# Calculate Phase and Chemical Equilibria

Direct minimization of the Gibbs free energy can be used to calculate phase equilibria, chemical equilibria, and/or simultaneous physical and chemical equilibria. This method may be preferred for systems where multiple liquid phases can coexist and/or where retrograde phase behavior is possible during depressuring or pressure relief.

Some of the advantages of direct minimization of the Gibbs free energy include:

• The atom matrix can be constrained to ensure that inert liquids and/or inert gases are only present in their respective phases. For example, this can be useful for systems containing hydrogen and heavy polymers.
• The atom matrix can be constrained to ensure partial or user defined conversion of one or more chemical species.
• Multiple liquid, vapor, and solid phases can be handled simultaneously with simplicity.
• Phase splitting can be determined a priori.

Some of the disadvantages include:

• The Gibbs free energy minimum can be very flat and requires high precision estimates.
• The Formation energies of all chemical species have to be thermodynamically consistent and calculated at the system temperature using reference elements.
• Reasonable initial guesses for phase splitting are required. Normally, the most non-ideal liquid component in the mixture will likely form the dominant component in one liquid phase while the second most non-ideal liquid component will likely form the dominant component is the second liquid phase.

## Minimization Algorithm

The computation of the equilibrium state of a system is one of constrained optimization. The minimization of the Gibbs free energy is subject to mass, element balance constraints, and where applicable, user defined constraints. Recent advances in the field of nonlinear optimization have greatly simplified the solution of this problem. Process Safety Office® SuperChems™ software uses the Wilson-Han-Powell successive quadratic programming (SQP) algorithm to directly minimize the Gibbs free energy for nonideal multiphase systems. Advantages of this algorithm include its low number of function and gradient evaluations and its ability to handle simple bounds on variables, such as non-negativity constraints which eliminates the need to transform variables in order to avoid singularities.

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