Multicomponent 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 ... 

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 ... 

An alternative form of the Fenske equation that is very convenient for multicomponent calculations is easily derived. Equation (6-62) can also be written as... 

For multicomponent systems, calculations with the Fenske equation are straightforward if fractional recoveries of the two keys, A and B, are specified. If the relative volatility is not constant, the average defined in equation (6-60) can be approximated by... 

For multicomponent mixtures, all components distribute to some extent between distillate and bottoms at total reflux conditions. However, at minimum reflux conditions none or only a few of the nonkey components distribute. Distribution ratios for these two limiting conditions are shown in Fig. 12.14 for the debutanizer example. For total reflux conditions, results from the Fenske equation in Example 12.3 plot as a straight line for the log-log coordinates. For minimum reflux, results from the Underwood equation in Example 12.5 are shown as a dashed line. 

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... 

Minimum stages for total reflux. Just as in binary distillation, the minimum number of theoretical stages or steps, can be determined for multicomponent distillation for total reflux. The Fenske equation (11.4-23) also applies to any two components in a multicomponent system. When applied to the heavy key H and the light key L, it... 

Fenske equation for the number of total plates at total reflux In Section 8.1.3.2, the number of ideal equilibrium stages needed to separate a binary mixture under total reflux conditions was obtained via the Fenske equation (8.1.169). A similar procedure will be followed here, except we focus on two different species i, k among ra species present in the multicomponent system. We start with the bottonunost plate and the reboiler. Under total reflux condition with total reboil, we know that, for stage N at the column bottom,... 

The Fenske equation in the form of (8.1.226) may be used to estimate how other components in the multicomponent system are present at the top and bottom of the column, provided N rdi, has been determined for two key components, i and k (light key and heavy key). Convert equation (8.1.226) to the following form ... 

Truly multicomponent solutions based on continuous distillation shortcut methods have been proposed for batch distillation. The Fenske, Underwood, and Gilliland equations or correlations are commonly used in conjunction with each other to solve continuous distillation problems as described in Section 12.3. Diwekar and Madhavan (1991) describe how these techniques may be modified for the design of batch distillation columns for variable and constant reflux cases. 

Owing to the availability of high-speed computers, short cut methods for designing distillation processes (e.g. McCabe-Thiele and Ponchon-Savarit for binary systems or the equations of Fenske, Underwood and Gilliland for multicomponent mixtures, see Gmehling and Brehm, 1996 and Satder, 2001 for details) are no longer required. 

For multicomponent systems, an approximate value of the minimum number of stages (at total reflux) may be obtained from the Fenske relationship [Eq. (5.3-28)]. In the use of this relationship for multicomponent mixtures, the mole fractions and the relative volatility refer to the light and heavy keys only. However, values for the nonkey components may be inserted in the equation to determine thtir distribution t er the number of minimum stages has been determined through the use of the key conmonents. For a more rigorous approach to the determination of minimum stages, see the paper Iqr Chien. ... 

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