Binary saddle

Let s illustrate the location of trajectory bundles of reversible distillation by the example of three-component acetone(l)-benzene(2)-chloroform(3) mixture with one binary saddle azeotrope with a maximum boiling temperature (Fig. 4.7)... 

Kiva, V. R, Marchenko, I. M., Garber, Yu. N. (1993). Possible Composition Distillation Products of Three-Component Mixture with Binary Saddle. Theor. Found. Chem. Eng., 27,373-380. 

Schematic DRD shown in Fig. 13-59 are particularly useful in determining the imphcations of possibly unknown ternary saddle azeotropes by postulating position 7 at interior positions in the temperature profile. It should also be noted that some combinations of binary azeotropes require the existence of a ternaiy saddle azeotrope. As an example, consider the system acetone (56.4°C), chloroform (61.2°C), and methanol (64.7°C). Methanol forms minimum-boiling azeotropes with both acetone (54.6°C) and chloroform (53.5°C), and acetone-chloroform forms a maximum-boiling azeotrope (64.5°C). Experimentally there are no data for maximum or minimum-boiling ternaiy azeotropes. The temperature profile for this system is 461325, which from Table 13-16 is consistent with DRD 040 and DRD 042. However, Table 13-16 also indicates that the pure component and binary azeotrope data are consistent with three temperature profiles involving a ternaiy saddle azeotrope, namely 4671325, 4617325, and 4613725. All three of these temperature profiles correspond to DRD 107. Experimental residue cui ve trajectories for the acetone-...

Liquid-Fluid Equilibria Nearly all binary liquid-fluid phase diagrams can be conveniently placed in one of six classes (Prausnitz, Licntenthaler, and de Azevedo, Molecular Thermodynamics of Fluid Phase Blquilibria, 3d ed., Prentice-Hall, Upper Saddle River, N.J., 1998). Two-phase regions are represented by an area and three-phase regions by a line. In class I, the two components are completely miscible, and a single critical mixture curve connects their criticsu points. Other classes may include intersections between three phase lines and critical curves. For a ternary wstem, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature due to the compressibility of the solvent. 

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. 

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. 

In Figure 5 two landscapes are shown that start at a binary master sequence (v = 50) and are followed up to the 12-error copies. One of the landscapes (left side) is the low-value plateau that has been considered already in previous examples, while the other resembles a mountain saddle as typical for any fractal type of hill country. The population numbers of mutants, relative to the population number of the wild type, were calculated by means of second-order perturbation theory [Eqs. (11.17) and (11.18)] for... 

Occasionally we can posit more than one structure possible, based on knowing only the temperatures for the pure species and the binary azeotropes. Since such structures can occur, care must be taken to see that all possible structures are discovered. Based on topological arguments, Zharov and Serafimov (1975 see also Serafimov, 1987) developed an equation among the number of nodes and saddles appearing in a residue curve map. Independently, Doherty and... 

Figure 18 shows the diagram with the required ternary saddle where the binary azeotrope is a node. It has four distillation regions (labeled I to IV) separated by distillation boundaries, each of which has its own set of maximum and minimum temperatures within it. For example, region I has a maximum temperature of 170° and a minimum of 120°. It is this property of having its own unique minimum temperature (from which all trajectories emanate) and maximum temperature (at which all terminate) that characterizes a region. 

As.suming the binary azeotrope is a saddle (as. shown in Fig. 19), we note that there is a trajectory that starts at the upper vertex and passes through the minimum-boiling azeotrope on the lower edge. This trajectory is a distillation boundary that splits the diagram into two distinct distillation regions, labeled 1 and II. Each region has the same minimum temperature but a different maximum temperature within it. 

Figure 2. Hyper-surface of the thermodynamic potential difference between the heterogeneous state consisting of a cluster in the otherwise homogeneous ambient phase and the homogeneous initial state (here demonstrated for the case of a binary system). The curve via the saddle corresponds to the generalized Gibbs approach while the classical Gibbs method corresponds to ridge crossing, nj and n2 are here the number of particles in the cluster.

FIGURE 11.10 Free energy of duster formation, AG(nA.nB). for binary nucleation (a) schematic diagram of saddle point in the AG surface, (b) AG surface for HjSO -HjO system at 298 K. 

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