Critical tie line

In the three-phase regime of equilibrium phase behavior, the diffusion path studies were based on a dimensionless parameter S which indicates the position of the system within the regime. Thus, S = 0 corresponds to the salinity where the three-phase region first appears via a critical tie line (14), and S = 1 corresponds to the salinity where it disappears into another critical tie line. The optimum salinity occurs at S 0.5. 

The enrichment (or the pressure) needed to develop miscibility between the injectant and the oil is determined experimentally in one-dimensional slim-tube tests.As the enrichment (or pressure) increases, the slim-tube recovery reaches a plateau long before first contact miscibility is developed. This enrichment is called the minimum miscibility enrichment, or MME, which is a function of reservoir pressure, temperature, and contaminants in the solvent. Similarly, a pressure called the minimum miscibility pressure, or MMP, can be identified for any solvent. Other experimental methods (e.g., rising bubble method) are also available to determine MMP or MME. In vaporizing three component systems, MMP (or MME) corresponds to the pressure (or enrichment) at which the critical tie line passes through the crude oil composition. In condensing three component systems, MMP (or MME) corresponds to the pressure (or enrichment) at which the critical tie line passes through the solvent composition. ... 

T is the temperature of the critical end point of the CPp curve. At T = T, the phase diagram is still a WI, but with a critical tie line from which the 3PT will appear with the slightest increase in temperature. 

It is important to remark on the shape of the line joining the three corners of the 3PT triangle (r lines). It is a single, continuous gauche line, with a minimum at 7j on the water side of the critical tie line and a maximum at Tj, on the oil side of the critical tie line. The branches can be identified (ra, Tb, and Tc), each corresponding to the compositions of each corner of the 3PT. It is important to note that Ta has nothing to do with CPa and that Tb has nothing to do with CPp. Ti are composition curves, and CP are critical point curves. At a critical end point, however, the critical point curve meets the extreme of the composition curve (CPa meets Ta at Tu and CPp meets Tb at 7j). 

From the temperature dependence of the phase behaviour the qualitative shape of the three interfacial tension curves can be deduced. As the two phases (a) and (c) are identical at the critical tie line at T the interfacial tension aac has to start from zero and increases monotonically with increasing temperature. Whereas the interfacial tension ubc decreases (monotonically) with increasing temperature and vanishes at Tu, because the two phases (c) and (b) become identical at the critical tie line at Tu. This opposite temperature dependence of crac and Ubc results in a minimum if one considers the sum of the two, crac + CTbc- In order to assure the stability of the water/oil interface... 

No invariant reaction exists inside the ternary system. A critical tie-line has been observed at 1220°C [1935Koe]. At this temperature the L-i-yi-i-y2 tie-line triangle degenerates to a straight line. The end points of this line are the critical point c and the upper point of the monovariant peritectic line Pc-ps- The... 

The van der Waals p., p. isothenns, calculated using equation (A2.5.3), are shown in figure A2.5.8. It is innnediately obvious that these are much more nearly antisynnnettic around the critical point than are the conespondingp, F isothenns in figure A2.5.6 (of course, this is mainly due to the finite range of p from 0 to 3). The synnnetry is not exact, however, as a carefiil examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area constniction makes the pressures equal. 

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

At the critical pohit (and anywhere in the two-phase region because of the horizontal tie-line) the compressibility is infinite. However the compressibility of each conjugate phase can be obtained as a series expansion by evaluating the derivative (as a fiuictioii of p. ) for a particular value of T, and then substituting the values of p. for the ends of the coexistence curve. The final result is... 

For T shaped curves, reminiscent of the p, isothemis that the van der Waals equation yields at temperatures below the critical (figure A2.5.6). As in the van der Waals case, the dashed and dotted portions represent metastable and unstable regions. For zero external field, there are two solutions, corresponding to two spontaneous magnetizations. In effect, these represent two phases and the horizontal line is a tie-line . Note, however, that unlike the fluid case, even as shown in q., form (figure A2.5.8). the symmetry causes all the tie-lines to lie on top of one another at 6 = 0 B = 0). 

Fig. 15. Isobaric vapor—liquid—liquid (VLLE) phase diagrams for the ethanol—water—benzene system at 101.3 kPa (D-D) representHquid—Hquid tie-lines (A—A), the vapor line I, homogeneous azeotropes , heterogeneous azeotropes Horsley s azeotropes, (a) Calculated, where A is the end poiat of the vapor line and the numbers correspond to boiling temperatures ia °C of 1, 70.50 2, 68.55 3, 67.46 4, 66.88 5, 66.59 6, 66.46 7, 66.47, and 8, the critical poiat, 66.48. (b) Experimental, where A is the critical poiat at 64.90°C and the numbers correspond to boiling temperatures ia °C of 1, 67 2, 65.5 3, 65.0 ...

Figure 8.4 Graph of temperature against molar volume (a), and density (b). for CO (gas) and C02 (liquid) in the temperature range from the triple point to the critical point. The dashed line in (b) is the average density. The area enclosed within the curves is a two-phase region, with the molar volume or the density of the gas and liquid at a particular temperature given by the horizontal (dotted) tie-lines connecting the gas and liquid sides of the curve.

As mentioned earlier, the physical properties of a liquid mixture near a UCST have many similarities to those of a (liquid + gas) mixture at the critical point. For example, the coefficient of expansion and the compressibility of the mixture become infinite at the UCST. If one has a solution with a composition near that of the UCEP, at a temperature above the UCST, and cools it, critical opalescence occurs. This is followed, upon further cooling, by a cloudy mixture that does not settle into two phases because the densities of the two liquids are the same at the UCEP. Further cooling results in a density difference and separation into two phases occurs. Examples are known of systems in which the densities of the two phases change in such a way that at a temperature well below the UCST. the solutions connected by the tie-line again have the same density.bb When this occurs, one of the phases separates into a shapeless mass or blob that remains suspended in the second phase. The tie-lines connecting these phases have been called isopycnics (constant density). Isopycnics usually occur only at a specific temperature. Either heating or cooling the mixture results in density differences between the two equilibrium phases, and separation into layers occurs. 

Binodials calculated by Tompa are shown in Fig. 123,a for the special case of a nonsolvent [l], a solvent [2], and a polymer [3] with Vi = V2, X23 = 0, and xi2 = Xi3 = 1.5. Otherwise stated, the nonsolvent-solvent and the nonsolvent-polymer segment free energies of interaction are taken to be equal, while that for the solvent and polymer is assumed to be zero. It is permissible, then, to take Xi = X2 = l and o 3 = V3/vi. The number of parameters is thus reduced for this special case from five to two. Binodial curves are shown in Fig. 123,a for 0 3 = 10, 100, and 00 tie lines are shown for the intermediate curve only. The critical points for each curve, shown by circles, represent the points at which the tie lines vanish, i.e., where the compositions of the two phases in equilibrium become identical. 

Fig. 123.—(a) Phase diagram calculated for three-component systems consisting of nonsolvent [1], solvent [2], and polymer [3] taking Xi==X2=l and Xz equal to 10 (dashed curve), 100 (solid curve), and °° (dotted curve) xi2 = xi3 = 1.5 and X23 =0. All critical points (O) are shown and tie lines are included for the xs = 100 curve. (Curves calculated by Tompa. ) (b) The binodial curve for a 3 = 100 and three solvent ratio lines. The precipitation threshold is indicated by the point of tangency X for the threshold solvent mixture. 

Ternary equilibrium curves calculated by Scott,who developed the theory given here, are shown in Fig. 124 for x = 1000 and several values of X23. Tie lines are parallel to the 2,3-axis. The solute in each phase consists of a preponderance of one polymer component and a small proportion of the other. Critical points, which are easily derived from the analogy to a binary system, occur at... 

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. 

Fig. 10.6. Schematic diagram of the energy, E, versus the number of particles, N, for a one-component fluid with a phase transition. Squares linked by dashed lines are coexisting phases joined by tie lines and the filled square indicates the critical point of the transition. Ellipses represent the range of particle numbers and energies sampled during different GCMC runs. Reprinted by permission from [6]. 2000 IOP Publishing Ltd...

A critical part of the calculations is to calculate the tie-line at the interface corresponding to local equilibrium, and Enomoto (1992) used the central atoms model to predict the thermodynamic properties of a and 7. Some assumptions were made concerning the growth mode and the calculation of this tie-line is dependent on whether growth occurred under the following alternative conditions ... 

In contrast, a heterogeneous solution of noncritical composition (e.g., v < xc, as shown by the arrow and dashed line in Fig. 7.11) shows a qualitatively different behavior as it is rises through the coexistence boundary and into the homogeneous region near and above Tc. For each increase in temperature along the dashed line in Fig. 7.11, a horizontal tie-line yields both the compositions of the A-rich and B-rich liquids (from the two ends of the tie-line), as well as the relative amounts of each phase (from the lever rule). Clearly, the critical composition xc remains near the middle of the tie-line as T increases toward Tc, whereas a noncritical composition x xc moves toward one or other terminus of the tie-line as the temperature is raised. 

Figure 7.12 Successive snapshots of a binary A/B system in the immiscible region of coexisting A-rich (lighter) and B-rich (darker) liquid phases (Fig. 7.11), comparing the T-dependent behavior for noncritical (upper sequence cf. dashed line in Fig. 7.11) versus critical composition (lower sequence cf. dotted line in Fig. 7.11), and showing how the meniscus rises out of the container in the first case but vanishes at the critical point Tc in the second case. [Both solutions are taken to be fairly comparable in the starting low-71 snapshot, but their deviations are readily apparent in the second snapshot at the temperature Ttie of the horizontal tie-line in Fig. 7.11.]...

Phase Relationships. The first systematic investigation of the two-phase behavior of polymer/polymer/solvent systems was probably made by Dobry and Boyer-Kawenoki (2) for a variety of polymer pairs, and more recently this work was extended by Kern and Slocombe (3) and Paxton (35) to a number of other systems including several vinyl polymers. Typically, the three-component phase behavior is as shown in Figure 19 for the polystyrene/polybutadiene/benzene system (2), where a one-phase (polystyrene/polybutadiene/benzene) region is separated by a phase boundary from a two-phase (polystyrene-rich/benzene and polybutadiene-rich/benzene) mixture. As with any three-component system of this type, a critical point exists somewhere near the maximum of the phase boundary, and appropriate tie lines give the compositions and amounts of the respective phases in the two-phase region. 

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