# Approximate Multicomponent Methods

Many years before the availability of computers for rigorous analysis, several simple approximate methods were developed for analyzing multicomponent systems. These methods are still quite useful for getting quick estimates of the size of a column Figure 2.13 Summary of calculated results.

(number of trays) and the energy consumption (reflux ratios and the corresponding vapor boilup and reboiler heat input).

### 2.3.1 Fenske Equation for Minimum Number of Trays

The minimum number of trays corresponds to total reflux operation (an infinite reflux ratio). The Fenske equation relates the compositions at the two ends of a column to the number of stages in the column under this limiting condition:

where Nmin is the minimum number of stages, xD LK is the mole fraction of the light-key

McCabe-Thiele 1

McCabe-Thiele 1 Liquid Composition

Figure 2.14 xy diagram.

Liquid Composition

### Figure 2.14 xy diagram.

component at the top of the column, xD HK is the mole fraction of the heavy-key component at the top of the column, xB HK is the mole fraction of the heavy-key component at the bottom of the column, xBLK is the mole fraction of the light-key component at the bottom of the column, and aLKHK is the relative volatility between the LK and HK components. Vapor Mole Fraction Benzene

Figure 2.15 Temperature and composition profiles.

Vapor Mole Fraction Benzene

Figure 2.15 Temperature and composition profiles.

This equation is applicable to multicomponent systems, but it assumes a constant relative volatility between the two components considered.

An example of the use of the Fenske equation is given in Chapter 4. Results of this approximate method will be compared with the results found from rigorous simulation.

### 2.3.2 Underwood Equations for Minimum Reflux Ratio

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

The feed composition Zj (mole fractions j = 1, NC), the desired distillate composition xDj (j = 1, NC), and the feed thermal condition q are specified. The relative volatilities aj (j = 1, NC) of the multicomponent mixture are known.

The first equation contains one unknown parameter 0. However, expanding the summation of NC terms and multiplying through by all the denominator terms (a,- — 0) give a polynomial in 0 whose order is NC; therefore there are NC roots of this polynomial. One of these roots lies between the two relative volatility values aLK and aHK. This is found using some iterative solution method. It is substituted into the second equation, which can then be solved explicitly for the minimum reflux ratio.

An example of the use of the Underwood equations is given in Chapter 4. The results of this approximate method will be compared with the results found from rigorous simulation.