Multicomponent Underwood equation

The multicomponent form of the Underwood equation can be used to calculate the vapor flow at minimum reflux in each column of the sequence. The minimum vapor rate in a single column is obtained by alternate use of two equations ... 

Equation (5.8) tends to predict vapor loads slightly higher than those predicted by the full multicomponent form of the Underwood equation. The important thing, however, is not the absolute value but the relative values of the alternative sequences. Porter and Momoh have demonstrated that the rank order of total vapor load follows the rank order of total cost. 

To aid in solving the tedious Underwood equation to ultimately arrive at (L/D)niin, Frank [100] has developed Figure 8-47, which applies for liquid feed at its bubble point and whether the system is binary or multicomponent, but does require that the key components are adjacent. Otherwise, the system must be solved for two values of 0 [74]. To obtain the necessary parameters for Figure 8-47, Frank recommends using the same overhead con-... 

Colburn (1941) and Underwood (1948) have derived equations for estimating the minimum reflux ratio for multicomponent distillations. These equations are discussed in Volume 2, Chapter 11. As the Underwood equation is more widely used it is presented in this section. The equation can be stated in the form ... 

The Underwood Equations can be used to predict the minimum reflux for multicomponent distillation9. The derivation of the equations is lengthy, and the reader is... 

Use the Underwood equations to determine the minimum reflux ratio for multicomponent distillation. 

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. 

The Underwood equations can be used to calculate the minimum reflux ratio in a multicomponent system if the relative volatilities of the components are constant. There are two equations. 

The Underwood equations (Underwood, 1948) provide a shortcut method for determining the minimum reflux ratio, ilmin, in multicomponent distillation under the following assumptions constant relative volatilities and constant molal overflows in the stripping section as well as in the enriching section. The minimum reflux ratio, i min> is obtained from a solution of the following two equations for n components ... 

Underwood equation A shortcut method used to estimate the minimum reflux ratio in a multicomponent distillation process. It was proposed by A. J. V. Underwood in 1948. 

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 ... 

Ultrafiltration, 631 applications, 633 membranes, 637-639 Underwood minimum reflux binaty, 387 multicomponent, 397 Units, conversion of, 671, 672 UMQUAC equation, 475 Upflow fixed beds, 609 Uranium recovery, 515 Utilities, typical characteristics, 15... 

For multicomponent separations, it is often necessary to estimate the minimum reflux ratio of a fractionating column. A method developed for this purpose by Underwood [10] requires the solution of the equation... 

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, the Underwood method may be used for estimating mininmim reflux ratio. Its limitations should be recognized (see Section 5.3-6). The general equations are... 

Qualitative leap to the second stage (i.e., to the distillation theory of ideal multicomponent mixtures) was realized by Underwood (1945,1946a, 1946b, 1948). Underwood succeded in obtaining the analytical solution of the system of distillation equations for infinite columns at two important simplifying assumptions -at constant relative volatilities of the components (i.e., which depend neither on the temperature nor on mixture composition at distillation column plates) and at constant internal molar flow rates (i.e., at constant vapor and liquid flow rates at all plates of a column section). The solution of Underwood is remarkable due to the fact that it is absolutely rigorous and does not require any plate calculations within the limits of accepted assumptions. 

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