Finding the Minimum Reflux Ratio Using the Underwood Equations
Consider a distillation column with a partial reboiler and a total condenser. This column is used to separate three hypothetical components , , and with relative volatilities and (i.e., the reference component is ) to be determined by the user. The calculation assumes that the reference component is the intermediate-boiling component, , and that the lightest and heaviest components are and , respectively. The feed to the column has a thermal quality, , also determined by the user. The feed composition is 40 mole% , 30 mole% , and 30 mole% . The fractional recoveries in the distillate of components and are 98% and 95%, respectively. The fractional recovery in the bottom of component is 95%. The distillate rate, , can be computed from the equations and for , where stands for fractional recovery. One can use as a basis a feed flow rate equal to 100 kmol/hr. In such a case, the distillate rate kmol/hr. The Demonstration applies the Underwood equations  in order to determine the minimum reflux ratio, .[more]
The first and second Underwood equations are:
The relevant root (between and 1) of the first Underwood equation is shown by the blue dot in the figure. The red curve is a plot of the function
Finally, the green region indicates where the appropriate root, , of the first Underwood equation is expected.[less]
 P. C. Wankat, Separation Process Engineering, 2nd ed., Upper Saddle River, NJ: Prentice Hall, 2007.