Curved equilibrium line

Note that the graphical integration is never exact and hence the correction often makes little difference except for cases of curved equilibrium lines. 

FIGURE 6.3-13 Diagram of curved equilibrium line showing inflection point and effective avemge slopes for both sections- (From Wilke nad von Stockar,10 copyright 1976. Used with permission of John Wiley Sons, New York.)... 

Complicated systems (high concentrations, heat effects, curved equilibrium lines, multicomponents, etc.) are considered elsewhere (3). 

Figure 4.1.1(c) illustrates the equilihrium vapor and liquid compositions for the henzene-toluene system at a constant total pressure of 1 atmosphere. The vapor phase is enriched in the lighter component, benzene, at all compositions, except in the case of pure benzene and pure toluene. If such vapor-phctse enrichment of benzene did not occur, the straight reference line (xi = Xk) would have represented the VLB. The actual curved equilibrium line relating X to xu may be obtained in the following manner. Consider first Raoult s law expressions for i = 1 and i = 2 ... 

From the Helmoltz free energy versus volume curves, equilibrium lines between the three phases (5-Pu, 5 -Pu and 6-Pu ) are calculated and the phase diagram plotted. The corresponding GGA and LDA+U phase diagrams are reported in figures 2 and 3 and could be compared to the experimental pressure-temperature diagram drawn in figure 1. 

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Rapid Approximate Design Procedure for Curved Operating and Equilibrium Lines. If the operating or the equihbrium line is nonlinear, equation 56 is of Httie use because will assume a range of values over the tower. The substitution of effective average values for m and... 

Fig. 8. Correlation of the effective average slopes m of the equilibrium line (4). (a) Equilibrium line curved concave upward (b) equilibrium line curved...

The end points of the operating line on an XY plot (Fig. 15-13) are X., Y, andXy, Y., and the number of theoretical stages can be stepped off graphically. The equilibrium curve is taken from the Hand type of correlation shown earlier (Fig. 15-9). When the equilibrium line is straight, its intercept is zero, and the operating line is straight, the number of theoretical stages can be calculated with one of the Kremser equations [Eqs. (l5-14) and (15-15)]. When the intercept of the eqnihbrinm line is not zero, the value of YJK, should be used... 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

The shape of the equilibrium line, or solubility curve, is important in determining the mode of crystallization to be employed in order to crystallize a particular substance. If the curve is steep, i.e. the substance exhibits a strong temperature dependence of solubility (e.g. many salts and organic substances), then a cooling crystallization might be suitable. But if the metastable zone is wide (e.g. sucrose solutions), addition of seed crystal might be necessary. This can be desirable, particularly if a uniformly sized product is required. If on the other hand, the equilibrium line is relatively flat (e.g. for aqueous common salt... 

For a plot of xp = 0.750, slope = 0.7875, read xyy at the equilibrium line for each theoretical tray and plot similar to Figure 8-38. Then determine the area under the curve between the selected x and the product x. Then ... 

Phase diagram of water (not to scale). The curves and line represent the temperatures and pressures at which phases are in equilibrium. The triple point is at 0.0rC, 4.56 mm Hg the critical point is at 374°C. 

Increasing the pressure may either increase or decrease the UCST and, thus, raise or lower the (liquid + liquid) equilibrium line. Figure 8.19, for example, shows the effect of pressure on the (liquid + liquid) equilibrium curves for (xin-CftHu + A2CH3OH).3 In this example, we see that increasing p increases the UCST. 

Figure 8.23. The solid and monotonically declining line to the right represents the equilibrium curve. The four curves represent lines with constant rates in the same plot. Since we want to operate at points where of maximum ammonia concentration, the optimal operation line is defined as the line running parallel to the equilibrium curve, passing through all the maxima. [Adapted from I. Dybkjaer, in Ammonia Catalysis and Manufacture (1995) Ed. A. Nielsen. Springer-Verlag, Berlin/Heidelberg, p. 199.]...

The equilibrium curve is plotted in Figure lie. As the feed is at its boiling-point, the q-line is vertical and the minimum reflux ratio may be found by joining the point (xj, xj) with the intersection of the q- m and the equilibrium curve. This line when produced to the y-axis gives an intercept of 0.475. 

The equilibrium line for the liquid-liquid transition given in Figure 5.12(a) is also given in Figure 5.12(b). This line goes through the intersections of all pairs of lines that satisfy xj g + x g =1, e.g. the intersection of the lines for xsi.B = 0.1 and 0.9 and that for xsi.b = 0.2 and 0.8. These two specific examples represent two discrete points on the equilibrium curve. 

The equilibrium adsorption characteristics of gas or vapor on a solid resemble in many ways the equilibrium solubility of a gas in a liquid. Adsorption equilibrium data are usually portrayed by isotherms lines of constant temperature on a plot of adsorbate equilibrium partial pressure versus adsorbent loading in mass of adsorbate per mass of adsorbent. Isotherms take many shapes, including concave upward and downward, and S-curves. Equilibrium data for a given adsorbate-adsorbent system cannot generally be extrapolated to other systems with any degree of accuracy. 

Figure 7.8 Equilibrium curves (dotted lines) for the WGS reaction with or without methanation reaction for feed gas composition with 50% H2. The filled squares and circles are CO conversions over G-66 A and Cu02Ce08O2 y, catalysts, respectively. The solid lines are model fits assuming first-order reversible kinetics. (Reprinted from [51], With permission from Elsevier.)...

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