Fenske equation minimum

Equation-Based Design Methods Exact design equations have been developed for mixtures with constant relative volatility. Minimum stages can be computed with the Fenske equation, minimum reflux from the Underwood equation, and the total number of stages in each section of the column from either the Smoker equation (Trans. Am. Inst. Chem. Eng., 34, 165 (1938) the derivation of the equation is shown, and its use is illustrated by Smith, op. cit.), or Underwoods method. A detailed treatment of these approaches is given in Doherty and Malone (op. cit., chap. 3). Equation-based methods have also been developed for nonconstant relative volatility mixtures (including nonideal and azeotropic mixtures) by Julka and Doherty [Chem. Eng. Set., 45,1801 (1990) Chem. Eng. Sci., 48,1367 (1993)], and Fidkowski et al. [AIChE /., 37, 1761 (1991)]. Also see Doherty and Malone (op. cit., chap. 4). 

However, the total number of equilibrium stages N, N/N,n, or the external-reflux ratio can be substituted for one of these three specifications. It should be noted that the feed location is automatically specified as the optimum one this is assumed in the Underwood equations. The assumption of saturated reflux is also inherent in the Fenske and Underwood equations. An important limitation on the Underwood equations is the assumption of constant molar overflow. As discussed by Henley and Seader (op. cit.), this assumption can lead to a prediction of the minimum reflux that is considerably lower than the actual value. No such assumption is inherent in the Fenske equation. An exact calculational technique for minimum reflux is given by Tavana and Hansen [Jnd. E/ig. Chem. Process Des. Dev., 18, 154 (1979)]. A computer program for the FUG method is given by Chang [Hydrocarbon Process., 60(8), 79 (1980)]. The method is best applied to mixtures that form ideal or nearly ideal solutions. 

Fenske Equation Overall Minimum Total Trays with Total Condenser... 

Because the feed tray is essentially non-effective it is suggested that an additional theoretical tray be added to allow for this. This can be conveniently solved by the nomographs [21] of Figures 8-16 and 17. If the minimum number of trays in the rectifying section are needed, the)t can be calculated by the Fenske equation substituting the limits of xpi for x jj and x i, and the stripping section can be calculated by difference. 

Assume xi values of bottoms compositions of light key for approximate equal increments from final bottoms to initial feed charge. Calculate L/V values corresponding to the assmned xi values by inserting the various xi values in the Fenske equation for minimum reflux ratio of l-(d). The xi values replace the x b of this relation as the various assumptions are calculated. The actual (L/D) are calculated as in l-(d) keeping the minimmn number of trays constant. Complete the table values. 

Minimum Trays at Total Reflux Fenske Equation ... 

The Fenske equation (Fenske, 1932) can be used to estimate the minimum stages required at total reflux. The derivation of this equation for a binary system is given in Volume 2, Chapter 11. The equation applies equally to multicomponent systems and can be written as ... 

Minimum number of stages Fenske equation, equation 11.58 ... 

For binary systems or systems that approach binary, the Fenske-Underwood-Erbar/Maddox Method is recommended. For minimum stages, use the Fenske equation.12... 

To solve Equation 9.51, it is necessary to know the values of not only a ,-j and 9 but also x, d. The values of xitD for each component in the distillate in Equation 9.51 are the values at the minimum reflux and are unknown. Rigorous solution of the Underwood Equations, without assumptions of component distribution, thus requires Equation 9.50 to be solved for (NC — 1) values of 9 lying between the values of atj of the different components. Equation 9.51 is then written (NC -1) times to give a set of equations in which the unknowns are Rmin and (NC -2) values of xi D for the nonkey components. These equations can then be solved simultaneously. In this way, in addition to the calculation of Rmi , the Underwood Equations can also be used to estimate the distribution of nonkey components at minimum reflux conditions from a specification of the key component separation. This is analogous to the use of the Fenske Equation to determine the distribution at total reflux. Although there is often not too much difference between the estimates at total and minimum reflux, the true distribution is more likely to be between the two estimates. 

Another approximation that can be made to simplify the solution of the Underwood Equations is to use the Fenske Equation to approximate xitD. These values of XitD will thus correspond with total reflux rather than minimum reflux. 

Having obtained the minimum number of stages from the Fenske Equation and minimum reflux ratio from the Underwood Equations, the empirical relationship of Gilliland10 can be used to determine the number of stages. The original correlation was presented in graphical form10. Two parameters (X and Y) were used to correlate the data ... 

The Underwood and Fenske equations may be used to find the minimum number of plates and the minimum reflux ratio for a binary system. For a multicomponent system nm may be found by using the two key components in place of the binary system and the relative volatility between those components in equation 11.56 enables the minimum reflux ratio Rm to be found. Using the feed and top compositions of component A ... 

Calculate the minimum number of trays using the Fenske Equation. 

This measure was based upon the ratio of the minimum necessary number of plates, A min (averaged over the reboiler composition) in a column to the actual number of plates in the given column, Nj. Christensen and Jorgensen assumed that the mixture has a constant relative volatility a and the column operates at total reflux using constant distillate composition (x o) strategy (section 3.3.2) and evaluated Nmin using the Fenske equation ... 

Table 13-6 shows subsequent calculations using the Underwood minimum reflux equations. The a and Xo values in Table 13-6 are those from the Fenske total reflux calculation. As noted earlier, the % values should be those at minimum reflux. This inconsistency may reduce the accuracy of the Underwood method but to be useful, a shortcut method must be fast, and it has not been shown that a more rigorous estimation of x values results in an overall improvement in accuracy. The calculated firnin is 0.9426. The actual reflux assumed is obtained from the specified maximum top vapor rate of 0.022 kg- mol/s [ 175 lb-(mol/h)] and the calculated D of 49.2 (from the Fenske equation). 

Calculate the minimum number of equilibrium stages, Nm, from the Fenske equation. Equation 6.27.2. 

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