Balance equations for vapor-liquid-liquid equilibrium (VLLE)

Energy balance

The following equation gives the enthalpy of a liquid phase:

or

where

This is the sum of two terms, the enthalpy of an ideal solution and the excess enthalpy of a non-ideal solution, which is also known as the heat of mixing.

The following equation relates excess enthalpy to the excess Gibbs energy:

or

The following equation gives the entropy:

We substitute this into the enthalpy equation to get the following equation:

The following equation is an algebraically equivalent expression of the preceding equation:

We use either expression depending on which one is most convenient to compute.

The following equation relates the activity coefficients to the excess Gibbs energy:

Since is a partial molar property, we can rewrite this equation as follows:

Therefore, the heat of mixing is inseparable from the liquid activity model.

Other balances

To solve an isothermal vapor-liquid-liquid equilibrium (VLLE) flash, we need the equilibrium condition where the chemical potential for each component is the same in all phases:

Equivalently, the fugacity of each component is the same in all phases:

For equations solved with Newton methods, we use the following material balance equation for each component:

The phase fractions must sum to unity:

The mole fractions in the liquid and vapor phase must sum to unity:

The mole fraction in the liquid 1 and liquid 2 phase must sum to unity:

Therefore, the following equations give the equilibrium between the vapor and liquid 1 phase and the liquid 1 and liquid 2 phase, respectively: