Phase Boundary Calculations

Bubble points or dew points can be determined by flash calculations with t t in 

Equation 2.7 or 2.13 set to either zero or one. Although the general flash calculation 

At the bubble point, the following condition must, therefore, be satisfied  

Similarly, at the dew point, the vapor composition equals the feed composition, and the composition of a liquid drop at equilibrium with the vapor is given by Equation 2.13a as 

A bubble point or dew point calculation consists of determining the pressure at a given temperature, or the temperature at a given pressure, which would result in X S that satisfy Equation 2.16 or 2.17. In addition, the phase equilibrium condition. Equation 2.12, must be satisfied. 

---

Since Chen used the second virial approximation in his phase boundary calculation, it is adequate to compare his results with the phase boundary concentrations calculated from Onsager s expression of S (cf. Sect. 2.5) and cj(N) of Eq. (18) together with the OTF. Chen expressed the phase boundary concentrations in terms of reduced quantities Cj = L2dc j and cA = L2dc A, which depend only on N. As shown in Fig. Al, the relative differences in both Q and cA between the two procedures are less than 13% over the whole range of N. This confirms the relevance of the OTF for semiflexible polymer systems. 

Table 7.5-1 Thermodynamic Properties of Oxygen Along the Vapor-Liquid Phase Boundary Calculated Using the Peng-Robinson Equation of State...

Use of equations (8.11) (constant AC°) and (8.12) (AC° - 0) results in the data in Table 8.4, which are plotted in Figures 8.10 and 8.11. As expected, including a constant AC° gives somewhat better results than assuming AG° = 0, but the errors introduced can be quite serious for phase boundary calculations, as shown. For a AT of only 400°C (from 298.15 to 700 K) the phase boundary is in error by more than a kilobar whether AC° is constant or zero. 

Bubble points or dew points can be determined by flash calculations with yr in Equations 2.7 or 2.13 set to either zero or one. Although the general flash calculation method described in the previous section may be applied to bubble points and dew points, phase boundary calculations are handled more efficiently by other methods. At the bubble point, the liquid composition equals the feed composition since all the feed remains in the liquid phase. The composition of a vapor bubble at equilibrium with the liquid is given by Equation 2.13b, which reduces to... 

Phase boundaries calculated from the above equations are shown in Fig. 2. Few data exist on the solubility of N in (aFe)-Al alloys. They have been obtained when studying the precipitation kinetics of AIN in ferrite eontaining 0.19 at.% A1 and 0.04 at% N [19720ga] or 2 at.% A1 and 0.02 at% N, the maximum amount of nitrogen dissolved in the ferrite matrix at 575°C [1988Lan]. An appreciable rate of precipitation was found between 450 and 620°C flie rate was greater in cold-woiked samples flian in as-quenched ones. 

In the case, where all 3 phases are present, the detector measurements reveal the amounts of tracers in each phase and the position of the boundaries between the phases The cross section area of each phase is calculated fi-om the latter. From this the tracer concentrations and hence the volume flows of the 3 phases are calculated. 

Critical Temperature The critical temperature of a compound is the temperature above which a hquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. 

Figure 12.6. (a) Calculated (a + a) phase boundary for Fe-V together with experimental boundaries (After Spencer and Putland 1973). (b) Comparison between calculated and experimental values of the concentration of Al, V and Fe in the two phases in Ti-6AMV alloy (after Saunders and... 

Some results reported by Zandbergen and Beenakker are shown in Fig. 26. Considering the severe simplifying assumptions made, the calculated phase boundary is in gratifying agreement with that found experimentally. Because of symmetry with respect to mole fraction in the three-liquid model, the calculated T-y curve shown is necessarily a parabola whose maximum is at... 

The Phase Boundaries. Our interest in the phase diagram is solely to define the method by which the calculation of the partial pressures should be done. Because (1) there are only small... 

Trasatti has calculated the potentials of several organic, solvents from Volta potentials and the partial surface potentials on the mercury solution phase boundaries at the potential of zero charge. ... 

Comparison of the calculated and observed changes in the EMF is shown in Table 1. It can be seen that the calculated changes in the phase boundary potential of membranes with 1.0 mM 1-3 in contact with 0.1 and 0.01 M aqueous KCl or RbCl were in good agreement with the corresponding observed values. Such an agreement indicates that it is reasonable to apply the present surface model to explain that the phase boundary potential is, in fact, determined by the amount of the primary cation permeated into or released out of the membrane side of the interface. 

The case of the prescribed material flux at the phase boundary, described in Section 2.5.1, corresponds to the constant current density at the electrode. The concentration of the oxidized form is given directly by Eq. (2.5.11), where K = —j/nF. The concentration of the reduced form at the electrode surface can be calculated from Eq. (5.4.6). The expressions for the concentration are then substituted into Eq. (5.2.24) or (5.4.5), yielding the equation for the dependence of the electrode potential on time (a chronopotentiometric curve). For a reversible electrode process, it follows from the definition of the transition time r (Eq. 2.5.13) for identical diffusion coefficients of the oxidized and reduced forms that... 

To calculate free energies of solvation for several organic molecules, Fortunelli and Tomasi applied the boundary element method for the reaction field in DFT/SCRF framework173. The authors demonstrated that the DFT/SCRF results obtained with the B88 exchange functional and with either the P86 or the LYP correlation functional are significantly closer to the experimental ones than the ones steming from the HF/SCRF calculations. The authors used the same cavity parameters for the HF/SCRF and DFT/SCRF calculations, which makes it possible to attribute the apparent superiority of the DFT/SCRF results to the density functional component of the model. The boundary element method appeared to be very efficient computationally. The DFT/SCRF calculations required only a few percent more CPU time than the corresponding gas-phase SCF calculations. 

In the strong segregation phase boundaries were calculated for xN 100 and compared to the experimental results for Pl-arm-PS stars. The results are shown in Fig. 36. 

The Thiele modulus and the effectiveness factor, respectively, were calculated for the three CO conversions X = 5,40, and 80%. The H2, CO, and H20 gas phase concentrations as well as the respective H2 concentration at the gas-wax phase boundary were taken from Table 12.3. The value of the diffusion coefficient Dm, is listed in Table 12.1. 

Figure 6.9 shows how the calculated phase boundaries compare with the experimental observations of Bowen and Schairer (1935) on the same binary join. The satisfactory reproduction of phase assemblages clearly indicates that the... 

Hume-Rothery was to prove a fair, if demanding, editor, and the result was an important review on the stability of metallic phases as seen from the CALPHAD viewpoint (Kaufman 1969). The relevant correspondence provides a fascinating insight into his reservations concerning the emerging framework fiiat Kaufman had in mind. Hume-Rothery had spent most of his life on the accurate determination of experimental phase diagrams and was, in his words (Hume-Rothery 1968), ... not unsympathetic to any theory which promises reasonably accurate calculations of phase boundaries, and saves the immense amount of work which their experimental determination involves . 

This has added mathematical complexity but it can be a very sensitive parameter for convergence which can be achieved quite rapidly. The calculation of the phase diagram is then achieved by calculating phase equilibria at various temperatures below 900 K and plotting the phase boundaries for each temperature. 

---