# 17.6: The Stability Criteria

- Page ID
- 577

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

(K_{i}’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).

- Calculate the mixture fugacity (f
_{zi}) using overall composition z_{i}. - Create a
*vapor-like second phase,*- Use Wilson’s correlation to obtain initial K
_{i}-values. - Calculate second-phase mole numbers, Y
_{i}: (17.15) - Obtain the sum of the mole numbers, (17.16)
- Normalize the second-phase mole numbers to get mole fractions:(17.17)
- Calculate the second-phase fugacity (f
_{yi}) using the corresponding EOS and the previous composition. - Calculate corrections for the K-values:

(17.18)

(17.19) - Check if:
- Convergence is achieved:(17.20)
- 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).

- Use Wilson’s correlation to obtain initial K
- 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 (S_{L}< 1 and S_{V}< 1), - Or both tests converge to trivial solution,
- Or one test converges to trivial solution and the other gives S
__<__1.

- Both tests yield S
- 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.

## Contributors and Attributions

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.