Fenske equations

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 ... 

Which components are the key components will normally be clear, but sometimes, particularly if close boiling isomers are present, judgement must be used in their selection. If any uncertainty exists, trial calculations should be made using different components as the keys to determine the pair that requires the largest number of stages for separation (the worst case). The Fenske equation can be used for these calculations see Section 11.7.3. 

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 ... 

Winn (1958) has derived an equation for estimating the number of stages at total reflux, which is similar to the Fenske equation, but which can be used when the relative volatility cannot be taken as constant. 

A limitation of the Erbar-Maddox, and similar empirical methods, is that they do not give the feed-point location. An estimate can be made by using the Fenske equation to calculate the number of stages in the rectifying and stripping sections separately, but this requires an estimate of the feed-point temperature. An alternative approach is to use the empirical equation given by Kirkbride (1944) ... 

The graphical procedure proposed by Hengstebeck (1946), which is based on the Fenske equation, is a convenient method for estimating the distribution of components between the top and bottom products. 

Hengstebeck and Geddes (1958) have shown that the Fenske equation can be written in the form ... 

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... 

The Fenske Equation can be used to estimate the composition of the products. Equation 9.35 can be written in a form ... 

Either Equation 9.47 or 9.49 can be used to obtain a more realistic relative volatility for the Fenske Equation. 

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 second column in the distillation train of an aromatics plant is required to split toluene and ethylbenzene. The recovery of toluene in the overheads must be 95%, and 90% of the ethylbenzene must be recovered in the bottoms. In addition to toluene and ethylbenzene, the feed also contains benzene and xylene. The feed enters the column under saturated conditions at a temperature of 170°C, with component flowrates given in Table 9.10. Estimate the mass balance around the column using the Fenske Equation. Assume that the K-values can be correlated by Equation 9.68 with constants A , 5 and C, given in Table 9.10. 

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 ... 

This is found from the relative volatility and the distribution of the keys between the overhead and bottoms by the Underwood-Fenske equation... 

Particularly when the number of trays is small, the location of the feed tray has a marked effect on the separation in the column. An estimate of the optimum location can be made with the Under-wood-Fenske equation (13.116), by applying it twice, between the overhead and the feed and between the feed and the bottoms. The ratio of the numbers of rectifying Nr and stripping Ns trays is... 

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 ... 

The Fenske equation applies not only to the light key and heavy key components. It can also be applied to any pair of components. 

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