Line, equilibrium

Equation 28 and its liquid-phase equivalent are very general and valid in all situations. Similarly, the overall mass transfer coefficients may be made independent of the effect of bulk fiux through the films and thus nearly concentration independent for straight equilibrium lines ... 

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 recommended design procedure uses the values of (E /and m from Figures 7 and 8 in equation 56 and yields a very good estimation of Alp despite the curvature of the operating and the equilibrium lines. This value differs from A/q obtained by equation 49 because of the /(I — y) term in the latter equation. A convenient approach for purposes of approximate design is to define a correction term AA/q which can be added to equation 55 ... 

Fig. 12. Correlatioa of AT. The three lines represeat the best fit of a mathematical expressioa obtaiaed by multidimensional nonlinear regressioa techniques for 99, 95, and 90% recovery the poiats are for 99% recovery. = mean molar heat capacity of Hquid mixture, average over tower AY = VA2 slope of equiHbrium line for solute, to be taken at Hquid feed temperature mg = slope of equilibrium line for solvent.

Fig. 13. Definition of effective average slopes of equilibrium line (45).

Fig. 14. Coiielation of effective average slope m of equilibrium line (dilute part of absorber) equilibrium line concave upward. gas mole fraction at...

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local hquid composition Xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. 

In many practical situations involving nearly complete cleanup of the gas, an approximate result can be obtained from the equations just presented even when solutions are concentrated or when absorption heat effects are present. In such cases the driving forces in the upper part of the tower are very much smaller than those at the bottom, and the value of mGM/LM used in the eqiiations should be the ratio of the slopes of the equilibrium line m and the operating line Lm/Gm iu the low-concentration range near the top of the tower. 

For dilute solutions in which both the operating and the equilibrium lines are straight and in which heat effects can be neglected, the integral term in Eq. (14-27) is... 

Evidently a temperature rise of 7 C would not be a safe design because the equilibrium line nearly touches the operating line near the bottom of the tower, creating a pinch. A temperature rise of 6 C appears to give an operable design, and for this case Lki = 349 kmol per 100 kmol of feed gas. 

Thus, the equilibrium line for each component passes through the origin with slope K, where... 

Equations (14-168) and (14-170) have been developed for binaiw mixture separations and hold for cases where the operating hne and equilibrium line are straight. Thus, when there is curvature, the equations should be used for sections of the column where hnearity can be assumed. When the eqiiihbriiim line and operating line have the same slope, HETP = Hog and Nog = (theoretical stages). 

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

For low concentrations in which the equilibrium line is linear the value of K is equal to m. 

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)]. 

For a symmetrical separation of component h from c, Brian Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, N.J., 1972) reported that the ratio of wash solvent to extraction solvent W /S should be set equal to the geometric mean of the two slopes of the equilibrium lines [Eq. (15-35)]. 

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. 

On a y-x (McCabe-Thiele) diagram, this equation represents the operating line which extends between the points (y , x" ) and (yf , x") and has a slope of Lj/Gi, as shown in Fig. 2.5. Furthermore, each theoretical stage can be represented by a step between the operating line and the equilibrium line. Hence, NTP can be determined by stepping off stages between the two ends of the exchanger, as illustrated by Fig. 2.5. 

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