## Rigorous Distillation Calculations

Joe R. Haas

! 'OP, Des Plaines, Illinois

Before the 1950s, column calculations were performed by hand. Although rigorous calculation procedures were available, they were difficult to apply for all but very small columns. Shortcut methods were therefore the primary design tool. Rigorous procedures were only used for small columns or for final design checks. Inaccuracies and uncertainties in the shortcut procedures were usually accommodated by overdesign.

The introduction of computers has entirely reversed the design procedure. Rigorous calculations, once taking several days, sometimes weeks, for even a relatively simple column, can now be performed quickly and efficiently using a computer. No longer is there a need to tolerate the inaccuracies and uncertainties inherent in the shortcut procedures. In modern distillation practice, rigorous methods are the primary design tool.

The use of computers also led to a rapid development of better rigorous procedures. The rigorous methods developed in the 1930s were replaced by more efficient methods. Further, developments took place to permit application of rigorous methods to many complex fraction-ators, some of which could not be adequately modeled by shortcut methods.

With the superior accuracy and capabilities of modern rigorous methods, a column should not be designed without them. A shortcut calculation is inferior in accuracy, and in some cases may give mis leading results. In most modern column design work, the role of shortcut calculations is restricted to eliminating the least-desirable design options, providing the designer with an initial estimate for the rigorous step and for troubleshooting the final design. The rigorous methods are used as the primary design and optimization tool.

4.1 Basic Concepts

### 4.1.1 Stage and column models

Column model. A rigorous method describes a column as a group of equations and solves these equations to calculate the operating conditions of the column. All flows are usually expressed in terms of moles/ hour. Also, when a rigorous calculation is performed, the following is usually specified:

■ Rate, composition, and condition of each feed

■ Number of stages in the column

■ The stage for each feed, product, heat exchanger, and pumparound

■ Separation specifications (see See. 3.1.1)

Column design and performance calculations present the column at steady state. What enters the column matches with what exits, for example,

2 (molar feed flow rates) = 2 (molar product flow rates)

1 (moles of any component in the feeds) = X (moles of the component in the products)

Feeds enthalpy + heat added = products enthalpy + heat removed

Some definitions

MESH equations. All of the equations used to describe the steady-state operation of a distillation column (Sec. 4.1.2). MESH stands for:

Material or flow rate balance equations, both component and total.

Equilibrium equations including the bubble-point and dew-point equations.

Summation or stoichiometric equations or composition constraints. Heat or enthalpy or energy balance equations.

Rigorous method: The mathematical method used to solve the MESH equations.

Solution: A solution is reached when all of the MESH equations are satisfied.

Simple stage: An equilibrium stage that contains no feeds, side products, or heat exchangers (top condenser/receiver, bottom reboiler, interreboiler, or intercondenser).

Simple column: A column with one feed, a top product and a bottom product, and no side products. The column can have a top condenser or receiver and bottom reboiler but no interreboiler or intercondenser.

Complex column: A column with more than one feed, or having one or more of the following features: side products; interreboilers, intercondensers, or pumparounds; and sidestrippers.

Figure 4.1a shows a complex column with two feeds and two side products. The top stage of the column is a partial condenser, with a vapor product D and a liquid product d. The reflux is the liquid, Llt and the reflux ratio is L^iD + d). The bottoms product B leaves stage .V. the reboiler. The stages are numbered from the top, with the condenser as stage 1 and the reboiler as stage N. Components are numbered from 1 to the last, C.

The simple stage model. The ideal or equilibrium stage was defined and discussed earlier (Sec. 2.1). The material and energy flows in and out of a simple stage, with no feeds or side products, is depicted in Fig. 4.16.

In this chapter, j represents the stage number and i the component number. The enthalpy terms, Hj and hj, are molar enthalpies of the vapor and liquid leaving the stage, respectively. These molar enthalpies are multiplied by the total flow rates, Vj and Lh leaving the stage to give the total energy leaving the stage in each phase.

Feed stage model. Piping arrangements encountered in most commercial columns were described by Kister (14). The feed stage model of Fig. 4.1c simulates the mixing pattern induced by these piping arrangements at the feed region. The feed stage model assumes that the feed liquid mixes with the liquid entering the feed stage while feed vapor mixes with vapor leaving the stage. If the feed is all liquid, it is added to the liquid flow entering the stage. If the feed is all vapor, it is added to the vapor flow entering the stage. A two-phase feed is distributed between the vapor leaving the feed stage and the liquid entering the feed stage. The distribution is found by an adiabatic flash of

vj.ihjt)

Lih,

Stage i -1

Stage]

Figure 4.1 Stage and column models, (a) Overall column model; (ib) simple stage model

Stage f-l

Stage f is