Unstable nodes

Residue curves can only originate from, terminate at, or be deflected by the pure components and azeotropes in a mixture. Pure components and azeotropes that residue curves move away from are called unstable nodes (UN), those where residue curves terminate are called stable nodes (SN), and those that deflect residue curves are called saddles (S). 

As an example, consider the residue curve map for the nonazeotropic mixture shown in Eigure 2. It has no distillation boundary so the mixture can be separated into pure components by either the dkect or indkect sequence (Eig. 4). In the dkect sequence the unstable node (light component, L) is taken overhead in the first column and the bottom stream is essentially a binary mixture of the intermediate, I, and heavy, H, components. In the binary I—H mixture, I has the lowest boiling temperature (an unstable node) so it is recovered as the distillate in the second column and the stable node, H, is the corresponding bottoms stream. The indkect sequence removes the stable node (heavy component) from the bottom of the first column and the overhead stream is an essentially binary L—I mixture. Then in the second column the unstable node, L, is taken overhead and I is recovered in the bottoms. 

If A,i and X.2 are real numbers and both or one of the roots are positive, the system response will diverge with time and the steady-state solution will therefore be unstable, corresponding to an unstable node. ... 

Unsaturation, in oils, 10 826 Unsensitization phenomenon, 19 237 Unshaped refractories, 6 491 Unslaked lime, 15 29 Unstabilized liquid sulfur trioxide, 23 517 Unstable angina, 5 109 Unstable flows, 11 761-765 Unstable node, in separating nonideal liquid mixtures, 22 303 Unstable nodes, residue curve maps, 8 790 Unstable reagents, measurement strategies for, 14 621... 

Real, both + ve Unstable node (monotonic divergence)... 

FiC. 3.4. Representations of the different singular points in the concentration phase plane (a) stable node, sn (b) stable focus, sf (c) unstable focus uf (d) unstable node, un (e) saddle. point, sp. 

With 0SS > 1, the lower root in (4.46) describes the small lower loop, which corresponds to the conditions for which the stationary state regains nodal character. Inside this region, the eigenvalues Al 2 are real and are positive, so we have an unstable node. This curve has a maximum at k = (3 — y/S) exp [ — i(3 + v/5)] w 0.0279, so this response is not to be found over a wide range of experimental conditions. 

The loci typically drawn out by these equations as 0 varies are shown in Fig. 4.8(a). For y closed loop near the origin. Inside this loop, the stationary state is an unstable node. The larger outer loop separates stable focal character (inside curve) from stable nodal states (outside curve). As y increases beyond i the small inner loop shrinks to zero size the outer loop still exists. Stable focal character exists over some values of the parameters n and k for any value of y. 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

If we consider the well-stirred system, the stationary state has two Hopf bifurcation points at /r 2, where tr(U) = 0. In between these there are two values of the dimensionless reactant concentration /r 1>2 where the state changes from unstable focus to unstable node. In between these parameter values we can have (tr(U))2 — 4det(U) > 0, so there are real roots to eqn (10.76). 

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

Note here, without proof, one of the synergetic theorems about limit cycles [14, 15] a stable limit cycle contains at least one singular point or the unstable node of focus-type exists. 

Fig. 2.5. An unstable node is obtained as a formal solution of the Lotka equations (2.1.22)—(2.1.23) with time inversion, t —> -t, and parameter pK/f32 = 2. Note that these equations cannot be associated with a set of mono- and bimolecular reactions.

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

Fig. 2. Types of stationary points on the plane, (a), (c), (e) Stable nodes (b), (d), (f) unstable nodes (g) saddle point (h) stable focus (i) unstable focus, (k) whirl.

Phase trajectories extend far from the singular point. It is an unstable node... 

Let Bt be an unstable node or a focus. Then will be a trace of the matrix for the linear approximation at this point. 

As S continues to increase we reach a point marked by an open circle. Here, the equilibrium point undergoes a saddle-node bifurcation. Somewhere before this bifurcation, the unstable focus point has turned into an unstable node with two positive real eigenvalues. In the saddle-node bifurcation, one of these eigenvalues... 

Fig. 4.7(c). For Dar> 0.166 (Fig. 4.7(d)), only three singular points remain in the system pure MeOH which is a stable node at any Damkohler number pure isobutene which is a saddle point at any Damkohler number and the above-mentioned unstable node which is located outside the triangle. 

Figure 4.27 shows residue curve maps for the reactive reboiler at three different Damkohler numbers. In the nonreactive case (Da = 0 Fig. 4.27(a)), the map topology is structured by one unstable node (pure B), one saddle point (pure C), and one stable node (pure A). Since pure A is the only stable node of nonreactive distillation, this is the feasible bottom product to be expected in a continuous distillation process. 

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

Residue curve maps of the THF system were predicted for reactive distillation at different reaction conditions (Fig. 4.29). The topology of the map at nonreactive conditions (Da = 0) is structured by a binary azeotrope (unstable node) between water and THF. Pure water and pure THF are saddle nodes, while the 1,4-BD vertex is a stable node. 

At Da = 0.4 (Fig. 4.29(b)), the two saddle points from the pure vertices move into the composition triangle. The stable node from the 1,4-BD vertex moves to the kinetic azeotrope at x = (0.0328, 0.6935). Pure water and pure THF now become stable nodes. The unstable node between water and THF remains unmoved, and forms two separatrices with the two saddle points. Thereby, the whole composition space is divided into three subspaces which have each a stable node, namely pure water, pure THF and the kinetic azeotrope. 

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