Relative volatility geometric average

For tiie equations listed in Table 6.27, it is assumed that the relative volatility is constant, but short cut methods are frequently used when the relative volatility varies. In this case, an average relative volatility is used. King [30] shows that the most appropriate average is the geometric average, defined by Equations 6.27.19. The equations listed in Table 6.27 are restricted to solutions fliat contain similar confounds, such as alaphatic or aromatic hydrocarbons. 

Calculate the column geometric-average relative volatility, (aj)ayg, (feed, distillate, and bottom product) of the light and heavy key components from Equation 6.27.19 (i = LK and i = HK). 

After calculating the temperature of the top and bottom products, obtain a new estimate of the colmnn relative volatility for each component. Find the relative volatihty of each conponent in the bottom and top product. Assuming that we have a total condenser, the conposition of the vapor rising above the top tray is equal to the conposition of the top product. The calculation for the dew-point tenperature will give the composition of the hquid on the top tray as well as the temperature. The temperature and hquid composition at the bottom tray is obtained from a bubble point calculation. Next, calculate the relative volatility of each conponent at the top and bottom tray. Using these values of the relative volatihty and the values for the feed, calculate the geometric average volatihty, (oCj)avg, of each component from Equation 6.26.19. This calculation is summarized in Table 6.7.2... 

Table 6.7.2 Summary of the Calculation for the Geometric-Average Relative Volatility... 

For multicomponent systems calculation with the Fenske equation is straightforward if fractional recoveries of the two keys, A and B, are specified. Equation (7-15) can now be used direcdy to find Nn. The relative volatility can be approximated by a geometric average. Once is known, the fractional recoveries of the non-keys (NK) can be found by writing Eq. (7=15) for an NK conponent, C, and either key conponent. Then solve for (FRc) or (FRc). When this is done, Eq. (7=15) becomes... 

Accurate use of the Fenske equation obviously requires an accurate value for the relative volatility. Smith (1963) covers in detail a method of calculating a by estimating tenperatures and calculating the geometric average relative volatility. Winn (1958) developed a modification of the Fenske equation that allows the relative volatility to vary. Wankat and Hubert (1979) modified both the Fenske and Winn equations for nonequilibrium stages by including a vaporization efficiency. 

The results of the Underwood equations will only be accurate if the basic assumption of constant relative volatility and CMO are valid. For small variations in a a geometric average calculated as... 

Conventional Process. In each of the distillation columns of the conventional flowsheet, geometric average relative volatilities are calculated from the reflux-drum and base temperatures. The operating pressure is fixed by specifying the reflux-dmm temperature at 320 K so that cooling water can be used in the condenser. The vapor pressures of the pure components and liquid compositions in the reflux drum and column base are known, so the relative volatilities can be calculated at both locations and averaged. 

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