Critical pseudocritical point

The above trend is valid for members of a homologous series. For components which are not members of a homologous series, the reverse trend may occur over a limited temperature range, causing relative volatility to increase as the equilibrium temperature is raised [Eq. (1.12)]. However, as temperature is raised further and approaches the critical point, relative volatility eventually diminishes and will reach unity at the pseudocritical point of the mixture. 

The heat transfer to supercritical carbon dioxide was measured in horizontal, vertical and inclined tubes at constant wall temperature for turbulent flow at Re-numbers between 2300 and lxl 05. The influence of the variation of physical properties due to the vicinity of the critical point was examined, as well as the influence of the direction of flow. Therefore most of the measurements were conducted at pseudocritical points. At those supercritical points the behaviour of the physical properties is similar to the behaviour at the critical point, but to a lesser degree. At such points the heat capacity shows a maximum density, viscosity and heat conductivity are changing very fast. [1]... 

At the pseudocritical points that are nearest the critical point, this behaviour is most accentuated. 

Figure 3. Heat transfer at a pseudocritical point at 120 bar at a small distance from the critical point.

We note from Figures 10.3-2 and 10.3-3 that the shapes of the critical loci of mixtures are complicated and that, in general, the critical temperature and/or pressure of a binary mixture is not intermediate to those properties of the pure fluids. It is of interest to note that, in analogy with the properties of a pure fluid, a pseudocritical point of a mixture of a fixed composition is defined by the mechanical stability inflection point,... 

Figure A33 Photos of carbon dioxide during transition through (a) critical and pseudocritical points and (h) corresponding pressure—temperature diagram (Gupta et al., 2013).

Pseudocritical point (characterized with P and pc) is the point at a pressure above the critical pressure and at a temperature (Tpc > Ter) corresponding to the maximum value of specific heat at this particular pressure. 

Critical parameters of selected fluids are listed in Table A3.1. For better understanding of general trends and specifics of various thermophysical properties near critical and pseudocritical points, it was decided to show these properties in comparison with subcritical properties for water (see Figs. A3.5—A3.12). Also, thermophysical... 

The specific heat of water (see Fig. A3.9(b)) (as well as of other fluids, for example, for carbon dioxide, see Fig. A3.18 and Fig. A3.26 for helium) has a maximum value at the critical point. The exact temperature that corresponds to the specific heat peak above the critical pressure is known as the pseudocritical temperature (see also Figs. A3.23 and A3.24, and Table A3.2 for water and carbon dioxide). For water at pressures approximately above 300 MPa and for carbon dioxide at pressures above 30 MPa (see Fig. A3.24), a peak (here, it is better to say a hump ) in specific heat almost disappears therefore, the term such as a pseudocritical point no longer exists. The same applies to the pseudocritical line. 

Table A3.3 Peak values of specific heat, volume expansivity, and thermal conductivity in critical and near pseudocritical points (a) water and (h) carhon dioxide... 

These heat transfer regimes and special phenomena appear to be due to significant variations of thermophysical properties near the critical and pseudocritical points (see Appendix A3) and due to operating conditions. 

The majority of empirical correlations were proposed in the 1960s—1970s (Pioro and Duffey, 2007), when experimental techniques were not at the same level (ie, advanced level) as they are today. Also, thermophysical properties of water have been updated since that time (eg, a peak in thermal conductivity in critical and pseudocritical points within a range of pressures from 22.1 to 25 MPa for water (see Appendix A3) was not officially recognized until the 1990s). 

Eq. [A4.4] is applicable for subcritical and supercritical pressures. However, adjustment of this expression to conditions of supercritical pressures, with singlephase dense gas and significant variations in thermophysical properties near the critical and pseudocritical points, was the major task for the researchers and scientists. In general, two major approaches to solve this problem were taken an analytical approach (including numerical approach) and an experimental (empirical) approach. 

Reduced Equations of State. A simple modification to the cubic van der Waals equation, developed in 1946 (72), uses a term called the ideal or pseudocritical volume, to avoid the uncertainty in the measurement of volume at the critical point. 

The Law of Corresponding States has been extended to cover mixtures of gases which are closely related. As was brought out in Chapter 2, obtaining the critical point for multicomponent mixtures is somewhat difficult therefore, pseudocritical temperature and pseudocritical pressure have been invented. 

It would be convenient if the critical temperature of a mixture were the mole weighted average of the critical temperatures of its pure components, and the critical pressure of a mixture were simply a mole weighted average of the critical pressures of the pure components (the concept used in Kay s rule), but these maxims simply are not true, as shown in Fig. 3.22. The pseudocritical temperature falls on the dashed line between the critical temperatures of CO2 and SO2, whereas the actual critical point for the mixture lies somewhere else. The solid line in Fig. 3.22 illustrates how the locus of the actual critical points diverges from the locus of the pseudo critical points. 

The second approach is the phenomenological renormalization (PR) [24,70] method, where the sequence of the pseudocritical values of X can be calculated by knowing the first and the second lowest eigenvalues of the matrix for two different orders, N and N1. The critical Xc can be obtained by searching for the fixed point of the phenomenological renormalization equation for a finite-size system [70],... 

The critical point of a mixture is determined by the critical points of the constituents however, no explicit relationship is available relating the former to the latter. Mixture pseudocritical... 

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