Three-component systems can be represented in two-dimensional ternary diagrams. There are three components, but the sum of the mole fractions must add to unity. Therefore, specifying two mole fractions completely defines the composition.
A typical rectangular ternary diagram is given in Figure 1.10. The mole fraction of component 1 is shown on the abscissa; the mole fraction of component 2, on the ordinate. Both of these dimensions run from 0 to 1. The three corners of the triangle represent the three pure components.
Since only two compositions define the composition of a stream, the stream can be located on this diagram by entering the appropriate coordinates. For example, Figure 1.10 shows the location of stream F that is a ternary mixture of 20 mol% n-butane (C4), 50 mol% n-pentane (C5), and 30 mol% n-hexane (C6).
One of the most useful and interesting aspects of ternary diagrams is the "ternary mixing rule," which states that if two ternary streams are mixed together (one is stream D with composition xD1 and xD2 and the other is stream B with composition xB1 and xB2), the mixture has a composition (zi and z2) that lies on a straight line in a x1-x2 ternary diagram that connects the xD and xB points.
Figure 1.11 illustrates the application of this mixing rule to a distillation column. Of course, a column separates instead of mixes, but the geometry is exactly the same. The two products D and B have compositions located at point (xD1-xD2) and (xB1-xB2), respectively. The feed F has a composition located at point (z1 -z2) that lies on a straight line joining D and B.
This geometric relationship is derived from the overall molar balance and the two overall component balances around the column:
Substituting the first equation in the second and third gives
Rearranging these two equations to solve for the ratio of B over D gives
Equating these two equations and rearranging give
XB1 - Zi XB2 - Z2 XD1 - zi _ Zi - XB1 Z2 - XD2 Xb2 - Z2
Figure 1.12 shows how the ratios given above can be defined in terms of the tangents of the angles 61 and U2. The conclusion is that both angles must be equal, so the line between D and B must pass through F.
As we will see in subsequent chapters, this straight-line relationship is quite useful in representing what is going on in a ternary distillation system.
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