FIG. 13-20 Liquid-phase activity coefficients for an ethanol-n-hexane system. [Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1931; data of Sinor and Weber, J. Chem. Eng. Data, 5, 243-247(1960).]

or she utilizes the single degree of freedom represented by the repetition variable (Nr = 1.0). If the distillation column contains more than one section (such as above and below a feed stage), the number of stages in each section must be specified and as many repetition variables exist as there are sections, that is, Nr = 2.

The various restricting relationships Nc can be classified as:

1. Inherent

2. Mass-balance

3. Energy-balance

4. Phase-distribution

5. Chemical-equilibrium

The inherent restrictions are usually the result of definitions and take the form of identities. For example, the concept of the equilibrium stage involves the inherent restrictions that TV = TL and PV = PL where the superscripts V and L refer to the equilibrium exit streams.

The mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with non-reactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C - 1) component balances and one overall mass balance.

The phase-distribution restrictions reflect the requirement that fV =fiL at equilibrium where f is the fugacity. This may be expressed by Eq. (13-1). In vapor-liquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vapor-liquid-liquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the number of restricting relationships due to the distribution phenomenon will be C(Np - 1), where Np is the number of phases present.

For the analysis here, the forms in which the restricting relationships are expressed are unimportant. Only the number of such restrictions is important.

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