Efficiencies Modified Murphree

This presentation is limited to two models, the Murphree plate efficiency and the modified Murphree plate efficiency, and their applications to columns in the service of separating both binary and multicomponent mixtures. For convenience of application, these efficiencies are restated in terms of the vaporization plate efficiency. Definitions of the vaporization point and plate efficiency follow immediately and the Murphree plate efficiencies are defined in a subsequent section as they arise in the development of the perfectly mixed liquid phase model. 

In the Murphree and modified Murphree plate efficiencies, it is supposed that the vapor and liquid leaving each plate are in thermal equilibrium Tj = Tjf = Tj. Since the sum of the y s over all components is equal to unity, it is evident from Eq. (13-39) that... 

Modified Murphree Plate and Point Efficiencies for the Perfectly Mixed Liquid Phase Model... 

First the equations for the modified Murphree plate efficiency are developed and then the corresponding equations for the Murphree plate efficiency are developed. Expressions for the vaporization efficiencies for each of these efficiencies are then presented. The section is concluded by the presentation of numerical examples for binary mixtures which demonstrate that essentially the same compositions of the vapor leaving a plate are predicted by vaporization efficiencies corresponding to the modified Murphree efficiency model as are predicted by the Murphree plate efficiencies. This result suggests that the correlations for the film coefficients developed by others4 for use with Murphree efficiencies may be used in the prediction of vaporization efficiencies for multi-component systems. 

THE MODIFIED MURPHREE PLATE EFFICIENCY AND THE CORRESPONDING VAPORIZATION EFFICIENCY... 

The modified Murphree plate efficiency model consists of the following set of assumptions. 

Then, it follows from Eqs. (13-46) and (13-48) that one should be able to predict the modified Murphree plate efficiency on the basis of the total number of overall transfer units, namely,... 

Since K,x( does not change with Z at a given W, this expression may be integrated and rearranged to give the definition of the modified Murphree point efficiency. 

The assumptions for the Murphree plate efficiency model are the same as those for the modified Murphree plate efficiency model except for the additional assumption that the liquid leaving each stage is at its bubble-point temperature. The beginning equation for the development of the expression for the Murphree plate efficiency is obtained by replacing Rt in Eq. (13-23) by the expression given by Eq. (13-29) to give... 

In this case an assumption in addition to those stated for the modified Murphree plate efficiency model is made. This assumption is that the liquid is perfectly mixed vertically but not horizontally, and that the liquid is at its bubble-point temperature at each point on the plate. Thus, along any one vertical line, the bubble-point temperature is the same, but it varies from point to point in the horizontal direction W. 

By following the same general approach as that demonstrated for the modified Murphree point efficiency, one obtains the following expression for the Murphree point efficiency... 

Comparison of the Murphree Plate Efficiencies and the Vaporization Efficiencies Corresponding to the Modified Murphree Efficiency Model... 

Comparison of the Murphree efficiency [Eq. (13-59)] with the modified Murphree efficiency [(13-48) and (13-49)] shows that the Murphree efficiency depends on riOGMi [Eq. (13-59)] while the modified Murphree efficiency depends on n OGi [Eq. (13-49)]. The number of overall transfer units riOGi and n 0Gui for the two efficiencies depend on the same set of transfer units (riGi and n Li) and are the same except that the quantity yfK. /yf appears in riOGi in the same position that mi appears in riOGMi. 

The vaporization plate efficiency (based on the modified Murphree efficiency model) and the Murphree plate efficiency have two points [(xji,yj+i, i) and (xfl9 yfji)] in common as may be seen from Fig. 13-3. The expression for the line connecting these two points is... 

After the interface point has been reached, the paths for the two models differ. In the case of the vaporization plate efficiency (based on the modified Murphree efficiency model), a projection is made from the point (xjf, yj,) to the point (xji, yji) with the slope Kjt that satisfies Eq. (13-40). In the case of the Murphree plate efficiency, a projection is made from the point (xjf, yj,) to the point (x i, yft) along the tangent line, yf = mjXj + blm Since the equilibrium curve is seldom straight, an appropriate value of mi lying between the values at (xf, y ) and (xjj, yft) is selected. The values of yjt predicted by use of vaporization efficiencies are calculated by use of Eqs. (13-40), (13-47), and (13-51). 

The following calculational procedure was used to compute the vaporization plate efficiencies (corresponding to the modified Murphree plate efficiency model) ... 

Examples 13-1 and 13-2 demonstrate that for the same set of film coefficients (w L, riG) and compositions xjU yj+ approximately the same values of yn are obtained by use of vaporization plate efficiencies (based on the modified Murphree model) as are obtained by use of Murphree plate efficiencies for binary mixtures. This result suggests that the existing correlations for riL and riG given in the Bubble-Tray Design Manual4 may be used to estimate the Ej-s for the modified Murphree model for multicomponent mixtures. This statement arises from the fact that the formulas for the vaporization plate efficiencies which were used in the computations for binary mixtures are precisely the same ones that... 

Table 13-6 Vaporiation efficiencies for the modified Murphree efficiency model...

When it is desired to compute, with rigorous methods, actual rather than equilibrium stages, Eqs. (13-69) and (13-94) can be modified to include the Murphree vapor-phase efficiency T ij, defined by Eq. (13-29). This is particularly desirable for multistage operations involving feeds containing components of a wide range ol volatility and/or concentration, in which only a rectification (absorption) or stripping action is provided and all components are not sharply separated. In those cases, the use of a different Murphree efficiency for each component and each tray may be necessary to compute recovery accurately. 

The Murphree efficiency can be rearranged to form a modified equilibrium equation... 

The efficiency homotopy. Taylor, Wayburn, and Vickery (80) propose a physical homotopy based on a pseudo-Murphree efficiency. As stated earlier, the efficiency can be entered into the MESH by modifying the... 

The equations are the Equilibrium relations, here modified to include the Murphree efficiencies defined by Eq. 13.1.5... 

FIGURE 12.11 Modified McCabe-Thiele plot to show approach to equihbrium on real stages. The relative height of the real stage to the equilibrium stage height denotes Murphree stage efficiency. 

The Kremser equation and its various modified forms consider only the number of theoretical plates, sot the number of actual plates in absorbers and strippers. For cases where the Murphree stage efficiency is known and relatively constant over the length of the tower, a correlation proposed by Nguyen can be used to estimate the total required number of actual plates NT as follows ... 

E equations—phase equilibrium relation for each component, here modified to include the Murphree efficiency defined by equation (4-55) (C equations for each stage) ... 

Although the equilibrium-based model, modified to incorporate stage efficiency, is adequate for binary mixtures and for the major components in nearly ideal multicomponent mixtures, that model has serious deficiencies for more general multicomponent vapor-liquid mixtures. Murphree himself stated clearly the limitations... 

If desired, (15-54) can be modified to permit real rather than theoretical stages to be computed. Values of the Murphree vapor-phase plate efficiency must then be specified. These values are related to phase compositions by the definition... 

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