*Perturbation Theory and Phase Behavior Calculations Using Equation of State Models DOI: http://dx.doi.org/10.5772/intechopen.93736*

fluid is physically single phase. They showed that the phase split equations can still be satisfied, this time with negative *β* values at pressures above the bubble point or with *β* values above unity at pressures above the upper or below the lower dew point. The more is the distance from the phase boundary the more is the absolute value of the molar fraction, eventually approaching �∞ and þ∞ at the convergence pressure *pk*. At convergence pressure, the equilibrium coefficients become equal to unity whereas beyond *pk* the flash equations have only one solution, the trivial one. Algorithms to compute the locus of the convergence pressure over a temperature range, known as "convergence locus" (CL), have been developed [31]. The negative flash area between the regular phase envelope and the CL is often referred to as the "shadow region" [32].

They also showed that stability tests can also be interpreted outside the phase envelope. Each of the two trial phases converges to a nontrivial solution (i.e. the TPD distance is positive) up to a locus in the shadow region, known as "stability test limit locus", STLL) which is enclosed by the CL. Such stability test results can be used to initialize negative flash calculations. Beyond STLL, the stability test only converges to the trivial solution. The regions discussed are shown in **Figure 3** for a black oil, where the phase envelope interior is shown in red, cyan and yellow color and the latter corresponds to the spinodal. The shadow regions above the bubble point and the dew point lines are shown in pink and blue color respectively. Green color indicates the area outside the CL where the trivial solution is the only one to the phase split problem.

To interpret physically the results of a negative flash we firstly need to note that a molar fraction value of 0 <*β* <1 in a regular flash calculation implies that *β* moles of gas of composition *yi* need to be added to 1 � *β* moles of liquid of composition *xi* to reconstruct the original feed composition *zi*. In a negative flash with *β* < 0, j j *β* ¼ �*β* moles of gas need to be removed from 1 � *β* ¼ 1 þ j j *β* moles of liquid to reconstruct one mole of the original feed composition. Similarly, when *β* > 1, *β* � 1 moles of liquid need to be removed from *β* moles of gas.

Clearly, negative flash solutions are not of any direct use in fluid flow calculations. However, they can significantly improve the convergence properties of the regular flash calculations close to the phase boundary by allowing the solution at some iteration to escape temporarily outside the phase envelope while trying to arrive to the exact solution.

**Figure 3.** *Regular phase envelope, shadow region and trivial solution region.*
