Inverse lever rule

The inverse lever rule indicates that when feed and solvent are mixed, their average composition lies on a point M (Fig. 2) such that M lies on a straight line between the points F (feed composition) and S (solvent composition) and that... 

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 ... 

The binodal branches do not coincide with the phase diagram axes. This means that the biopolymers are limitedly cosoluble. For instance, on mixing a protein solution A and a polysaccharide solution B a mixture of composition C can be obtained. This mixed solution spontaneously breaks down into two liquid phases, phase D and phase E. Phase D is rich in protein and E is rich in polysaccharide. These two liquid phases form a water-in-water (WIW) emulsion. Hie phase volume ratio is estimated by the inverse lever rule. The phase D/phase E volume ratio equals the ratio of the tieline segments EC/CD. Point F represents the phase separation threshold, that is, the minimal critical concentration of biopolymers required for phase separation to occur. 

Any point on Fignre 10.1 defines one ternary composition. Points interior to the mntnal solnbility diagram form two phases as shown in Figure 10.2, where a mixture with composition M in this diagram is shown to separate into two phases with extract composition E and raffinate composition R. The inverse lever rule defines the relative amounts of the two phases. So, given the line segment EM and MR, relative amounts are... 

Graphical solutions to material balance problems involving equilibrium relationships offer the advantages of speed and convenience. Fundamental to all graphical methods is the so-called inverse lever rule, which is derived in Example 3.3 and applied in Example 3.4. 

To illustrate the thermodynamic behavior required by the process in Fig. 7.8-5, consider Fig. 7.8-6. If the LLE step is carried out at temperature T, to achieve an extract with composition 1, then cooling this hot extract to temperature 7 2 wiil result in two liquid phases with the compositions at points 2 and 3, which are connected by the equilibrium tie line (points 2 and 3 at T ), that also intersects the hot extract at composition 1. The relative amounts of raffinate to solvent phase at temperature are given by the inverse lever rule (i.e., the length from point 1 to 2 divided by the length from point 1 to 3). 

If the composition and temperature position is located within a two-phase region, things are more complex. The tie line must be used in conjunction with a procedure that is often called the lever rule (or the inverse lever rule), which is applied as follows ... 

Further reduction of volume does not require an increase of pressure the piston moves downwards, but the pressure remains constant. We notice, however, that more and more liquid is formed. We eventually reach an indicator reading of 170 cm, where all the i-butane is liquid except for a few bubbles. Note also that in this two-phase region (L+ V) the relative amounts of liquid and vapor - for a mixture of specified volume -are given by the well known inverse lever rule. 

Furthermore, compositions may be found graphically by a lever rule. All mixtures of pure C02 and pure H2 fall along the ordinate, the distance from pure C02 being inversely proportional to the amount of pure C02... 

Step 2 Locate point M on a straight line connecting L and V using the inverse lever-arm rule ... 

Equation (1.5) represents the so-called lever rule points xip, Xio, and xib are located on one straight line, and the lengths of the segments [xif, xid and [xib, Xip] are inversely proportional to the flow rates D and B (Fig. 1.1b). Mixture with a component number n > 5 cannot be represented clearly. However, we wiU apply the terms simplex of dimensionality (n -1) for a concentration space of n-component mixture C , hyperfaces C i of this simplex for (n - l)-component constituents of this mixture, etc. 

The amounts of coexisting melt and solid can be determined by the lever rule, using the simple geometric relation that the amounts of the coexisting phases are inversely proportional to their distance from the point of the global composition measured along a conode. For example, at T4 the relationship holds... 

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