Page 283 - Separation process engineering

α HK−ref and α LK−ref . Thus use Case C (discussed later) for sandwich components. The method of Shiras et
al. (1950) can be used to check for distribution of NKs.

Case B. Assume that the distributions of NKs determined from the Fenske equation at total reflux are also
valid at minimum reflux. In this case the Dx NK,dist values are obtained from the Fenske equation as

described earlier. Again solve Eq. (7-33) for the φ value between the relative volatilities of the two keys.
This φ, the Fenske values of Dx NK,dist, and the Dx LK,dist and Dx HK,dist values obtained from Eqs. (7-36) and
(7-37) are used in Eq. (7-29) to find V . Then Eqs. (7-38) and (7-35) are used to calculate D and L .
min
min
This procedure is illustrated in Example 7-2.
Case C. Exact solution without further assumptions. Equation (7-33) is a polynomial with C roots. Solve
this equation for all values of φ lying between the relative volatilities of all components,

α LNK,1−ref < φ < α LNK,2−ref < φ < α LK−ref < φ < α HK−ref < φ < α HNK,1−ref
2
1
4
3
This gives C-1 valid roots. Now write Eq. (7-29) C-1 times; once for each value of φ. We now have C-1
equations and C-1 unknowns (V and Dx i,dist for all LNK and HNK). Solve these simultaneous equations
min
and then obtain D from Eq. (7-38) and L from Eq. (7-35). A sandwich component problem that must
min
use this approach is given in Problem 7.D15.
In general, Eq. (7-33) will be of order C in φ where C is the number of components. Saturated liquid and
saturated vapor feeds are special cases and, after simplification, are of order C-1. If the resulting
equation is quadratic, the quadratic formula can be used to find the roots. Otherwise, a root-finding
method should be employed. If only one root, α LK−ref > φ > α HK−ref , is desired, a good first guess is to

assume φ = (α LK−ref + α HK−ref )/2.

The results of the Underwood equations will only be accurate if the basic assumption of constant relative
volatility and CMO are valid. For small variations in α a geometric average calculated as

(7-39)

can be used as an approximation. Application of the Underwood equations to systems with multiple feeds
was studied by Barnes et al. (1972).

Example 7-2. Underwood equations

For the distillation problem given in Example 7-1 find the minimum reflux ratio. Use a basis of 100
kmol/h of feed.

Solution

A. Define. The problem was sketched in Example 7-1. We now wish to find (L/D) min .

B. Explore. Since the relative volatilities are approximately constant, the Underwood equations can
easily be used to estimate the minimum reflux ratio.

C. Plan. This problem fits into Case A or Case B. We can calculate Dx i,dist values as described in
Cases A or B, Eqs. (7-36) and (7-37), and solve Eq. (7-33) for φ where φ lies between the
relative volatilities of the two keys 0.21 < φ < 1.00. Then V can be found from Eq. (7-29), D
min
from Eq. (7-38) and L from Eq. (7-35).
min
D. Do It. Follow Case B analysis. Since the feed is a saturated vapor, q = 0 and ΔV feed = F (1 − q) =