Binodal curve/line

A line is drawn from Rn through M to give Ex on the binodal curve and ExF and SR to meet at the pole P. It may be noted that P represents an imaginary mixture, as described for the leaching problems discussed in Chapter 10. 

In an ideal stage, the extract Ex leaves in equilibrium with the raffinate Rx, so that the point Rx is at the end of the tie line through Ex. To determine the extract E2, PRi is drawn to cut the binodal curve at E2. The points R2, E3, R3, E4, and so on, may be found in the same way. If the final tie line, say ER4, does not pass through R , then the amount of solvent added is incorrect for the desired change in composition. In general, this does not invalidate the method, since it gives the required number of ideal stages with sufficient accuracy. 

The binodal curve has been calculated and is shown in Fig. 10.2 as a solid line. Furthermore, the calculated distribution of acetic acid between both phases is shown in Fig. 10.3. From these figures, it can be seen that the fitting to experimental data is good. 

Data could be obtained from such a graph to construct a McCabe-Thiele type diagram. Tie line points on the binodal curve would provide data for an equilibrium curve, and data could be obtained to construct an operating line (from lines drawn from 0 through the binodal curve). 

Figure 14.6. Construction of points on the distribution and operating curves Line oh is a tieline. The dashed line is the tieline locus. Point e on the equilibrium distribution curve, obtained as the intersection of paths be and ade. Line Pfg is a random line from the difference point and intersecting the binodal curve in /and g. Point j is on the operating curve, obtained as the intersection of paths gj and fhj.

Both maximum and minimum limits exist of the solvent/feed ratio. The maximum is the value that locates the mix point M on the binodal curve near the solvent vertex, such as point Mmla on Figure 14.7(b). When an operating line coincides with a tieline, the number of stages will be infinite and will correspond to the minimum solvent/feed ratio. The pinch point is determined by the intersection of some tieline with line RnS- Depending on whether the slopes of the tielines are negative or positive, the intersection that is closest or farthest from the solvent vertex locates the operating point for minimum solvent. Figure 14.9 shows the two... 

The operating curve is drawn similarly with horizontal projections from pairs of random points of intersection of the binodal curve by lines drawn through the difference point P. Construction of these curves also is explained with Figure 14.6. 

Figure 14.9. Minimum solvent amount and maximum extract concentration. Determined by location of the intersection of extended tielines with extended line RNS. (a) When the tielines slope down to the left, the furthest intersection is the correct one. (b) When the tielines slope down to the right, the nearest intersection is the correct one. At maximum solvent amount, the mix point Mm is on the binodal curve.

Other nomenclature is identified on the flowsketch. The solvent-free reflux point R0 s located on the extension of line SEx. Operating point Q is located at the intersection of lines SRC and RnM. Lines through Q intersect the binodal curve in compositions of raffinate and reflux related by material balance for instance, R and E +l. When the line QF is crossed, further constructions are... 

Fig. 1. Phase diagram of a polymer solution. The upper curve shows LOST and the lower curve UCST behavior. Solid lines binodal, dotted lines spinodal T. temperature, x polymer concentration...

Phase diagram asymmetry can be evaluated by (i) the ratio of the biopolymer concentrations at a critical point, (ii) the angle made by the tie-lines with the concentration axis of one of the biopolymers and (iii) the length of the segment of a binodal curve between the critical point and the phase separation threshold. Association of macromolecules usually changes both their excluded volume and the affinity for the solvent water. This results in nonparallel tie-lines on the phase diagram. Normally, the tie-lines can be nonparallel since an increase in concentration of biopolymers is usually accompanied by their self-association. Equilibrium between the phases is not achievable when phase separation is accompanied by gelation. 

Point Q is at the intersection of lines R M and EiSe, where Se refers to the solvent that is removed from the final extract, and may or may not be of the same composition as the fresh solvent S. Depending on the shape of the curve, point Q may be inside the binodal curve as in Example 14.8, or outside as in Figure 14.8. 

For conditions of constant pressure, or when pressure effects are negligible, binary LEE is conveniently displayed on a solubihty diagram, a plot of T vs. xi. Figure 14.12 shows binary solubility diagrams of three types. The first diagram [Fig. 14.12(a)] shows curves (binodal curves) that define an "island." They represent the compositions of coexisting phases curve UAL for the a phase (rich in species 2), and curve UBL for the P phase (rich in species 1). Equilibrium compositions jc and at a particular T are defined by the intersections of a horizontal tie line with the binodal curves. Temperature Tl is a lower consolute temperature, or... 

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