# 17.6: The Stability Criteria

• • Vice Provost Emeritus for Global Program & Professor Emeritus (Petroleum and Natural Gas Engineering) at Pennsylvania State University
• Sourced from John A. Dutton: e-Education Institute

Interestingly enough, one of the most difficult aspects of making VLE calculations may not be the two-phase splitting calculation itself, but knowing whether or not a mixture will actually split into two (or even more) phases for a pressure and temperature condition.

A single-phase detection routine has to be simultaneously introduced at this stage to detect whether the system is in a true single-phase condition at the given pressure and temperature or whether it will actually split into two-phases. Several approaches may be used here: the Bring-Back technique outlined by Risnes et al. (1981), and Phase Stability Criteria introduced by Michelsen (1982), among others. Here we describe Michelsen’s stability test.

Michelsen (1982) suggested creating a second-phase inside any given mixture to verify whether such a system is stable or not. It is the same idea behind the Bring-Back procedure (Risnes et al., 1981), but this test additionally provides straightforward interpretation for the cases where trivial solutions are found
(Ki’s —> 1). The test must be performed in two parts, considering two possibilities: the second phase can be either vapor-like or liquid-like. The outline of the method is described below, following the approach presented by Whitson and Brule (2000).

1. Calculate the mixture fugacity (fzi) using overall composition zi.
2. Create a vapor-like second phase,
1. Use Wilson’s correlation to obtain initial Ki-values.
2. Calculate second-phase mole numbers, Yi: (17.15)
3. Obtain the sum of the mole numbers, (17.16)
4. Normalize the second-phase mole numbers to get mole fractions: (17.17)
5. Calculate the second-phase fugacity (fyi) using the corresponding EOS and the previous composition.
6. Calculate corrections for the K-values: (17.18) (17.19)
7. Check if:
1. Convergence is achieved: (17.20)
2. A trivial solution is approached: (17.21)

If a trivial solution is approached, stop the procedure.

If convergence has not been attained, use the new K-values and go back to step (b).

3. Create a liquid-like second phase,

Follow the previous steps by replacing equations (17.15), (17.16), (17.17), and (17.18) by (17.22), (17.23), (17.24), and (17.25) respectively. (17.22) (17.23) (17.24) (17.25)

The interpretation of the results of this method follows:

• The mixture is stable (single-phase condition prevails) if:
• Both tests yield S < 1 (SL < 1 and SV < 1),
• Or both tests converge to trivial solution,
• Or one test converges to trivial solution and the other gives S < 1.
• Only one test indicating S > 1 is sufficient to determine that the mixture is unstable and that the two-phase condition prevails. The same conclusion is made if both tests give S > 1, or if one of the tests converges to the trivial solution and the other gives S > 1.