dy _ Kpoafa pMiG

(7.106j

m j* is the vapor composition in equilibrium with the bulk liquid composition x, and therefore y* = mx + b Equilibrium is assumed at the interface, i.e., yi = mxt + b Combining Eqs. (7.11) with Eq. (7,6) gives

^OG kL pMx

Combining Eq. (7.12) with Eq. (7.10) gives where X is defined by Eq. (7.5). Equation (7.13) is the famous relationship used for adding vapor and liquid mass transfer resistances.

hfpM Gl{_\2GM) and hjpML/(12LM) are the residence times, tG and tL, respectively, of the vapor and liquid in the froth. Equations (7.10a and b) can be rewritten

The above derivation assumes that vapor flows upward in plug flow and that there is no horizontal vapor mixing, while liquid flows horizontally in plug flow and there is no vertical mixing. Lockett and Uddin (12,122) and Standart (123,124) showed that these liquid flow assumptions are poor, unnecessary, and lead to incorrect implications regarding tray efficiency. By modifying the definition of NL, Lockett derived a fundamentally superior equation analogous to Eq. (7.13). Most theoretical models, however, use Eq. (7.13). Equation (7.13) is also the equation used for packed columns, but for packed columns, it is based on sounder assumptions (12).

For most distillation systems, the resistance to mass transfer is concentrated in the vapor phase [i.e., the 1 fNG term in Eq. (7.13) dominates], Lockett (12) showed that the liquid phase mass transfer resistance increases with liquid rates. At high liquid rates, the liquid phase resistance typically constitutes 25 to 50 percent or more of the total mass transfer resistance.

In order to express the point efficiency in terms of transfer units, Eq. (7.10c) is integrated from point n - 1 to point n (Fig. 7.1a). The integration assumes that in the vertical direction liquid is perfectly mixed and vapor is in plug flow, and gives

Combining with Eq. (7.2) gives

Standart et al. (123,124) questioned the validity of these assumptions. In one case (124), they observed a point efficiency greater than unity, which is inconsistent with the assumptions and Eq. (7.16). They derived a model void of these assumptions, but its complexity precludes its use for design. Lockett (12) stated that the assumption of perfect liquid mixing in the vertical direction was verified to be a very good assumption, and that the assumption of no vapor backmixing is likely to be good for the spray and froth regimes but less satisfactory for the emulsion regime. An in-depth coverage of the fundamentals of point efficiency is available in several texts (12,29,84,125,126).

Figure 7.3 shows the sequence of steps converting phase resistances into a tray efficiency. Gas and liquid film resistances are added to give the point efficiency (Sec. 7.1.2). Had both vapor and liquid on the tray been perfectly mixed, the Murphree tray efficiency would have equaled the point efficiency (see Sec. 7.1.1). Since the phases are not perfectly mixed, a model of the vapor- and liquid-mixing patterns is

needed for converting point efficiency into tray efficiency. Liquid mixing patterns are plug flow, backmixing, and stagnant zones, while vapor-mixing patterns are perfect mixing and plug flow.

Lewis (121) was the first to derive quantitative relationships between the Murphree and the point efficiency. He defined three mixing cases, assuming plug flow of liquid in all:

Lewis case 1 Vapor perfectly mixed between trays {Note: in this case the direction of liquid flow on successive trays is immaterial).

Lewis case 2 Vapor unmixed between trays. Liquid flows in the same direction on successive trays.

Lewis case 3 Vapor unmixed between trays. Liquid flows in alternate direction on successive trays.

The Lewis cases give the maximum achievable tray efficiency. In practice, efficiency is lower due to liquid and vapor nonuniformities and liquid mixing. The countercurrent nature of contact is greatest in case 2 and least in case 3, causing tray efficiency to decline from case 2 to case 1 to case 3. The potential for reaching higher efficiency has enticed many specialty tray designs to use case 2 as the basis.

Most tray efficiency models are based on Lewis case 1, and fewer on case 3. Since it is uncommon to have liquid flowing in the same direction on successive trays, Lewis case 2 is seldom used. For case 1, Lewis derived the following relationship:

Most theoretical models incorporate the effects of vapor and liquid nonuniformity into the relationship between the Murphree and the point efficiency. Developing models for vapor-liquid contact on trays has been a fertile research area in the last couple of decades, with literally hundreds, maybe thousands of papers published on the subject. A thorough review is given by Lockett (12).

The "dry" Murphree efficiency calculated thus far takes into account the vapor and liquid resistances and the vapor-liquid contact patterns, but is uncorrected for the effects of entrainment and weeping. This correction converts the dry efficiency into a "wet" or actual Murphree tray efficiency. Modeling the effect of entrainment and weeping on tray efficiency is based on presumed mixing patterns of liquid on the trays (Fig. 7.3). Colburn (127) derived an equation for the effect of entrainment on efficiency, assuming perfect mixing of liquid on the tray. Although this assumption is questionable, Colburn's equation gives a reasonable approximation to the effect of entrainment on efficiency provided \ is close to unity (12,128). Colburn's equation is

Lockett et al. (12,128,129) present rigorous methods for allowing for the effects of entrainment and weeping. Colwell and O'Bara (58) and Banik (57) also present rigorous methods for allowing for the effects of weeping on tray efficiency.

The Murphree tray efficiency obtained from Eq. (7.18) can be converted into a column efficiency using Eq. (7.4).

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