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# Ternary phase diagrams

Phase diagrams are graphical representations of the liquid, vapor, and solid phases that co-exist at various ranges of temperature and pressure within a reservoir. Ternary phase diagrams represent the phase behavior of mixtures containing three components in a triangular diagram.

## Properties of ternary diagrams

Phase behavior of mixtures containing three components is represented conveniently on a triangular diagram such as those shown in **Fig. 1**. Such diagrams are based on the property of equilateral triangles that the sum of the perpendicular distances from any point to each side of the diagram is a constant equal to the length of any of the sides. Thus, the composition of a point in the interior of the triangle can be calculated as

where

Several other useful properties of triangular diagrams are a consequence of this fact. For mixtures along any line parallel to a side of the diagram, the fraction of the component of the corner opposite to that side is constant (**Fig. 1b**). In addition, mixtures lying on any line connecting a corner with the opposite side contain a constant ratio of the components at the ends of the side (**Fig. 1c**). Finally, mixtures of any two compositions, such as A and B in **Fig. 1d**, lie on a straight line connecting the two points on the ternary diagram. Compositions represented on a ternary diagram can be expressed in volume, mass, or mole fractions. For vapor/liquid equilibrium diagrams, mole fractions are most commonly used.

## Features of ternary diagrams

**Fig. 2** shows the typical features of a ternary phase diagram for a system that forms a liquid and a vapor at fixed temperature and pressure. Mixtures with overall compositions that lie inside the binodal curve will split into liquid and vapor. Tie lines connect compositions of liquid and vapor phases in equilibrium. Any mixture with an overall composition along a tie line gives the same liquid and vapor compositions. Only the amounts of liquid and vapor change as the overall composition changes from the liquid side of the binodal curve to the vapor side. If the mole fractions of Component *i* in the liquid, vapor, and overall mixture are *x*_{i}, *y*_{i}, and *z*_{i}, the fraction of the total moles in the mixture in the liquid phase is given by

## Plait point

**Eq. 3** is another lever rule similar to that described for binary diagrams. The liquid and vapor portions of the binodal curve meet at the plait point, a critical point at which the liquid and vapor phases are identical. Thus, the plait-point mixture has a critical temperature and pressure equal to the conditions for which the diagram is plotted. Depending on the pressure, temperature, and components, a plait point may or may not be present.

## Phase behavior representation

Any one ternary diagram is given for fixed temperature and pressure. As either the temperature or pressure is varied, the location of the binodal curve and slopes of the tie lines may change. **Fig. 3** shows the effect of increasing pressure on ternary phase diagrams for mixtures of C_{1}, butane (C_{4}), and decane (C_{10}) at 160°F.^{[1]}^{[2]} The sides of the ternary diagram represent a binary system; therefore, the ternary diagram includes whatever binary tie lines exist at the temperature and pressure of the diagram. **Fig. 4** shows the corresponding binary phase diagrams for the C_{1}–C_{4} and C_{1}–C_{10} pairs. The C_{4}–C_{10} pair is not shown because it forms two phases only below the vapor pressure of C_{4}, approximately 120 psia at 160°F (see **Fig. 8.9**).

As **Fig. 3** shows, at 1,000 psia the two-phase region is a band that stretches from the C_{1}–C_{10} side of the diagram to the tie line on the C_{1}–C_{4} side. If the pressure is increased above 1,000 psia, the liquid composition line shifts to higher methane concentrations; methane is more soluble in both C_{4} and C_{10} at the higher pressure (see **Fig. 4**). The two-phase region detaches from the C_{1}–C_{4} side of the diagram at the critical pressure of the C_{1}–C_{4} pair (approximately 1,800 psia). As the pressure increases above that critical pressure, the plait point moves into the interior of the diagram (**Fig. 3**, lower diagrams). With further increases in pressure, the two-phase region continues to shrink. It would disappear completely from the diagram if the pressure reached the critical pressure of the C 1 –C 10 system at 160°F (nearly 5,200 psia).

According to the phase rule, three phases may coexist at a fixed temperature and pressure for some ternary systems. **Fig. 5** shows the general structure of such systems. The three-phase region (3*Φ*) on a ternary diagram is represented as a triangle in **Fig. 5**. Any overall composition lying within the three-phase region splits into the same three phases (I, II and III). Only the amounts of each phase change as the overall composition varies within the three-phase region. Given 1 mole of an overall mixture in the three-phase region, the geometrical relations

determine the fraction of each phase. The edges of the three-phase region are tie lines for the associated two-phase (2*Φ*) regions; thus, there is a two-phase region adjacent to each of the sides of the three-phase triangle. Three-phase regions can exist in several phase diagrams applied in the design of EOR processes.

## References

- ↑ Reamer, H.H., Fiskin, J.M., and Sage, B.H. 1949. Phase Equilibria in Hydrocarbon Systems. Ind. Eng. Chem. 41 (12): 2871-2875. http://dx.doi.org/10.1021/ie50480a052.
- ↑ Sage, B.H. and Lacey, W.N. 1950. Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen: Monograph on API Research Project 37. New York: American Petroleum Institute.

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## See also

Phase diagrams for EOR processes