## Theory of Distillation

introduce the separation factor:

Fundamentals

where x denotes mole fraction of a component, subscript L denotes light component, H heavy component, T denotes the top of the section, and B the bottom.

It is relatively straightforward to derive models of distillation columns based on almost any degree of detail, and also to use such models to simulate the behaviour on a computer. However, such simulations may be time-consuming and often provide limited insight. The objective of this article is to provide analytical expressions that are useful for understanding the fundamentals of distillation and which may be used to guide and check more detailed simulations:

1. minimum energy requirement and corresponding internal flow requirements

2. minimum number of stages

3. simple expressions for the separation factor

The derivation of analytical expressions requires the assumptions of:

1. equilibrium stages

2. constant relative volatility

### 3. constant molar flows

These assumptions may seem restrictive, but they are actually satisfied for many real systems, and in any case the resulting expressions yield invaluable insights, also for systems where the approximations do not hold.

### The Equilibrium-Stage Concept

The equilibrium (theoretical)-stage concept (Figure 1) is central in distillation. Here we assume vapour-liquid equilibrium (VLE) on each stage and that the liquid is sent to the stage below and the vapour to the stage above. For some trayed columns this may be a reasonable description of the actual physics, but it is certainly not for a packed column. Nevertheless, it is established that calculations based on the equilibrium-stage concept (with the number of stages adjusted appropriately) fits data from most real columns very well, even packed columns.

One many refine the equilibrium stage concept, e.g. by introducing back-mixing or a Murphee efficiency factor for the equilibrium, but these 'fixes' often have relatively little theoretical justification, and are not used in this article.

For practical calculations, the critical step is usually not the modelling of the stages, but to obtain a good description of the VLE. In this area there has been significant advances in the last 25 years, especially after the introduction of equations of state for VLE prediction. However, here we will use simpler VLE models (constant relative volatility) which apply to relatively ideal mixtures.

### Vapour-Liquid Equilibrium

In a two-phase system (PH = 2) with Nc nonreacting components, the state is completely determined by Nc degrees of freedom ( f), according to Gibb's phase rule:

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