Joining trajectory parameter

The first of the two trajectory or interaction rules is the joining trajectory parameter, J(AB), which defines the propensity of movement of an ingredient A toward or away from a second ingredient B, when the two are separated by... 

A joining trajectory parameter, /(AB), describes the movement of a molecule at A to join with a molecule at B, when an intermediate cell is vacant (see Figure 8). This rule follows the rule to move or not to move described above. The parameter / is a nonnegative real number. When J = 1, molecule A has the same probability of movement toward or away from B as for the case when the B cell is empty. When / > 1, molecule A has a greater probability of movement toward an occupied cell B than when cell B is empty. When 0 [Pg.216]

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

For spUts without distributed components and with one distributed component, the conditions of joining can always be valid at sufficiently large values of parameters L/V and V/L in the top and bottom sections, respectively, if the sizes of trajectory bundles unlimitedly grow at the increase of these parameters. 

But we saw in Section 5.4 that the values of parameters L/V and V/L and the sizes of trajectory bundles of adiabatic colunms sections are limited, if for product point there are two reversible distillation trajectory tear-off points. Therefore, necessary conditions of separability in adiabatic columns can be insufficient if for one or for both product points there are two reversible distillation trajectory tear-off points from boundary elements to which points 5 belong. In these cases, to check separabiUty it is necessary to verify whether corresponding separatrix sections trajectory bundles join at the maximum possible value of the parameter L/V or V/L. 

Usage of nonadiabatic columns (i.e., column with intermediate at height inputs or outputs of heat), broadens conditions of separability of mixtures, having two reversible distillation traj ectory tear-off points. If, for example, heat is brought and drawn off in feed cross-section (Fig. 5.36), as it was offered in the work (Poellman Blass, 1994), then it is possible in one of the sections of the column, where there is limitation at the value of the parameter L/V or V/L, to keep the corresponding allowed value of this parameter, and joining of trajectories of the sections can be maintained at the expense of increase of vapor and liquid flows in the second section (Petlyuk Danilov, 1998). In a more general case, when there are limitations of the values of the parameters L/V and V/L in both sections, it is possible to use intermediate inputs and outputs of heat in the middle cross-sections of both sections. 

The rest of the algorithm is similar to that for columns with one feed. Conditions of joining of trajectories in cross-sections of both feeds are checked at various values of the parameter (LlV)r and the value (LIV), at which there is a joining in the cross-section of control feed, is found. Both feasible cases of joining described above (see Subsection 6.3.3) are checked. The first case corresponds to direct or indirect split in the column with one feed, and the second case corresponds to intermediate split. 

We now examine the conditions of joining of sections trajectories at a set flow rate of entrainer (i.e., at set value of the parameter E/D) for a three-component mixture in the mode of minimum reflux. Each of two feeds can be the control one, and the intermediate section trajectory in the mode of minimum reflux in both cases should pass through the saddle point Sm because this trajectory passes through the node point not only in the mode of minimum reflux, but also at reflux bigger than minimum (point arises at the boundary element of the concentration simplex because the extractive distillation under consideration is sharp). 

Trajectory bundles of bottom and intermediate sections in the mode of minimum reflux should join with each other in the concentration space of dimensionality (n - 1). Therefore, joining is feasible at some value of the parameter (L/T) if the summary dimensionality of these bundles is equal to (n - 2). 

Therefore, extractive distillation at three and more components in the top product is feasible in principle, but requires a search for an allowed composition of the pseudoproduct. If it is necessary to design a sharp extractive distillation column at ntr = 3, then it is not allowed to set the rate of the entrainer arbitrarily, but it is necessary to determine it from the conditions of joining of trajectories of the top and intermediate sections. The parameter E/D ensures an additional degree of freedom, which allows to increase by one the number of product components in the top product ntr. 

Minimum reflux mode is determined by the conditions of joining of trajectories of two sections adjacent to the feed cross-section. Therefore, the interconnected parameters (L/V) " and (V/L) " are determined initially for these two sections. Compositions in points x and x g are calculated preliminarily for these sections at set requirements to compositions of all the products at the conditions of sharp or quasisharp separation in each section. Minimum reflux mode is calculated in the same way as for the simple column that separates initial raw materials into products of compositions x jy and x g. Liquid and vapor flow rates for the other sections are calculated at the obtained values of (L/V) " and (L/L)f" with the help of material balance equations (strictly speaking, with the help of equations of material and thermal balance). 

Figure 6.15b shows the trajectories of sections n and yi (part of the trajectory from point to tear-off point 5ri is hctitious). The flow rate of vapor Vn and its composition (equilibrium composition in point 5ri), and also the flow rate and composition of liquid (flow rate L i and composition in point 5ri), are becoming clear after the determination of parameter L/V). It is accepted here that trajectories of sections ri, r2, and S2 join in point 5 1. At such optimum joining, the maximum concentration of component 3 in section ri is achieved (in point... 

As far as the joining of trajectories of the top and the intermediate sections is concerned, the composition point at the first tray below the cross-section of the entrainer input Xe should be located quite close to the boundary element of the concentration simplex that contains components of the top product and of the entrainer (in Fig. 7.12 - to side 1-2). Allowable concentration of impurity components in point Xe is determined by the requirements to the purity of the top product. Therefore, the composition in point Xe is not an optimized parameter and the composition in point Xe i at the first tray above the cross-section of entrainer input is determined by the conditions of material balance in this cross-section. 

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