Eliminating (Lmin + D) and solving gives (Dxd)C4 = 9.22 Now
D = Z(Dxd\ = 26 + 9 + 24.5 + 9.22 + 0.11 = 68.831b mole/h
Lmia 72.243 9.22
3.2.4 Extension of the minimum reflux equations
Varying relative volatilities. When relative volatility varies throughout the column, the average relative volatility is estimated by one of the criteria in Sec. 3.2.1.
Nonconstant molar overflow. When the assumption of constant molar overflow (Sec. 2.2.2) does not apply, the Underwood method still gives a good estimate of the minimum liquid flow in the zones above and below the feed, assuming that constant molar overflow applies in these regions (9). The minimum reflux flow to the top stage can then be obtained by solving combined enthalpy and mass balances written for an envelope cut by the flows above the feed and those at the top of the column. A more accurate method involves using modified molecular weights and modified mole fractions; the method is described in detail elsewhere (9).
Complex columns. A number of distinct approaches have been used for calculating minimum reflux in complex columns. Barnes et al. (41) extended the Underwood equation to multifeed columns. The minimum reflux in each section between two adjacent feeds is considered separately. The light key in the vapor leaving each section is used in Eq. (3.10), and can be calculated by subtracting the amount of light keys entering in the feed above a section from the light key in the total overhead product. A complicating factor here is the presence of distributed components (41).
Yaws et al. (42-44) proposed an alternative "factor" method for extending the Underwood equation to columns containing multiple feeds or side products. This method calculates an apparent minimum reflux for one feed as if no other feed is present and assigns a factor to convert each other feed (or side product) to an add-on minimum reflux term.
The most popular reflux-stages relationships are by Gilliland (45) and Erbar and Maddox (46). Many designers (9,11,29,47) recommend both, some (10,23,28,30,32,33,48) prefer Gilliland's, while others (13,49) prefer Erbar and Maddox's. The Erbar and Maddox correlation is considered more accurate (22,26,29,33,49), especially at low reflux ratios (49); however, the accuracy of Gilliland's equation for shortcut calculations is usually satisfactory. The single curve in Gilliland's correlation is easier to computerize.
It is important to use a consistent set of minimum reflux/minimum stages/reflux-stages correlation (27). Both the Gilliland and the Erbar and Maddox methods are consistent with the popular Fenske (Sec. 3.2.1) and Underwood (Sec. 3.2.2) methods.
Gilliland plot (45; Fig. 3.9a). This plot correlates reflux and stages by
When R - Rmia, X = 0, and Y approaches unity. When N - Nmin, Y = 0, and X approaches unity. The curve therefore stretches from the coordinates (0,1) at minimum reflux to (1,0) at total reflux.
Gilliland (45) used the Fenske method (Sec. 3.2.1) to compute minimum stages, and his own method for computing minimum reflux. However, it was shown (11,48) that the Underwood method (Sec. 3.2.2) for minimum reflux can also be used.
Numerical Gilliland equations. Since Gilliland derived his original plot (45), several authors (5,37,48,50-53) developed numerical equations to represent it. Chang (52) shows that Hengstebeck's equation (5) gives the best fit to the Gilliland plot. However, there is some scatter in the fit of data to the Gilliland plot, and the expression that best fits the plot is not necessarily the best reflux-stages correlation (48,52).
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