Phase diagrams closed-loop

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Fig. 10 Phase diagrams of dPS/PnPMA blends, o UCST at ambient pressure, LCST at ambient pressure. Closed-loop phase behaviours are observed at higher pressure A 97 bar 117 bar 138 bar 166 bar 186 bar. Dashed line Prediction of Tg, blend by Fox equation at ambient pressure. From [59]. Copyright 2004 American Chemical Society... 

Figure 5.26. Iron binary alloys. Examples of the effects produced by the addition of different metals on the stability of the yFe (cF4-Cu type) field are shown. In the Fe-Ge and Fe-Cr systems the 7 field forms a closed loop surrounded by the a-j two-phase field and, around it, by the a field. Notice in the Fe-Cr diagram a minimum in the a-7 transformation temperature. The iron-rich region of the Fe-Ru diagram shows a different behaviour the 7 field is bounded by several, mutually intersecting, two (and three) phase equilibria. The Fe-Ir alloys are characterized, in certain temperature ranges, by the formation of a continuous fee solid solution between Ir and yFe. Compare with Fig. 5.27 where an indication is given of the effects produced by the different elements of the Periodic Table on the stability and extension of the yFe field.

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

The static bifurcation characteristics of the resulting closed loop system have also been discussed in the previous section and we have seen that the bifurcation diagram of the reactor dense-phase dimensionless temperature, namely a plot of Yrd versus the controller gain Kc is a pitchfork. Such bifurcations are generally structurally unstable when any of the system parameters are altered, even very slightly. 

Figure 15.5. Pressure-temperature phase diagram of pure C02 with a superimposed flow diagram for a closed-loop supercritical fluid treatment process.

Li Unf. Ins. Be Far. 5-8% Cl. 7 (—23 n) Order of Entries 1. Size factor (far. = favorable if within 15% atomic size of Fe unf. = unfavorable mar. = marginal) 2. Atomic % solid solubility in a-Fe. 3. Characteristics of phase diagram (cl. y — closed y loop a far. a phase favored y fav. — y phase favored no fav. = neither phase favored ins. = insoluble int. = interstitial). 4. Measured rate of change in magnetisation per solute atom of disordered alloy. 5. Predicted rate of change in magnetisation per solute atom (in parentheses). B <0.1% Small ions enter iuterstitially ... 

For the hydrocarbon--CO2 systems studied here, at pressures above the critical pressure (7.383 MPa) and above the critical temperature (304.21 K) of C02 the isobaric x,T coexistence plots of liquid and vapor phases form simple closed loops. The minimum occurs at the lower consolute point or the Lower Critical Solution Temperature (LCST). Since pressure is usually uniform in the vicinity of a heat transfer surface, such diagrams serve to display the equilibrium states possible in a heat transfer experiment. 

There is another type of phase diagram enconntered in some polymer solutions, where the LCST lies below the UCST. This is called closed-loop and appears for polymer solutions where hydrogen bonding effects are dominant snch as polyethylene gly-col/water and polyvinyl alcohol/water (Figure 16.4). 

Di erent polymer-solvent systems may have completely different phase diagrams. For some systems, such as polystyrene-cyclohexanone, UCST < LCST [Fig. 3.13(a)] but for others, e.g., highly polar systems like polyoxyethylene-water, UCST > LCST and closed solubility loop is found [Fig. 3.13(b)]. 

The phase diagram of a nonionic amphiphile-water binary system is more complicated (see Figure 3.12). A classic upper critical point exists, but it is usually located below 0°C. At higher temperatures most nonionic amphiphiles show a miscibility gap, which is actually a closed loop with an upper as well as a lower critical point. The lower critical point CPp is often referred to as the cloud point temperature. The upper critical point often lies above the boiling temperature of the mixture (at 0.1 MPa). The position and the shape of the loop depend on... 

The closed loop is not the only characteristic of the nonionic surfactant-water binary phase diagram. Like the ionic surfactant-water mixture, nonionic surfactants, at higher concentration in water, exhibit lyotropic mesophases. Figure 3.14 shows a typical binary phase diagram exhibiting the full lyotropic mesophase sequence II, cubic isotropic phase HI, direct hexagonal phase (middle phase) VI, special cubic ( viscous phase) La, lamellar phase (neat phase). Note the presence of the two-phase domains surrounding each mesophase, the critical point on top of each, and the zero-variant three-phase feature. 

Some substances exhibit both upper and lower consolute temperatures. The diagram for the system nicotine-water is shown schematically in Fig. 15.3(b). The lower consolute temperature is about 61 °C, the upper one about 210 °C. At all points in the closed loop two phases are present, while the points outside the loop represent homogeneous states of the system. 

The hydrophobic interaction results in the existence of a lower critical solution temperature and in the striking result that raising the temperature reduces the solubility, as can be seen in liquid-liquid phase diagrams (see Figure 5.2a). In general, the solution behaviour of water-soluble polymers represents a balance between the polar and the non-polar components of the molecules, with the result that many water-soluble polymers show closed solubility loops. In such cases, the lower temperature behaviour is due to the hydrophobic effects of the hydrocarbon backbone, while the upper temperature behaviour is due to the swamping effects of the polar (hydrophilic) functional groups. 

In the normal FI extraction mode the enrichment factor is limited by the phase ratio which, in turn, is restricted by practical factors. A relatively complex extraction system was described by Atallah et al.[26] in which the continuously pumped sample is extracted by a small volume of oiganic phase trapped in a closed loop which incorporates the segmentor, extraction coil, phase separator and detector flow-cell. A simplified schematic diagram of the circulated extraction part of the manifold in shown in Fig. 

According to the type of T versus q> diagram (Fig. 25.4), the binary solution can exhibit an upper critical solution temperature (UCST), a lower critical solution temperature (LCST), or both (close-loop phase behavior). Above the UCST or below the LCST the system is completely miscible in all proportions [82], Below the UCST and above LCST a two-phase liquid can be observed between cp and cp". The two-phase liquid can be subdivided into unstable (spontaneous phase separation) and metastable (phase separation takes some time). These two kinds of mixtures are separated by a spinodal, which is outlined by joining the inflexion points (d AGIdcp ) of successive AG versus cp phase diagrams, obtained at different temperatures (Fig. 25.3b). Thus, the binodal and spinodal touch each other at the critical points cp and T. ... 

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