Lever rule application

Himmelbau (1995) or any of the general texts on material and energy balances listed at the end of Chapter 2. The Ponchon-Savarit graphical method used in the design of distillation columns, described in Volume 2, Chapter 11, is a further example of the application of the lever rule, and the use of enthalpy-concentration diagrams. 

Note that the resulting fractional amounts are in weight percent, because the abscissa axis of the phase diagram reports the fractional weights of the two components (similar application of baricentric coordinates to a molar plot of type 7.2 would have resulted in molar fractions of phases in the system). Applying the lever rule at the various T, we may quantitatively follow the crystallization behavior of the system (i.e., atT = 1350 °C, = 0.333 and Xl = 0.666 atT = 1300... 

Answer Application of the lever rule in the 1 -2-3 -F Liquid region shows that even in the best case (at the left edge of the hatched region) only about 3% of the 1-2-3 phase is formed, the rest being liqnid of different composition. 

Since the point M lies in the two-phase region of the triangular diagram, the term mixture applies only on a scale larger than the size of the droplets formed. The droplet dispersion formed by agitation has sufficient interfacial area (see Section I.C) for equilibrium to be reached quickly, so that point M represents the mean of the extract composition (point E) and the raffinate composition (point R) which are connected by the appropriate tie-line. A further application of the inverse lever rule permits calculation of the relative amounts of extract and raffinate. In this example, the material balance based on 1 kg of feed is summarized as follows ... 

Direct measurements of Ai c, the molecular area that corresponds to the LC side of the LE-LC coexistence region, cannot be made unequivocally because of the proximity to collapse and because of the steepness of the coexistence curve. It can be determined, however, by application of the lever rule. If a monolayer is prepared at a molecular area A located within the LE-LC two-phase region, then the mole fraction of the system that is in the LE phase is given by... 

Flash calculations and the application of the lever rule (overall mass balance relating the feed, distillate and bottoms product streams) to predict feasible sharp splits for a given feed condition. 

Fig. 13.9 Application of the lever rule to a mixture of two incompatible components A and B.

Fig. 14.10 Melting point diagram for a system of two components that show incompatible behavior in their solid states as well as application of the lever rule .

Characterizes a homogeneous or heterogeneous mixture of intermediate composition, the support point by the application of the lever rule (348)... 

Numerical Solution The graphical solution is quite simple and fast, but reading numbers off the graph can only be done approximately. For more accurate answers, we must perform a numerical solution. This is done more easily once the graphical solution has been obtained. First, application of the lever rule gives... 

The heat of crystallization is the heat that has to be supplied or removed during crystallization at constant temperature. It is equal to the negative value of the heat of solution during the dissolution of crystals in an almost saturated solution. The heat of crystallization is accounted for in the enthalpy values. Processes in crystallizers can easily be tracked, if an enthalpy concentration diagram is available for the respective system. The pure component s enthalpy is zero at reference temperature, not the enthalpy of real mixtures however. In such diagrams, the lever rule is applicable. This is shown for the system calcium chloride/water in Fig. 8.3-5, where the specific enthalpy is plotted vs. the mass fractions. 

The lever rule finds its most notable application in the use of tie-lines to establish the compositions and relative proportions of the two solvent layers in an extraction process. Suppose, for example, that two liquid solutions with overall composition represented by M in Figure 6.9 are contacted and allowed to settle into a two-phase equilibrium. The compositions will then be given by the tie-line end points L and N, and the relative amounts of L and N will be in the ratio. Here again the geometrical construction... 

A special case of the lever-arm rule, which renders it applicable to extraction analysis, is an equilibrium-limited stage for a three-phase system (Figure 3.22). Everything stated for the lever-arm rule still applies here since the mass balances around the control volume (equilibrium stage) are still the same. The compositions of the three streams will still lie on a straight line, and stream ratios can still be calculated as before. 

The existence of these straight lines and the applicability of the lever-arm rule can be proved by deriving Eq. (13z44). 

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