# Equations Required To Describe Conventional Distillation Columns

A conventional distillation column is defined as one that has one feed and two product streams, the distillate D and the bottoms B. Such a column has the same configuration as the one shown in Fig. 1-5. First consider the case where the following specifications are made for a column at steady-state operation: (1) the number of plates in each section of the column, (2) the quantity, composition, and thermal condition of the feed at the column pressure, (3) the type of overhead condenser (total or partial), (4) the column pressure (or the pressure at a given point in the column where the variation of the pressure throughout the column is not negligible), (5), the reflux of ratio, Li/D, or V2 or Lx, and (6) the temperature of the distillate or the total distillate rate. (The first three of these are specifications of the geometry of the column and the feed, and the second three are the specification of operating variables.) Steady-state operation means that no process variable changes with time. For this set of operating conditions, the problem is to find the compositions of the top and bottom products. The set of equations required to represent such a system fbr all components (i = 1, 2,..., c) are as follows

Equilibrium relationships y a â€” Kjixji !>'), = i

Material balances

Vj+lyj+ui = LjxJt + DXm (j = 1, 2,...,/- 2) l /V'/i + vf)'H = I/- ixf-1, ,â€¢ + DXm

FX, = DXm + Bxm Vj+,Hj+, = Ljhj + DHD+QC (j= 1, 2, ...,/- 2)

VJ+1 H]+1 = Ljhj â€” BhB + Qr (j = f,f + l,...,N-l) FH = Bhâ€ž + DHd + Qc~Qr

balances

Inspection of this set of equations shows that they are a logical extension of those stated in Chap. 1, Eq. (1-43), for the binary system. A schematic representation of the component-material balances is shown in Fig. 2-1. The behavior assumed on the feed plate is demonstrated by model 2, which is shown in Fig. 2-2.

The above enthalpy balances may be represented by the same enclosures

Figure 2-1 Representation of the component-material balances given by Eq. (2-1). (Taken from Holland: Introduction to the Fundamentals of Distillation, Proc. 4th Ann. Educ. Symp. Instrument Soc. Am., Apr. 5-7, 1972, Wilmington, Delaware.)
Figure 2-2 Model 2 for the behavior of the feed plate. (Taken from Holland: Introduction to the Fundamentals of Distillation, Proceedings of the Fourth Annual Education Symposium of the Instrument Society of America, April 5-7, 1972, Wilmington, Delaware.)

shown in Fig. 2-1. As in the case of the material balances for any one component, the number of independent energy balances is equal to the number of stages (j = 1, 2, ..., N - 1, N). In this case the total number of independent equations is equal to N(2c + 3), as might be expected from the fact that an adiabatic flash is represented by 2c + 3 equations.

For a column whose geometry [the total number of stages, the feed plate locations, and the type of condenser (partial or total)] and feed have been specified, the remaining variables to be specified are as follows:

 Variable Number Vapor and liquid mole fractions 2 cN Total-flow rates

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