7.1.1 Definitions
Overall column efficiency is defined by
Eoc = NJNa
.V, is calculated by any of the methods in Chaps. 2 to 5. Once the tray efficiency is known, the number of actual trays can be obtained from Eq. (7.1).
Since efficiencies vary from one section to another, it is best (12) to apply Eq. (7.1) separately for each section (e.g., rectifying and stripping). In practice, efficiency data and prediction methods are often too crude to give a good breakdown between the efficiencies of different sections, and Eq. (7.1) is applied over the entire column.
Alternative definitions of tray efficiency are sometimes used. Lockett (12) reviewed the pros and cons of several efficiency definitions. He cited the industry's experience that the more rigorous and theoretically correct a definition is, the more difficult it is to use. For instance, the Standart efficiencies are often considered the soundest fundamentally, but apparently have never been used for a design. For the design and operation engineer, the overall column (or section) efficiency is by far the most important.
Point efficiency is defined by (Fig. 7.1a).
v* is the composition of vapor in equilibrium with the liquid at point n. y„ is the 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
Vn-1
Vn-1
Figure 7.1 Point and Murphree efficiencies, (a) Point; Murphree, given point cannot exceed the equilibrium composition, point efficiencies are always lower than unity. If there is a concentration gradient on the tray, point efficiency will vary from point to point on the tray.
Murphree tray efficiency (120) is the same as point efficiency, except that it applies for the entire tray instead of to a single point (Fig. 7.1), i.e., y* is the composition of vapor in equilibrium with the liquid leaving the tray. yn is the actual composition of vapor leaving the tray. The Murphree tray efficiency is the ratio of the change of composition on the tray to the change that would occur on a theoretical stage.
If both liquid and vapor are perfectly mixed, liquid composition on the tray is uniform and so is vapor composition. The Murphree tray efficiency will then 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'„ on the tray (Fig. 7.16). This frequently lowers y* relative to y„, thus enhancing tray efficiency [Eq. (7.3)] compared to point efficiency, y* may even drop below _v„; in this case, exceeds 100 percent [Eq. (7.3)].
Overall column efficiency can be calculated from the Murphree tray efficiency using the relationship developed by Lewis (121).
where
Equation (7.4) is based on the assumption of constant molar overflow • Sec. 2.2.2) and a constant value of ¿?mv from tray to tray. It needs to be applied separately to each section of column (e.g., rectifying and stripping) because GM/LM, and therefore X, varies from section to section. Where molar overflows or Murphree efficiencies vary throughout a section of column, the section needs to be divided to subsections small enough to render the variations negligible. Equation (7.4) can then be applied to each subsection.
The point and Murphree efficiency definitions above are expressed in terms of vapor concentrations. Analogous efficiency definitions can be expressed in terms of liquid concentrations. Further discussion is in Refs. 12 and 122.
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