- 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
7.3 Gilliland Correlation for Number of Stages at Finite Reflux Ratios
A general shortcut method for determining the number of stages required for a multicomponent distillation at finite reflux ratios would be extremely useful. Unfortunately, such a method has not been developed. However, Gilliland (1940) noted that he could empirically relate the number of stages N at finite reflux ratio L/D to the minimum number of stages Nmin and the minimum reflux ratio (L/D)min. Gilliland did a series of accurate stage-by-stage calculations and found that he could develop a graphical correlation of the function
with the function
Since 1940, a number of investigators have developed equations to fit Gilliland’s data so that the correlation can easily be used with calculators and computers (Coker, 2010; Davis, 2020a). The latest and possibly the best equation was developed by Davis (2020a, 2020b) using a rational function instead of a polynomial. Davis’s correlation (2020b) is
Figure 7-3 shows Gilliland’s data points and Davis’s rational function. The data points are the result of Gilliland’s stage-by-stage calculations and show the scatter inherent in this correlation.
Sometimes the values of N and Nmin, and thus Y, will be known and we will want to determine X and L/D. One advantage of Eq. (7-35) is that the inverse equation is easily determined:
Note that in Davis’s paper (2020a), the value 0.99357 is truncated to 0.99. Equations (7-35) and (7-36) are extremely sensitive to this value, and the value 0.99 does not fit the curve.
)
FIGURE 7-3. Gilliland correlation with equation (7-35) developed by Davis (2020a). Reprinted with permission from Chemical Engineering Education, 54(4), 219 (2020), copyright 2020, Chemical Engineering Education
Another advantage of Eq. (7-35) is that the function has the correct limiting behavior. As X → 0, Y → 1; and as X → 1, Y → 0 (see Problem 7.A7).
To use the graphical Gilliland correlation or Davis’s fit to the Gilliland correlation, we proceed as follows:
Calculate Nmin from the Fenske equation.
Calculate (L/D)min from the Underwood equations or analytically for a binary system.
Choose actual (L/D). This is usually done as some multiplier (1.05 to 1.5) times (L/D)min.
Calculate the abscissa X.
Determine the ordinate value Y.
Calculate the actual number of stages, N.
The Gilliland correlation should be used only for rough estimates. The calculated number of stages can be off by ±30%, although they are usually within ±7%. Because L/D is usually a multiple of (L/D)min, L/D = M(L/D)min, the abscissa can be written as
The abscissa is not very sensitive to the (L/D)min value but does depend on the multiplier M.
The optimum feed plate location can also be estimated. First, use the Fenske equation to estimate where the feed stage would be at total reflux. This can be done by determining the number of stages required to go from the feed concentrations to the distillate concentrations for the keys.
Now assume that the relative feed location is constant as we change the reflux ratio from total reflux to a finite value. Thus,
The actual feed stage can now be estimated from Eq. (7-38b). Because the estimate is based on an assumption that certainly may not be true and it does not include the effect of feed quality, it is not too accurate. Best practice is to use these estimates as first guesses of the feed location for simulations. Erbar and Maddox (1961; see King, 1980 or Coker, 2010) developed a somewhat more accurate correlation that uses more than one curve.
A rough heuristic is to estimate N = 2.5 Nmin. This estimate then requires only a calculation of Nmin and is useful for very preliminary estimates.
EXAMPLE 7-3. Gilliland correlation
Estimate the total number of equilibrium stages and the optimum feed plate location required for the distillation problem presented in Examples 7-1 and 7-2 if the actual reflux ratio is set at L/D = 2.
Define. The problem was sketched in Examples 7-1 and 7-2. F = 100, L/D = 2, and we wish to estimate N and NF.
Explore. An estimate can be obtained from the Gilliland correlation, while a more exact calculation could be done with a process simulator.
Plan. Calculate the abscissa X from Eq. (7-34b), determine the ordinate Y from Davis’s fit of the Gilliland correlation Eq. (7-35), and then find N from Eq. (7-34a). (L/D)min = 0.6663 was found in Example 7-2, and Nmin = 3.77 in Example 7-1. The feed plate location is estimated from Eqs. (7-38a) and (7-38b).
Do it.
where xD,LK and xD,HK were found in Example 7-1.
Check. A complete check requires solution with a process simulator.
Generalize. The Gilliland correlation is a rapid method for estimating the number of equilibrium stages in a distillation column. It should not be used for final designs because of its inherent inaccuracy.