The Stability Criteria

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).

1. Calculate the mixture fugacity (f_zi) using overall composition z_i.
2. Create a vapor-like second phase,
   1. Use Wilson’s correlation to obtain initial K_i-values.
   2. Calculate second-phase mole numbers, Y_i:
      $$Y_i=z_i{K}_i$$
      (17.15)
       
   3. Obtain the sum of the mole numbers,
      $$S_v={\displaystyle \sum _i^n{Y}_i}$$
      (17.16)
       
   4. Normalize the second-phase mole numbers to get mole fractions:
      $$y_i=\frac{Y_i}{S_v}$$
      (17.17)
       
   5. Calculate the second-phase fugacity (f_yi) using the corresponding EOS and the previous composition.
   6. Calculate corrections for the K-values:
      
      $$R_i=\frac{f_{xi}}{f_{yi}}\frac{1}{S_v}$$
      (17.18)
       
      $$K_i^{(n+1)}=K_i^{(n)}{R}_i$$
      (17.19)
       
   7. Check if:
      1. Convergence is achieved:
         $${\displaystyle \sum _i^n{(R_i-1)}^2<1\cdot {10}^{-10}}$$
         (17.20)
          
      2. A trivial solution is approached:
         $${\displaystyle \sum _i^n{(\ln K_i)}^2<1\cdot {10}^{-4}}$$
         (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.

   $$Y_i=z_i/K_i$$
   (17.22)
    
   $$S_L={\displaystyle \sum _i^n{Y}_i}$$
   (17.23)
    
   $$x_i=\frac{Y_i}{S_L}$$
   (17.24)
    
   $$R_i=\frac{f_{xi}}{f_{zi}}{S}_L$$
   (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.
• 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.