcational values of parameters [237], In the general case, the system of equations (104) has four singular points. The inflammation condition has the form... 

Usually reaction rates are the most common parameters in chemically reacting systems. Models discussed in the literature may be studied from the point of view of parameters. Particularly, in the bifurcation analysis of reaction models we refer to... 

Another example is the mechanism where efficiency of reactions is affected by the surrounding molecules. Suppose that, depending on the surrounding molecules, intersection of the stable and unstable manifolds of a reaction changes between (a) and (b) or between (b) and (c). Then, these molecules can enhance or decrease the reaction rate by switching the level of dynamical correlation. Moreover, when the surrounding molecules induce bifurcation of reaction paths, their existence can trigger new reactions. These possibilities would offer a clue to understand molecular functions from a dynamical point of view. 

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

We formulate the reactive flash modd for an equimolar chemistry. Next, we hypothesize a condition under which the trajectories of the flash cascade model lie in the feasible product regions for continuous RD. This hypothesis is tested for an example mixture at different rates of reaction. The fixed point criteria for the flash cascade are derived and a bifurcation analysis shows the sharp split products from a continuous RD. 

At the first bifurcation point, extinction point A, = 0 and the rates of the elementary reactions for the branch y = 0 are... 

At the bifurcation point of maximum complexity, Bi = 2 = 0, a further simplification for the reaction rate is... 

The designer should be aware that there is a critical reactor volume, which generally corresponds to a bifurcation point of the mass balance equations. For stable operation the reactor should be larger than this critical value. As example, for essentially first-order reaction with pure product and recycle, the feasibility condition is simply Da>. The definition of the plant Damkohler number includes reactor volume, reaction kinetics and fresh reactant feed flow rate. Similar expressions hold for more complex stoichiometry. 

Fig. 2. Stable steady states and the bifurcation point of the three-step reaction (4). The numerical values of the rate constants chosen are f = 4 [t c, f = 2 [t c ],...

The stationary state is unstable for p > p. Thus there will be a point of Hopf bifurcation provided the rate constants for reaction and the adsorption and desorption of the poison satisfy the condition... 

In our CSTR example the constants Kc and a have opposite physical effects. If a increases, the flow rate q decreases and thus the rate of reaction increases, as does the heat of reaction. On the other hand, if Kc increases, then the heat removal by heat transfer to the cooling jacket increases, reducing the rate of reaction and the production of heat. Note that the search directions in our respective lowhighkc and lowhighal sub-programs point in opposite directions for the 5-shaped a bifurcation curves and for the inverted 5-shaped Kc bifurcation curves. 

The proofs of Theorems 10.2, 10.3, and 10.4 are found in [348]. Equation (10.17) is of particular interest. Near the Takens-Bogdanov point, the frequency of the limit-cycle oscillations along the line of Hopf bifurcations, a = 0, is given by >h = see above. On the line of saddle-node bifurcations we have Aj = 0. An equation like (10.17) is expected from simple dimensional arguments. The only intrinsic length scales in reaction-diffusion systems come from the diffusion coefficients. The inverse time is determined by the rate coefficients of the reaction kinetics. Thus (10.17) provides an estimate of the intrinsic length of the Turing pattern near a double-zero point ... 

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