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332 Chapter 6

of the fractionator. To determine the height of a fractionator, the first step is to
identify and specify the recoveries of the heavy and light key components. The
light key component will be recovered to a significant extent in the top product,
whereas the heavy key component will be recovered to a significant extent in the
bottom product. After specifying recoveries of the key components, the next step
is to calculate the recoveries of all other components. The component recoveries
are estimated using the Geddes equation [34], Equation 6.27.1 in Table 6.27.
Yaws et al. [35] compared the percent recovery of each component, calculated
from Equation 6.27.1, with the percent recovery calculated using an exact method
and found that the maximum percent deviation was only 0.23% for any one com-
ponent.
For the equations listed in Table 6.27, it is assumed that the relative volatil-
ity is constant, but short cut methods are frequently used when the relative volatil-
ity varies. In this case, an average relative volatility is used. King [30] shows that
the most appropriate average is the geometric average, defined by Equations
6.27.19. The equations listed in Table 6.27 are restricted to solutions that contain
similar compounds, such as alaphatic or aromatic hydrocarbons.
In the short cut method, the number of equilibrium stages needed for a given
separation are correlated in terms of the minimum number of stages and the mini-
mum reflux ratio. Fenske [38] derived an expression for the minimum number of
actual stages, Equation 6.27.2, by a stage to stage analysis, assuming that the rela-
tive volatility is constant. This equation is applicable to multicomponent as well
as binary solutions, and is derived in a number of texts, such as by King [30].
Underwood [39] derived Equations 6.27.3 and 6.27.4 for estimating the
minimum reflux ratio for a specified separation of two key components. These
equations assume constant molar overflow and relative volatility. Underwood
showed that at minimum reflux the value of 0 in Equations 6.27.3 and 6.27.4 must
lie between the relative volatility of the heavy and light key components. If the
key components are not adjacent, there will be more than one value of 0. This
case is illustrated in an example by Walas [6]. Here, we will assume that the key
components are adjacent. As has been pointed out by Walas [6], the minimum
reflux ratio calculated by the Underwood equations could turn out to be negative,
which means that the equations do not apply for the given separation.

Table 6.27 Summary of Equations for Sizing Fractionators
Subscripts: i = the i component
F = feed — D = distillate — B = bottom product
LK = light key component — HK = heavy key component

Component Distribution
log (n;D /n ;B) = Ac + B c log (a i) avg (6.27.1 A)
n iF = n iD + n iB (6.27. IB)

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