Critical line

Anisimov M A, Povodyrev A A, Sengers J V and Levelt-Sengers J M H 1997 Vapor-liquid equilibria, scaling and crossover in aqueous solutions of sodium chloride near the critical line Physica A 244 298... 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

Fig. 6. Qualitative piessuie—tempeiatuie diagiams depicting ctitical curves for the six types of phase behaviors for binary systems, where C or Cp corresponds to pure component critical point G, vapor 1, Hquid U, upper critical end point and U, lower critical end point. Dashed curves are critical lines or phase boundaries (5). (a) Class I, the Ar—Kr system (b) Class 11, the CO2—CgH g system (c) Class 111, where the dashed lines A, B, C, and D correspond to the H2—CO, CH —H2S, He—H2, and He—CH system, respectively (d) Class IV, the CH —C H system (e) Class V, the C2H -C2H OH...

Evaluate ways to prevent backflow in critical lines between process equipment... 

Fig. 7. The (pt T) projection of a system in which the saturated vapor pressure curve does not cut the critical line, MN.

The critical line is similarly broken into MO and PN. Between 0 and P there is no critical point and no saturated vapor pressure... 

This theory is adequate to explain practically all oscillatory phenomena in relaxation-oscillation schemes (e.g., multivibrators, etc.) and, very often, to predict the cases in which the initial analytical oscillation becomes of a piece-wise analytic type if a certain parameter is changed. In fact, after the differential equations are formed, the critical lines T(xc,ye) = 0 are determined as well as the direction of Mandelstam s jumps. Thus the whole picture of the trajectories becomes manifest and one can form a general view of the whole situation. The reader can find numerous examples of these diagrams in Andronov and Chaikin s book4 as well as in Reference 6 (pp. 618-647). 

Essential steps have been 1) Accept only observational papers, whose line lists included with reasonably accuracy the critical lines to determine the nebular... 

From the family of AG (P, T) curves the projection on the (P, T) plane of the critical lines corresponding to the UCFT for these latexes can be calculated and this is shown plotted in Figure 4. It can be seen that the UCFT curve is linear over the pressure range studied. The slope of the theoretical projection is 0.38 which is smaller than the experimental data line. Agreement between theory and experiment could be improved by relaxing the condition that v = it = 0 in Equation 6 and/or by allowing x to be an adjustable parameter. However, since the main features of the experimental data can be qualitatively predicted by theory, this option is not pursued here. It is apparent from the data presented that the free volume dissimilarity between the steric stabilizer and the dispersion medium plays an important role in the colloidal stabilization of sterically stabilized nonaqueous dispersions. 

Figure Jh The characteristics of the flow equation (11) for Nf = 2 light flavors. The innermost characteristic line coincides with the prediction from lattice QCD [13] for the critical line at small g, which is represented by the dashed line with the hatched error band.

The characteristic line emanating from Tc is naturally related to the critical line Tc(fi) enclosing the hadronic phase. The comparison, in Figure 4, of our result for the curvature of the critical line at g - 0, which can be calculated in lattice QCD [13], is a nontrivial and successful test of the extension of the quasiparticle approach to g > 0. 

The critical line Ar(rii (O) may manifest itself in observations by a population clustering of compact objects in FMXBs [41], A population gap in the... 

The EoS HHJ - INCQM with 2SC quark matter phase has a type of hard -soft - hard EoS [40], Therefore critical line is mainly orthogonal to the mass axis and the expected population clustering seems to be not frequency but mass clustering. As already reported by M.C. Miller at this conference, there is observational evidence that the population of LMXB s is mainly homogeneous. 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

Most of the interaction parameters employed were taken from other studies (20, 21), and are reportedly obtained by minimizing errors in the match of phase equilibrium data. However, in (21), the SRK equation employed was slightly different from that used here. The parameters for CO2 - H2O were chosen because they had been shown to give a critical line which is qualitatively correct. The H2O - CO interaction parameter is the value given in (20) for H2S - CO. For H2O - H2, kij was taken to be -0.25 in the absence of any literature studies. 

The calculated critical lines for the binary pairs are shown in Figure 1. All these lines are discontinuous, indicating high density phase separations. For each binary pair the principal part of the critical line begins at the critical point for the component with the higher critical temperature. There is a second branch of each of the critical lines, beginning at the critical point of the component with the lower critical temperature, which terminates on intersecting a liquid-liquid-vapor three-phase line. 

Figure 1. Critical lines in water-gas shift binary pairs...

In Figure 1 the low temperature branches of the critical lines have been omitted, except for the CO2 - H2O binary. The reason for the omission is that these lines are extremely short and are not of much interest. 

The mole fraction of the component whose critical point is the origin of the critical line has been indicated as a parameter along each of the lines. It should be noted from these numbers that there are compositions in each of the binary pairs for which there is no critical point. It should also be noted that some C02 - CO mixtures have two critical points. 

The general shape of the binary critical lines dictates the shape of critical lines in the reacting mixtures. 

Critical points have been calculated for such mixtures with various initial H20/C0 ratios. The results are shown in Figure 2. In this figure, the line of zero conversion is the critical line for H20 - CO binary. For mixtures with initial H20/C0 ratios greater than about 4, it is possible to find critical points for all conversions. When the initial ratio is less, this is no longer possible. Lines of 25%, 50% and 75% conversion are also shown in Figure 2. 

Figure 2. Critical lines in reacting mixtures of CO with excess HtO...

The study described above for the water-gas shift reaction employed computational methods that could be used for other synthesis gas operations. The critical point calculation procedure of Heidemann and Khalil (14) proved to be adaptable to the mixtures involved. In the case of one reaction, it was possible to find conditions under which a critical mixture was at chemical reaction equilibrium by using a one dimensional Newton-Raphson procedures along the critical line defined by varying reaction extents. In the case of more than one independent chemical reaction, a Newton-Raphson procedure in the several reaction extents would be a candidate as an approach to satisfying the several equilibrium constant equations, (25). 

Fig. 5.12 Two different 3-D representations of the phase diagram of 3-methylpyridine plus wa-ter(H/D). (a) T-P-x(3-MP) for three different H2O/D2O concentration ratios. The inner ellipse (light gray) and corresponding critical curves hold for (0 < W(D20)/wt% < 17). Intermediate ellipses stand for (17(D20)/wt% < 21), and the outer ellipses hold for (21(D20)/wt% < 100. There are four types of critical lines, and all extrema on these lines correspond to double critical points, (b) Phase diagram at approximately constant critical concentration 3-MP (x 0.08) showing the evolution of the diagram as the deuterium content of the solvent varies. The white line is the locus of temperature double critical points whose extrema (+) corresponds to the quadruple critical point. Note both diagrams include portions at negative pressure (Visak, Z. P., Rebelo, L. P. N. and Szydlowski, J. J. Phys. Chem. B. 107, 9837 (2003))...

The values for the class A test samples are for the most part less than the critical value of 1.2. indicating that these test samples are correctly classified as belonging to class A. There are n o samples that are above the critical line, even though it is known that the samples belong in this class. At a 95% confidence level, it is not unusual to have 1 out of 20 samples incorrectly classified. but here there are 2 out of 12. This observation is considered below, when the values are partitioned into the contribution from the PCA residual and the distance from the SIMCA boundary. ... 

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