# Distillation Column Using the Francis Formula for Flow through Weirs

Consider a distillation column operating at atmospheric pressure with 10 stages, a partial reboiler, and a total condenser. An equimolar mixture composed of ethanol and water is to be separated by this distillation column. The feed is a saturated liquid with a flow rate equal to 10 kmol/min. Feed enters at stage 8, counting from the top. At normal operating conditions, the reflux and reboil ratios are set equal to 10 and 15, respectively. If you assume (1) a uniform tray spacing equal to 24 in and (2) an operating vapor-phase velocity equal to 80% of the value of the flooding velocity, then the column diameter can be calculated and is equal to 3.394 m. The active area is set equal to ​(i.e. a fraction of the total cross-sectional area of the column).
The momentum balance for each tray is neglected. The Francis weir formula is assumed and provides the additional equations used in the Demonstration in order to compute molar holdup of the trays. The weir height is set equal to 5 cm. In addition, condenser and reboiler volumes are taken equal to .
A step in either the reflux or reboil ratio is applied at . For every stage, plots of the composition and the temperature profiles as well as the molar holdup (all variables are versus time in minutes) for user-set values of the percent step are displayed by the Demonstration. The most drastic dynamic effects are observed in the lower part of the column, from the feed stage downwards.

### DETAILS

The Francis formula for flow through weirs is given by:
,
where is the molar holdup at stage in kmols, is the weir height in m, is the molar flow rate in kmol/min, is the gravitational acceleration, is the active area, is the weir length (calculated from the knowledge of the active area and from pure geometrical considerations), and is the liquid density at stage .
Expressions for pure component molar liquid densities and vapor and liquid enthalpies were adapted from Aspen HYSYS.
The mixture is assumed to obey modified Raoult's law, and activity coefficients are predicted using the Wilson model [1].
Reference
[1] G. M. Wilson, "Vapor-Liquid Equilibrium XI: A New Expression for the Excess Free Energy of Mixing," Journal of the American Chemical Society, 86(2), 1964 pp. 127–130.

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