Plait point

For systems of type II, if the mutual binary solubility (LLE) data are known for the two partially miscible pairs, and if reasonable vapor-liquid equilibrium (VLE) data are known for the miscible pair, it is relatively simple to predict the ternary equilibria. For systems of type I, which has a plait point, reliable calculations are much more difficult. However, sometimes useful quantitative predictions can be obtained for type I systems with binary data alone provided that... 

In most cases only a single tie line is required. When several are available, the choice of which one to use is somewhat arbitrary. However, our experience has shown that tie lines which are near the middle of the two-phase region are most useful for estimating the parameters. Tie lines close to the plait point are less useful, since no common models for the excess Gibbs energy can adequately describe the flat region near the... 

Figure 4-18. Calculated distribution for ternary liquid-liquid systems show good agreement with experiment except very near the plait point.

Liquid-liquid equilibrium separation calculations are superficially similar to isothermal vapor-liquid flash calculations. They also use the objective function. Equation (7-13), in a step-limited Newton-Raphson iteration for a, which is here E/F. However, because of the very strong dependence of equilibrium ratios on phase compositions, a computation as described for isothermal flash processes can converge very slowly, especially near the plait point. (Sometimes 50 or more iterations are required. )... 

Feed conditions in the region of the plait point of type I system. 

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. 

Outside the two-phase region, ELIPS yields a value of 0 for E/F on the R-phase side and 1 for E/F on the E-phase side. Con-, vergence to these values again requires about eight or fewer iterations, except near the plait-point region where convergence is somewhat slower. 

The criterion used for "too near the plait point" is that ratio of K s for the two "solvent" components is less than seven with the feed composition in the two-phase region. 

The calculational procedure employed in BLIPS, when used with the particular initial phase-composition estimated included in the subroutine, has converged satisfactorily for all systems we have encountered (except very near plait points as noted). 

Liquid phase compositions and phase ratios are calculated by Newton-Raphson iteration for given K values obtained from LILIK. K values are corrected by a linearly accelerated iteration over the phase compositions until a solution is obtained or until it is determined that calculations are too near the plait point for resolution. 

CHECK IF CALCULATION NEAR PLAIT POINT IS PROBABLY IN SINGLE PHASE... 

Fig. 3. Typical nonionic amphiphile—oil—water—temperature phase diagram, illustrating (a) the S-shaped curve of T, M, and B compositions, (b) the lines of plait points, (c) the lower and upper critical end points (at and respectively), and (d) the lower and upper critical tielines.

It is seen that as the concentration of C is increased, the tie-lines become shorter because of the increased mutual miscibility of the two phases at the plait point, P, the tie lines vanish. However, P does not necessarily represent the highest possible loading of C which can exist in the system under two-phase conditions. In Figure 2b the plait point Hes on the diagonal because the compositions of the two phases approach each other at P. 

Capacity. This property refers to the loading of solute per weight of extraction solvent that can be achieved in an extrac t layer at the plait point in a Type I system or at the solubihty hmit in a Type II system. 

Density. The difference in density between the two hquid phases in eqiiilibrium affects the countercurrent flow rates that can be achieved in extrac tion equipment as well as the coalescence rates. The density difference decreases to zero at a plait point, but in some systems it can become zero at an intermediate solute concentration (isopycnic, or twin-density tie line) and can invert the phases at higher concentrations. Differential types of extractors cannot cross such a solute concentration, but mixer-settlers can. 

Inteifacial tension. A high interfacial tension promotes rapid coalescence and generally requires high mechanical agitation to produce small droplets. A low interracial tension allows drop breakup with low agitation intensity but also leads to slow coalescence rates. Interfacial tension usually decreases as solubility and solute concentration increase and falls to zero at the plait point (Fig. 15-10). 

Faltungspunkt, m. plait point, point of plication or folding. 

As the tangent plane rolls on the primitive surface, it may happen that the two branches of the connodal curve traced out by its motion ultimately coincide. The point of ultimate coincidence is called a plait point, and the corresponding homogeneous state, the critical state. 

The conditions which must be satisfied at the plait point may be deduced as follows Expand by Taylor s theorem the expressions on the right of (9) and (10), omitting terms of higher orders than the second ... 

Now the plait point is on the spinodal curve, and any two corresponding points of the connodal curve adjacent to the plait point are on a part of the surface which is convex in every direction, and for which therefore... 

Thus the spinodal curve does not cut the connodal curve at the plait point, and it is simplest to assume the two curves to be tangent at that point. From (26) it follows that the direction of the tangent at any point of the spinodal curve is given by ... 

At the plait point this equation will give the same value for dS/du as equations (27) and (28), and hence at that point the... 

Again, the limiting position of the line joining points of the connodal curve and the direction of the common tangent to the connodal and spinodal curves at the plait point is given by ... 

With motion along the connodal curve towards the plait point the magnitudes Ui and U2, Si and S2, and ri and r2, approach limits which may be called the energy, entropy, and volume in the critical state. The temperature and pressure similarly tend to limits which may be called the critical temperature and the critical pressure. Hence, in evaporation, the change of volume, the change of. entropy, the external work, and the heat of evaporation per unit mass, all tend to zero as the system approaches the critical state ... 

The plait point is an ordinary point on the connodal curve, and hence it is immediately evident that the specific volume and entropy in the critical state are intermediate between those of adjectent liquid and vapour phases. 

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