Critical point, envelope

The experiment could be repeated at a number of different temperatures and initial pressures to determine the shape of the two-phase envelope defined by the bubble point line and the dew point line. These two lines meet at the critical point, where it is no longer possible to distinguish between a compressed gas and a liquid. 

When the two components are mixed together (say in a mixture of 10% ethane, 90% n-heptane) the bubble point curve and the dew point curve no longer coincide, and a two-phase envelope appears. Within this two-phase region, a mixture of liquid and gas exist, with both components being present in each phase in proportions dictated by the exact temperature and pressure, i.e. the composition of the liquid and gas phases within the two-phase envelope are not constant. The mixture has its own critical point C g. 

The initial condition for the dry gas is outside the two-phase envelope, and is to the right of the critical point, confirming that the fluid initially exists as a single phase gas. As the reservoir is produced, the pressure drops under isothermal conditions, as indicated by the vertical line. Since the initial temperature is higher than the maximum temperature of the two-phase envelope (the cricondotherm - typically less than 0°C for a dry gas) the reservoir conditions of temperature and pressure never fall inside the two phase region, indicating that the composition and phase of the fluid in the reservoir remains constant. 

For both volatile oil and blaok oil the initial reservoir temperature is below the critical point, and the fluid is therefore a liquid in the reservoir. As the pressure drops the bubble point is eventually reached, and the first bubble of gas is released from the liquid. The composition of this gas will be made up of the more volatile components of the mixture. Both volatile oils and black oils will liberate gas in the separators, whose conditions of pressure and temperature are well inside the two-phase envelope. 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

Figure 5. Three-dimensional isodensity envelopes of (a) SCI2, (b) H2O, and (c) Cl2. The outer envelope has the value of 0.001 au, the van der Waals envelope the inner one is the bond critical point density envelope (pb-envelope).

FIGURE 10. Three-dimensional isodensity envelopes of the bond critical point density (pb-envelope)... 

In addition to the critical points an important feature appearing in one-electron densities are the closed contour curves corresponding to a given value of the density [20]. These closed equidensity curves envelop each one of the nuclei up to the bond critical point located along the bond path joining two nuclei. Other closed equidensity contours envelop both nuclei, etc. This is shown schematically in Fig. 2 for the LiH molecule. 

In the P,7-section the two-phase envelope is tangent to the binary critical curve in the critical point. 

Figure 2-10 shows a more nearly complete pressure-volume diagram.2 The dashed line shows the locus of all bubble points and dew points. The area within the dashed line indicates conditions for which liquid and gas coexist. Often this area is called the saturation envelope. The bubble-point line and dew-point line coincide at the critical point. Notice that the isotherm at the critical temperature shows a point of horizontal inflection as it passes through the critical pressure. 

The definition of the critical point as applied to a pure substance does not apply to a two-component mixture. In a two-component mixture, liquid and gas can coexist at temperatures and pressures above the critical point, Notice that the saturation envelope exists at temperatures higher than the critical temperature and at pressures higher than the critical pressure. We see now that the definition of the critical point is simply the point at which the bubble-point line and the dew-point line join. A more rigorous definition of the critical point is that it is the point at which all properties of the liquid and the gas become identical. 

Figure 2-15 shows phase data for eight mixtures of methane and ethane, along with the vapor-pressure lines for pure methane and pure ethane.3 Again, observe that the saturation envelope of each of the mixtures lies between the vapor pressure lines of the two pure substances and that the critical pressures of the mixtures lie well above the critical pressures of the pure components. The dashed line is the locus of critical points of mixtures of methane and ethane. 

Figure 2-20 gives the pres sure-volume diagram for a mixture of n-pentane and n-heptane, showing several isotherms and the saturation envelope.4 Notice that at lower temperatures the changes in slope of the isotherms at the dew points are almost nonexistent. Also notice that the critical point is not at the top of the saturation envelope as it was for pure substances. 

When the temperature exceeds the critical temperature of one component, the saturation envelope does not go all the way across the diagram rather, the dew-point and bubble-point lines join at a critical point. For instance, when the critical temperature of a mixture of methane and ethane is minus 100°F, the critical pressure is 750 psia, and the composition of the critical mixture is 95 mole percent methane and 5 mole percent ethane. 

When the pressure of interest exceeds the critical pressures of both components, the phase envelope exhibits two critical points. For instance, mixtures of methane and ethane exhibit critical points at 900 psia and minus 62°F and at 900 psia and 46°F. 

Figure 2-27 gives the saturation envelope for mixtures of methane, propane, and n-pentane at the same temperature as Figure 2-26 but at a higher pressure. The bubble-point and dew-point lines join at a critical point. The critical point gives the composition of the mixture, which has a critical pressure of 1500 psia and a critical temperature of 160°F. 

Above this pressure, dot 6, all mixtures of methane and propane are single phase. Thus only the methane-n-pentane binaries have two-phase behavior, and only the methane-n-pentane side of the ternary diagram can show a bubble point and a dew point. The bubble-point and dewpoint lines of the saturation envelope do not intercept another side of the diagram, rather the two lines join at a critical point, i.e., the composition of the three-component mixture that has a critical pressure of 1500 psia at 160°F. 

Note the wide variety of critical pressures and critical temperatures and the different positions that the critical points take on the saturation envelopes. Also note the very large separation between the critical temperature and the cricondentherm in all instances and the separation between cricondenbar and critical pressure for the lighter hydrocarbon mixtures in Figures 2-35 and 2-36. 

Mixture 2 on Figure 2-37 illustrates a mixture containing a large quantity of the light component. The phase envelope is relatively small and is located at low temperatures. The critical point is located far down the left-hand side of the phase envelope and is fairly close to the critical point of the pure light component. There is a large area in which retrograde condensation can occur. 

As heavy component is added to the mixtures—lines 3 and 4, for instance—the phase envelope increases in size and covers wider ranges of temperature and pressure. The critical point moves up closer to the top of the envelope. 

Phase behavior of multicomponent reservoir fluids is similar. Reservoir gases, which are predominately methane, have relatively small phase diagrams with critical temperatures not much higher than the orSiranempnatiire of nietfiahe. The critical point is"fairdown the left slope of the envelope. 

Black oils consist of a wide variety of chemical species including large, heavy, nonvolatile molecules. The phase diagram predictably covers a wide temperature range. The critical point is well up the slope of the phase envelope. 

The phase diagram for a typical volatile oil, Figure 5-2, is somewhat different from the black-oil phase diagram. The temperature range covered by the phase envelope is somewhat smaller, but of more interest is the position of the critical point. The critical temperature is much lower than for a black oil and, in fact, is close to reservoir temperature. Also, the iso-vols are not evenly spaced but are shifted upwards toward the bubble-point line. 

The phase diagram of a retrograde gas is somewhat smaller than that for oils, and the critical point is further down the left side of the envelope. These changes are a result of retrograde gases containing fewer of the heavy hydrocarbons than do the oils. 

Point P is called the plait point. Only a single phase exists as the plait point is approached. The plait point may be considered akin to the vapor-liquid-phase envelope critical point from Chap. 2, where the vapor-liquid phases become one phase. 

The critical point of a binary mixture occurs where the nose of a loop in Fig. 12.4 is tangent to the envelope curve. Put another way, the envelope curve is the... 

The condition at which the liquid just begins to form is called the dew point. The condition at which the vapor just begins to form is called the bubble point. A curve can be plotted showing the temperature and pressure at which a mixture just begins to liquefy. Such a curve is called a dew-point curve or dew-point locus. A similar curve can be constructed for the bubble point. The phase envelope is the combined loci of the bubble and dew points, which intersect at a critical point. The phase envelope maps out the regions where the various phases exist. 

Phase envelopes for typical natural gas tend to be fairly broad. That is, they cover a large range of temperature and pressure. On the other hand, the phase envelopes for acid gas mixtures tend to be quite narrow. Figure 3.2 shows the phase envelope for a mixture containing 50 mol% H2S and 50 mol% C02. This phase envelope was calculated using the Peng-Robinson equation of state, and the bubble, dew, and critical points are labeled. 

Although the gross shape of the phase envelope predicted by the mean-field theory, as well as the regions of lamellar and hexagonal phases, are more or less in agreement with experiments on diblock copolymers (see Fig. 13-4), the predictions of the theory near the critical point at / = 0.5, /(V = 10.5 are incorrect. Fredrickson and Helfand (1987) showed that the second-order transition predicted by the mean-field theory is corrected to SL first-order transition when the effects of fluctuations on the free energy are accounted for using a so-called Brazovskii Hamiltonian (Brazovskii 1975). 

The critical point of a binary mixture occurs where the nose of a loop in Fig. 10.3 is tangent to the envelope curve. Put another way, the envelope curve is the critical locus. One can verify tliis by considering two closely adjacent loops and noting what happens to the point of intersection as their separation becomes infinitesimal. Figure 10.3 illustrates that the location of the critical point on the nose of the loop varies with composition. For a pure species the critical point is the highest temperature and highest pressure at which vapor and liquid phases can coexist, but for a mixture it is in general neither. Therefore under certain conditions a condensation process occurs as the result of a reduction in pressure. 

Fig. 7.4. Representations of the Laplacian distributions of methane and methylfluoride. The figures in (a) are displays of the zero envelope of V p(r), those in (b) of the atomic graphs. The envelope encompassing the inner shell charge concentration on carbon appears as a small sphere. The envelopes of the bonded maxima in the VSCC of carbon also encompass the protons in CH and CH3F. There is a transfer of charge from C to F in CHjF and the bonded maximum along the C F axis is reduced to the small region lying between the C nucleus and the envelope on F. An atomic graph displays the connectivity of the critical points in a VSCC. The carbon nucleus is denoted by a solid cross, the positions of the remaining nuclei by open crosses. There is a bonded maximum, a (3, — 3) critical point in — V p, at each of the four vertices.

---