This compares well with the value of 1.04 obtained from the Hengstebeck diagram (Sec. 2.3.5), and with a value of 1.07 obtained either from the Underwood method, corrected for nonconstant molar overflow (Sec. 3.2.4), or from extrapolation of computer simulation results (Fig. 3,8).
Graphical Underwood method. To eliminate the trial and error involved, Van Winkle and Todd (37) developed a graphical solution technique for obtaining 6. This technique is only applicable to bubble-point liquid feeds.
Other shortcut methods. Other methods are also available for minimum reflux calculations. Van Winkle (28) surveyed them and recommended Underwood's and two others:
1. The Brown-Martin method (38): This method is based on the observation that at the feed pinch point, the ratio of key components is essentially equal to their ratio in the liquid feed. This method is the least complicated and gives conservative estimates. Suitable mainly for hydrocarbons in situations where great accuracy is not required (28).
2. The Colburn method (39): This method calculates the minimum reflux ratio of the key components as if they formed a binary system, then corrects this value for light and heavy nonkeys. The Colburn method assumes constant molar overflow and constant relative volatility in each zone of constant composition in the column. This method is more elaborate, but has been recommended (28) as probably the most accurate shortcut method for minimum reflux.
3.2.3 Minimum reflux for systems containing distributed nonkeys
A component is said to be distributed (or distributing) at minimum reflux if it appears both in the distillate and the bottoms at minimum reflux. Usually, nonkeys are nondistributed (or nondistributing), that is, at minimum reflux the heavy nonkeys are totally contained in the bottoms and the light nonkeys in the distillate. A nonkey component may be distributed if
■ It has a volatility very close to that of one of the keys, or
■ If the specified separation of the keys is sloppy (i.e., not sharp), or
■ If the nonkey has a volatility intermediate between the keys.
Shiras et al. (40) developed an equation to determine whether or not a component is distributed at minimum reflux
The relative volatilities are based on a reference value of 1.0 for the heavy key component. The Shiras et al. criterion applies at minimum reflux as follows:
Dr > 1 Component is nondistributed; contained entirely in distillate.
0 < Dr < 1 Component distributed, DR is the recovery of the component in the distillate.
Dr < 0 Component is nondistributed; contained entirely in bottoms.
Hines and Maddox (26) stated that they solved literally hundreds of minimum reflux cases and are yet to find a case in which predictions from Eq. (3.12) are not correct.
Application of Underwood's equation to systems containing distributed nonkey components is as follows:
1. Find which components are distributed using Eq. (3.12).
2. For n distributed components (including the keys) solve Eq. (3.10) for n - 1 values of 0, so that each value of 0 is between the relative volatilities of adjacent components. For instance, if a system contains four distributed nonkey components, plus two key components, Eq. (3.10) needs to be solved for five values of 8. If
«dk1 > «lk > «dk2 > «dk3 > «hk > «dk4. then alk < flj < adk1; adk2 < < alk> adk3 < < adk2i ahk < < adk3! adk4 < h < «hk-
3. Treat the mole fraction of each distributed nonkey component in the distillate as an unknown. Write Eq. (3.11) for each value of 6 calculated above. (L/D)min is also unknown. Solve the equations simultaneously to get the mole fraction of each distributed component in the distillate and (L/D)min. In the above example, there are five values of 0 and therefore five equations. There are also five unknowns—the mole fractions of DK1, DK2, DK3, and DK4 in the distillate, and (L/D)min.
Example 3.3 Calculate the minimum reflux for a depropanizer similar to that of Example 2.4 using Underwood's method. In this case, butane is acceptable both in the top and bottom product, but it is required that 98 percent of the propane in the feed is recovered in the top product, and 99 percent of the pentane is to be recovered in the bottom product.
solution In this case, propane and n-pentane are the light key and heavy key, respectively. It is apparent that «.-butane is a distributed key, while other components (methane, ethane, and n-hexane) are nondistributed nonkeys. However, for the sake of the exercise, we will check this using Eq. (3.12). Relative volatilities based on a reference value of 1.0 for the heavy key are shown in the table below. From the statement of the problem,
This confirms the apparent observation that n-butane is the only distributed component. Equation (3.10) is now solved for two values of 9 such that 1.0 < eL < 2.11 and 2.11 < e2 < 4.08. Since the feed is 66 percent vapor, 1 - q = 0.66. The calculations are in Table 3.3, and give 0! = 1.264, 62 = 2.847. Equation (3.11) can be written as imin + D = (3.13)
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