## Tray Efficiency

Definitions

Overall Column Efficiency This is the ratio of the number of theoretical stages to the number of actual stages

Since tray efficiencies vary from one section to another, it is best to apply Eq. (14-132) separately for the rectifying and stripping sections. In practice, efficiency data and prediction methods are often too crude to give a good breakdown between the efficiencies of different sections, and so Eq. (14-132) is applied over the entire column. Point Efficiency This is defined by Eq. (14-133) (Fig. 14-40«):

where y * is the composition of vapor in equilibrium with the liquid at point n. The term yn is actual vapor composition at that point. The point efficiency is the ratio of the change of composition at a point to the change that would occur on a theoretical stage. As the vapor composition at a given point cannot exceed the equilibrium composition, fractional point efficiencies are always lower than 1. If there is a composition gradient on the tray, point efficiency will vary between points on the tray.

Murphree Tray Efficiency [Ind. Eng. Chem. 17, 747 (1925)] This is the same as point efficiency, except that it applies to the entire tray instead of to a single point (Fig. 14-40b):

If both liquid and vapor are perfectly mixed, liquid and vapor compositions on the tray are uniform, and the Murphree tray efficiency will coincide with the point efficiency at any point on the tray. In practice, a concentration gradient exists in the liquid, and xn at the tray outlet is lower than x'n on the tray (see Fig. 14-40b). This frequently lowers y* relative to yn, thus enhancing tray efficiency [Eq. (14-134)] compared with point efficiency. The value of y* may even drop below yn. In this case, Emv exceeds 100 percent [Eq. (14-134)].

FIG. 14-40 Point and Murphree efficiencies. (a) Point. (b) Murphree. (From H. Z. Kister, Distillation Design, copyright © 1992 by McGraw-Hill; reprinted by permission.)

FIG. 14-40 Point and Murphree efficiencies. (a) Point. (b) Murphree. (From H. Z. Kister, Distillation Design, copyright © 1992 by McGraw-Hill; reprinted by permission.)

Overall column efficiency can be calculated from the Murphree tray efficiency by using the relationship developed by Lewis [Ind. Eng. Chem. 28, 399 (1936)].

Equation (14-135) is based on the assumption of constant molar overflow and a constant value of EMV from tray to tray. It needs to be applied separately to each section of the column (rectifying and stripping) because GM/LM, and therefore X, varies from section to section. Where molar overflow or Murphree efficiencies vary throughout a section of column, the section needs to be divided into subsections small enough to render the variations negligible.

The point and Murphree efficiency definitions above are expressed in terms of vapor concentrations. Analogous definitions can be made in terms of liquid concentrations. Further discussion is elsewhere (Lockett, Distillation Tray Fundamentals, Cambridge University, Press, Cambridge, England, 1986).

Fundamentals Figure 14-41 shows the sequence of steps for converting phase resistances to a tray efficiency. Gas and liquid film resistances are added to give the point efficiency. Had both vapor and liquid on the tray been perfectly mixed, the Murphree tray efficiency would have equaled the point efficiency. Since the phases are not perfectly mixed, a model of the vapor and liquid mixing patterns is needed for converting point efficiency to tray efficiency. Liquid mixture patterns are plug flow, backmixing, and stagnant zones, while vapor-mixing patterns are perfect mixing and plug flow.

Lewis (loc. cit.) was the first to derive quantitative relationships between the Murphree and the point efficiency. He derived three mixing cases, assuming plug flow of liquid in all. The Lewis cases give the maximum achievable tray efficiency. In practice, efficiency is lower due to liquid and vapor nonuniformities and liquid mixing.

Most tray efficiency models are based on Lewis case 1 with vapor perfectly mixed between trays. For case 1, Lewis derived the following relationship:

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