Equilibrium line calculations

In shortcut calculations the slope of the equilibrium line in Bancroft (weight-ratio) coordinates m is also used [Eq. (15-4)]. 

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... 

Likewise, one knows that Y will be on the equilibrium line with X (see Fig. 15-12). One can therefore calculate a pseudo concentration of solute in the inlet extraction solvent Yf that 011 fall on the operating line [Eq. (15-12)] where=Xr [Eq. (15-20)]. 

The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase N r is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. 

This graphical representation is easier to use for nonideal systems than the calculation method. This is another limiting condition for column operation, i.e., below this ratio the specified separation cannot be made even with infinite plates. This minimum reflux ratio can be determined graphically from Figure 8-23, as the line with smallest slope from xp intersecting the equilibrium line at the same point as the q line for mixture following Raoul t s Law. 

Note that if the problem of accurate graphical representation occurs in the rectification end of the diagram, the corresponding relation to use to calculate the balance of the trays, assuming straight line operating and equilibrium lines in the region is [59] ... 

Ternary equilibrium curves calculated by Scott,who developed the theory given here, are shown in Fig. 124 for x = 1000 and several values of X23. Tie lines are parallel to the 2,3-axis. The solute in each phase consists of a preponderance of one polymer component and a small proportion of the other. Critical points, which are easily derived from the analogy to a binary system, occur at... 

If the operating and equilibrium lines are straight, and they usually can be taken as such when the concentrations are small, the number of stages required can be calculated using the equations given by Robinson and Gilliland (1950). 

Equation 11.66 is not of much practical use in distillation, as the slopes of the operating and equilibrium lines will vary throughout the column. It can be used by dividing the column into sections and calculating the slopes over each section. For most practical purposes, providing the plate efficiency does not vary too much, a simple average of the plate efficiency calculated at the column top, bottom and feed points will be sufficiently accurate. 

To use RXN to calculate the equilibrium lines above in general and specific forms, we type... 

The long command tells the program to show the reaction s equilibrium constant versus temperature and calculate its equilibrium equation pH = causes the program to render the equation in terms of pH instead of log aH+. To find the equilibrium lines written in terms of pe and Eh, we type... 

K is the linearized relationship between the perturbations in vapor composition y and liquid composition x,. Note that this is not the same K value" used in VLE calculations which relates total x and y variables. The K s in Table 12.2 are the slopes of the equilibrium line and relate perturbation variables. 

Figure 35. Total conductivity, a, and oxygen stoichiometry, 3-d, at 1000 °C of Lao.9Sro.iMn03-(5, from measurements by Kuo et al. The model calculations are based on a large polaron model with equilibrium constants as given in ref 216. Thick line calculated stoichiometry, thin line calculated conductivity. (Reprinted with permission from ref 216. Copyright 2000 Elsevier.)...

Figure 2. Left equilibrium geometries of the two lowest energy isomeric states of Au clusters obtained using LDA or GGA scalar relativistic pseudo-potentials. The ground state is Au for GGA and Auj for LDA (except for n=6, which LDA structure is also Aue). Right difference in the binding energy per atom of the planar and 3D structures given in the left panel for neutral gold clusters with 6

Another effect of aromatics is increased carbon formation, which has long been recognized as the primary means of catalyst deactivation in the ATR of hydrocarbons. Using carbon-forming reactions (6)-(8), an equilibrium line for carbon formation as a function of O2/C and S/C ratios can be calculated. Figure 9 shows the results of this calculation for n-Ci4, along with the experimental results for two Ni-based commercial catalysts. 

The fact that the catalyst can promote dehydrogenation as well as isomerization is shown in Figure 5, where the weight per cent benzene in the products of experiments on dehydrogenation of cyclohexane is shown plotted against the reaction temperature. These results are compared to Von Muffling s (7) calculated equilibrium line. The... 

The graphite-diamond equilibrium line up to 1200 K was calculated in 1938 (6) by using the observed heat, compressibility, and thermal expansion data of the two components (Fig. 1). Subsequendy, estimates of the diamond—graphite equilibrium line were refined and extended (9) and the extrapolation to higher temperatures fits the experimental data (10). It is evident that diamond is not thermodynamically stable below a pressure of about 1.6 GPa (16 kbar) and early investigators were using pressures in their experiments where diamond would have been unstable. 

Figure 15.2 Graph of log K, the thermodynamic equilibrium constant calculated from molecular data (solid line), against 1 /T. Also shown is a prediction of log K obtained by Nernst (dashed-dotted line).

Reaction (15.1) is known as the Haber reaction in recognition of the major role of Fritz Haberf in characterizing this process early in the twentieth century. At that time neither the molecular data nor the mathematical relationships were available for calculating the equilibrium condition, so that Haber had to rely upon experimental measurement. He determined the equilibrium concentration of NH3 in the (N2 + 3H2) mixture8 as a function of temperature. His measurements, graphed as mole percent NH3, were made at a total pressure of 1 atm (1.01 bar), and are also shown in Figure 15.3.1 The agreement with the prediction from the thermodynamic equilibrium constant calculated from the molecular parameters (solid line) is excellent. 

Transfer units may be calculated from Eq. 14-25, replacing mole fractions with ppm concentrations, and since the operating and equilibrium lines are straight,... 

Figure 4.16. Temperature dependence of the equilibrium value of the "rheological" degree of conversion. Points - experimental solid line calculated.

Having the K equilibrium constants calculated for each component of the mixture, the FlshRef program applies the equation shown on code line 1010 ... 

Since the top and bottom pinch points shown in Fig. 11 represent intersections of operating and equilibrium lines, they may be predicted analytically. At the top, the lean gas and the lean solvent are in equilibrium, i.e., yi = Kx0 (Fig. 9). Similarly, at the bottom the rich gas and the rich solvent are in equilibrium, i.e., y +i = mKxN. By material balance, the minimum solvent rate can be calculated. Frequently, the pinch occurs at the bottom. 

Tie lines tie together corresponding concentrations on the operating line (bulk concentrations) and the equilibrium line (interfacial concentrations). The tie line slope is determined from the final equality for TLS in Eq. (11.31). The values of Hg and Hi are the average values determined in step 5 (1.34 ft and 0.82 ft, respectively). The value of L M/G M was determined in step 3 (3.91). Various tie lines are to be constructed over the entire concentration range covered by the operating line. A particular tie line will cross the operating fine atx and y, the values of which are used in Eq. (11.31) to calculate yBm and xBM, and in the equation for TLS. 

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