Estimating Equilibrium Curves

In distillation work for binary systems with constant relative volatility, the equilibrium between phases for a given component can be expressed by the following equation  

The procedure to calculate the points for an equilibrium curve is to start with x = 0, and calculate the value of y for a given value of a. A second point is calculated by incrementing the value of x and repeating the calculation for y. This calculation is repeated until the value of x = 1. 

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With this coordinate system the operating line is straight. The equilihiinm curve may be based on actual data for the specific system at column operating conditions or nuy be approximated on the basis of related data. Rousseau and Staton outline steps fix estimating equilibrium curves based on the Hairy s law constant fix unreacted componoit A in the liquid and the equilibrium constant for the chonical reaction of A with the reactive con onoit. 

The equilibrium curve and the optimal operation line are again plotted in an ammonia concentration versus temperature plot for each of the two sets of conditions in Fig. 8.26, but now together with the optimal catalyst curves for a few selected nitrogen bonding energies. The right-hand panel also shows the operating line, and it is now possible to estimate which catalyst should be where in the reactor. 

The equilibrium relations of epidote-K-mica-K-feldspar-pyrite-chlorite, hematite S jq = pyrite -I- H2S, anhydrite-magnetite-pyrite-clinozoisite and pyrite-hematite-magnetite assemblage are shown in Fig. 2.27. Based on the equilibrium curves and analytical data on epidote and chlorite, /hjS of the epithermal Au-Ag vein ore fluids for some propylitic alterations is also estimated (Shikazono, 1985a). 

Figure 5.71A shows univariant equilibrium curves for various molar amounts of ferrous component in the orthopyroxene mixture. The P-T field is split into two domains, corresponding to the structural state of the coexisting quartz (a and j3 polymorphs, respectively). If the temperature is known, the composition of phases furnishes a precise estimate of the P of equilibrium for this paragenesis. Equation 5.277 is calibrated only for the most ferriferous terms, and the geobarometer is applicable only to Fe-rich rocks such as charnockites and fayalite-bearing granitoids. 

Selected entries from Methods in Enzymology [vol, page(s)] Computer programs, 240, 312 infrared S-H stretch bands for hemoglobin A, 232, 159-160 determination of enzyme kinetic parameter, 240, 314-319 kinetics program, in finite element analysis of hemoglobin-CO reaction, 232, 523-524, 538-558 nonlinear least-squares method, 240, 3-5, 10 to oxygen equilibrium curve, 232, 559, 563 parameter estimation with Jacobians, 240, 187-191. 

In the design of equipment, it is necessary to estimate the rate of mass transfer from known or predicted values of the mass-transfer coefficients and the bulk concentrations. This may be done by solving Equation (6.1-5) simultaneously with the equilibrium curve to obtain xi and yt. The rate of transfer may then be calculated from Equation (6.1-2). 

The number of ideal stages can be estimated by drawing steps between the equilibrium curve and the operating line as shown in Figure 10.11. The intermediate concentrations can be also read from the figure. 

Step 2. Select solvent inlet purity. The solvent (benzene) enters the extractor at a point that must be reasonably true and obtainable for the equilibrium curve. Please note that we cannot cross the equilibrium curve and that the McCabe-Thiele steps will be to the right of the curve. The left side would be for points of miscible liquids, not two-phase. Since the water is to contain 1.0% solute at exit, a good estimate for the solvent entering is 0.5 wt % acid. Reviewing the Fig. 7.2 equilibrium curve, in order to meet this treated water 1.0% acid specification, the concentration of acid in the inlet counterflow solvent must be below 1.0%. We may assume the solvent has been nearly stripped free of acetic acid. Thus, 0.5% acid in the inlet solvent (benzene) is a reasonable assumption. Plotting the McCabe-Thiele equilibrium steps will show that this is a good and reasonable estimate to achieve the treated water specification of 1.0% acid remainder. Then,... 

Figure 53A illustrates the Hill plot of the 02 equilibrium curves of (a+CNp)A(aP)cXL in 0.1 M phosphate buffer at various pH values. The 02 equilibrium curves converge at a high-saturation range, as seen in (ap)A(aP)cXL, whereas the lower asymptotes diverge more than those of (aP)A(aP)cXL (see Miura et al. (1987). The 02 binding curves show high cooperativity at low pH, with a Hill coefficient value of 1.8. The estimated Kj (i = 1, 2, or 3) values are plotted against pH as shown in Fig. 53B. K3 of (a+CNP)A( P)cXL is observed to be less pH dependent than A and K2. Its value is very similar to those of K4 of (aP)A(aP)cXL and Hb A, which are essentially pH independent. The K value of (a+CNP)A(aP)cXL shows a pH dependence similar to that of K2 or K3 of (ap)A(aP)cXL rather than that of K of (aP)A(ap)cXL. The slope of the plot of P30 of (a+CNP)A(aP)cXL versus pH is about 0.6 at pH 7.1, which is steeper than that of (aP)A(aP)cXL,...

The 02 equilibrium curves of asymmetric cyanomet hybrid Hbs with two cyanomet subunits—(a+CNp+CN)A(aP)cXL and (aP+CN)A (a+CNp)cXL—were also measured. However, it was not feasible to estimate reliable values of Kt and by curve fitting (Miura et al., 1987). We found that it was difficult to fit the 02 binding scheme with two binding constants over the entire saturation range of the... 

The equilibrium curve slope m can be estimated from Henry s law. 

The tray efficiency may generally be estimated on the basis of past experience with similar columns or from operating data of an existing column. Given the tray efficiency, a pseudo-equilibrium curve can be drawn, such as the dotted line in Figure... 

The number of stages between the solvent feed stage and the main feed stage, and between the main feed stage and the reboiler are estimated using the Y-X diagram of the acetone/chloroform solvent-free equilibrium curve T/ vs. X, ... 

The equilibrium curve used for estimating the number of stages in this section is a plot of 12 vs. 12 where and Xi2 = Xi -I- X. ... 

While thus the estimation of coexistence curves and equilibrium scattering functions S q) from Monte Carlo simulations is a well-controlled procedure, it is rather hard to estimate spinodal curves (Fig. 22). While is also possible to generate metastable states at different conditions and check whether they decay... 

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