- 7.0 Summary—Objectives
- 7.1 Total Reflux: Fenske Equation
- 7.2 Minimum Reflux: Underwood Equations
- 7.3 Gilliland Correlation for Number of Stages at Finite Reflux Ratios
- References
- Problems
This chapter is from the book
This chapter is from the book
This chapter is from the book
7.2 Minimum Reflux: Underwood Equations
For binary systems, the pinch point usually occurs at the feed plate. When this occurs, an analytical solution for the limiting flows can be derived (King, 1980) that is also valid for multicomponent systems as long as the pinch point occurs at the feed stage. However, multicomponent systems with nondistributing components will have separate pinch points in both the stripping and the enriching sections. If there are HNKs and/or LNKs, there will be nondistributing components unless the separation is sloppy, the NKs have volatilities that are very close to the keys, or a sandwich component is present. With nondistributing components, an analysis procedure developed by Underwood (1948) is used to find the minimum reflux ratio.
The development of the Underwood equations is quite complex and is presented in detail by Underwood (1948), Smith (1963), and King (1980). For most practicing engineers the details of the development are not as important as the use of the Underwood equations; we therefore follow the approximate derivation of Thompson (1981). Thus, we outline the important points but ignore the mathematical details of the derivation.
If there are nondistributing HNKs present, a pinch point of constant composition will occur at minimum reflux in the enriching section above where the HNKs are fractionated out. With nondistributing LNKs present, a pinch point will occur in the stripping section. For the enriching section in Figure 7-2, the mass balance for component i is
At the pinch point, where compositions are constant,
The equilibrium expression can be written in terms of K values as
Combining Eqs. (7-16) to (7-18) we obtain a simplified balance valid in the region of constant compositions.
)
FIGURE 7-2. Distillation column
Defining the relative volatility αi-ref = Ki/Kref and combining terms in Eq. (7-19),
Solving for the component vapor flow rate, Vmin yi,j+1, and rearranging
Equation (7-21) can be summed over all components to give the total vapor flow rate in the enriching section at minimum reflux:
In the stripping section, a similar analysis can be used to derive
Because the conditions in the stripping section are different than in the rectifying section, in general αi-ref ≠ 
i-ref and Kref ≠ K—ref.
Underwood (1948) describes generalized forms of Eqs. (7-22) and (7-23) that are equivalent to defining
Equations (7-22) and (7-23) then become polynomials in ϕ and 
with C roots. The equations are now
If we assume CMO and constant relative volatilities 
, Underwood showed there are common values of ϕ and 
that satisfy both equations. Equations (7-25a) and (7-25b) can now be added. Thus, at minimum reflux,
where α is now an average volatility.
Eq. (7-26) is simplified with the external column mass balance
to
ΔVfeed is the change in vapor flow rate at the feed stage. If q is known,
If the feed temperature is specified, a flash calculation on the feed can be used to determine ΔVfeed.
Equation (7-28) is known as the first Underwood equation. It can be used to calculate appropriate values of ϕ. Equation (7-25a) is known as the second Underwood equation and is used to calculate Vmin. The exact method for using the Underwood equations depends on what can be assumed. We consider three cases.
Case A. Assume that none of the NKs distribute. In this case, the amounts of NKs in the distillate are
while the amounts of the keys are
Solve Eq. (7-28) for the one value of ϕ between the relative volatilities of the two keys, αHK−ref <ϕ < αLK−ref. This value of ϕ can be substituted into Eq. (7-25a) to calculate Vmin. Then
and Lmin is found from mass balance
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 DxD,NK values are obtained from the Fenske equation as described earlier. Again solve Eq. (7-28) for the ϕ value between the relative volatilities of the two keys. This ϕ, the Fenske values of DxNK,dist, and the DxD,LK and DxD,HK values from Eqs. (7-30c) and (7-30d) are used in Eq. (7-25a) to find Vmin. Then Eqs. (7-31) and (7-32) are used to calculate D and Lmin. This procedure is illustrated in Example 7-2. Case C results are probably more accurate.
Case C. This case presents the exact solution without further assumptions. Equation (7-29) is a polynomial with C roots. Solve this equation for all values of ϕ lying between the relative volatilities of all components:
This gives C − 1 valid roots. Now write Eq. (7-25a) C − 1 times—once for each value of ϕ. We now have C − 1 equations and C − 1 unknowns (Vmin and DxD,i for all LNKs, sandwich components, and HNKs). Solve these simultaneous equations and then obtain D from Eq. (7-31) and Lmin from Eq. (7-32). Problem 7.D15 is a sandwich component problem that must use this approach.
In general, Eq. (7-28) is 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 or Goal Seek or Solver should be employed. If only one root, αLK−ref > ϕ > αHK−ref, is desired, a good first guess is to assume ϕ = (αLK−ref + αHK−ref)/2. If looking for multiple roots, a good first guess to find the φ value between two α values is the average of the two α values.
The results of the Underwood equations are accurate only if the assumptions of constant relative volatility and CMO are valid. For small variations in α, a geometric average calculated as
can be used as an approximation.
EXAMPLE 7-2. Underwood equations
For the distillation problem given in Example 7-1, find the minimum reflux ratio. Use a basis of 100.0 kmol/h of feed.
Define. The problem is sketched in Example 7-1. We now wish to find (L/D)min.
Explore. Because the relative volatilities are approximately constant, the Underwood equations can be used to estimate the minimum reflux ratio.
Plan. Because by most definitions in Problem 7.D10 benzene is distributing in Example 7-1, Case A does not apply. Either Case B or C can be used. We follow a Case B analysis and use the DxD,i values calculated in Example 7-1. We then solve Eq. (7-28) for ϕ value between the relative volatilities of the two keys 0.21 < ϕ < 1.00. Then Vmin can be found from Eq. (7-25a), D from Eq. (7-31), and Lmin from Eq. (7-32).
Do it. Because the feed is a saturated vapor, q = 0, ΔVfeed = F (1 − q) = F = 100, and Eq. (7-28) becomes
For Case B analysis, the fractional recovery of benzene is the value calculated in Example 7-1 at total reflux, DxD,ben = 100(0.4)(0.9985) = 39.94
The other values are DxD,tol = 100(0.3)(0.95) = 28.5 and DxD,cum = 100(0.3)(0.02) = 0.60.
From a mass balance, Lmin = Vmin − D = 46.00, and (L/D)min = 0.6663.
Check. The Case A calculation gives essentially the same result.
Generalize. The addition of more components does not make the calculation significantly more difficult as long as the fractional recoveries can be accurately estimated. The value of ϕ must be determined accurately because small errors can have a major effect on the results. Because this separation is easy, (L/D)min is quite small, and (L/D)min is not as dependent on the exact value of ϕ as it is when (L/D)min is large.