Locus Envelope

Before applying these ideas to special cases, we may note that the envelope locus may be a single curve (Fig. 96) or several (Fig. 97). For an exhaustive discussion of the properties of these discriminant relations, I must refer the reader to the text-books on the subject, or to M. J. M. Hill, On the Locus of Singular Points and Lines, Phil. Trans., 1892. To summarize ... 

The envelope locus satisfies the original equation but is not included in the general solution (see xx, Fig. 146),... 

Joule-Thomson Inversion Temperature. The Joule-Thomson coefficient is a function of temperature and pressure. Figure 5.8 shows the locus of points on a temperature-pressure diagram for which p,jx. is zero. Those points are at the Joule-Thomson inversion temperature 7). It is only inside the envelope of this... 

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

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. 

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. 

At pressures above the vapor pressure of propane and less than the critical locus of mixtures of methane and n-pentane, for instance 500 psia, dot 4, the methane-propane and methane-n-pentane binaries exhibit two-phase behavior, and propane-n-pentane mixtures are all liquid. Thus the saturation envelope appears as in Figure 2-28 (4). 

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

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. 

Figure 3.2 shows a phase envelope for an acid gas mixture. Note that the locus at lower pressure is the dew-point curve, whereas the one at higher pressure is the bubble-point curve. In fact, any point inside the phase envelope is a two-phase point. 

Figure 5.5 shows the phase envelope and two hydrate loci for this acid gas mixture. The first curve labeled "Saturated" assumes that there is plenty of water present. The other hydrate locus is labeled "3.2 g/m3[std]," and this is the hydrate curve for the acid gas containing the specified amount of water. 

For each porosity, there is a particular yield locus, a family of three 3deld loci is shown in Figure 12.38. Many experiments [72] have established that the envelope of the Mohr circles through the points Ei that lead to steady state flow for different porosities is, to a veiy close... 

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. 

A given yield locus generally has an envelope shape the initial density for all points forming this locus prior to shear is constant. That is, the locus represents a set of points all beginning at the porosity this critical state porosity is determined by the intersection with the effective yield locus. 

Bubble points and dew points may be generated as described above for a given mixture over ranges of temperature and pressure. The locus of bubble points is the bubble point curve and the locus of dew points is the dew point curve. The two curves together define the phase envelope. In addition to the bubble point curve (total liquid saturated) and the dew point curve (total vapor saturated), other curves may be drawn representing constant vapor mole fraction. All these curves meet at one point, the critical point, where the vapor and liquid phases lose their distinctive characteristics and merge into a single, dense phase. 

Figure 3.2c shows the familiar cigar-shaped vapor—liquid envelope found in many elementary textbooks on phase equilibria. At a fixed overall composition (denoted by x in this figure) there exists a single vapor phase at very low pressures. As the pressure is isothermally increased, the two-phase vapor-liquid envelope is intersected and a dew or liquid phase now appears. The locus of points that separates the two-phase vapor-liquid region from the one-phase vapor region is called the dew point curve. The concentration of the equilibrium vapor and liquid phases within the two-phase boundary of the vapor-liquid envelope is determined by a horizontal tie line similar to the one depicted in this figure. 

As the pressure is further increased at this fixed overall composition, the amount of the liquid phase increases while the amount of the vapor phase shrinks until only a small bubble of vapor remains. If the pressure is still further increased, the bubble of vapor finally disappears, then a single liquid phase exists. The locus of points that separates the two-phase vapor-liquid region from the one-phase liquid region is called the bubble point curve. This vapor-liquid envelope can now be inserted into the three-dimensional P-T-x diagram in figure 3.2a. 

In Eq. (5.3-10), if the vapor molar flow rate V and the liquid molar flow rate L +, are constant throughout the upper envelope, then die equation represents a straight line with slopa +, Vn whco y is plotted against x , . The line is known as the upper operating tine and is die locus of points coupling die vapor and liquid compositions of streams passing each other (see Fig. 5.3-5). 

At the critical temperature and pressure the liquid and vapor phases are indistinguishable. One such point exists for each envelope. The curve connecting the points for all envelopes is called the locus of criticals. ... 

Binodal points represent the points of contact of a common tangent to A vs. V at constant temperature and composition when a region of negative curvature exists between two regions of positive curvature. The locus of binodal points, known as the binodal curve or two-phase envelope, represents the experimentally observed phase boundary under normal conditions. For example, saturated liquid and saturated vapor represent states on the binodal curve. The binodal region exists between the binodal and spinodal curves, where p/ V)T,aa jv < 0. 

The products of the major histocompatibility locus have been shown to interact with viral antigens on the cell surface of thymocytes. A number of polypeptides bind to the major envelope glycoprotein of Rauscher murine leukaemia virus. The receptors for this glycoprotein, which probably involve lipoproteins, have been prepared from plasma membrane preparations from mouse cells. ... 

In Figure 8.12 the outer envelope is the locus of saturated equimolar liquid states and saturated equimolar vapor states. However, note that Figure 8.12 is not a phase-equilibrium diagram in Figure 8.12 every point on the two-phase line represents an equimolar mixture, but phases in vapor-liquid equilibrium generally do not have the same composition. Consequently, Figure 8.12 contains no tie lines across the two-phase region. Outside the saturation envelope, the mixtures are stable one-phase fluids. Underneath that envelope, the mixtures may be metastable one-phase fluids or they may be unstable to one phase (that is, they may exist as two-phases). 

The middle envelope is the spinodal the set of states that separate metastable states from unstable states. Recall from 8.3 that one-phase mixtures become diffusionally unstable before becoming mechanically unstable. Therefore, the mixture spinodal is the locus of points at which the diffusional stability criterion (8.3.14) is first violated that is, it is the locus of points having... 

The inner envelope in Figure 8.12 is the line of incipient mechanical instability the line separating states that are only diffusionally unstable from states that are both diffusionally and mechanically unstable. The line of incipient mechanical instability is the locus of points at which (8.1.31) is first violated that is, the points at which... 

---