Modeling of Phase Equilibria

8/11/2019 Modeling of Phase Equilibria

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Modeling of Phase Equilibria

(Aqueous-Carbonic fluid example)

It is well known that there is considerable immiscibility amongst H 2O and CO 2. Themutual solubility of the two species is enhanced at high temperatures and pressures thatcan be represented as a series of solvi in T-X diagrams constructed at different pressuressuch as the sketch below.

One can use the same principle of lever rule to calculate the composition of the two phases to which an original homogeneous aqueous-carbonic fluid at high temperaturewill split. Addition of NaCl to the system enhances the immiscibility and the fluid splitsat lower temperature and pressures compared to fluids free of NaCl. (refer to Boweersand Helgeson, 1983 to see the series of binary T-X diagrams at fixed wt% of NaClrelative to water. Although this is strictly a ternary system, phase equilibrium modelingcan be attempted on pseudobinaries with fixed NaCl contents.

The mathematical formalism of modeling phase equilibrist is as follows:

If the original homogeneous fluid splits into two phases that are in equilibrium, then thechemical potential (molar Gibbs free energy), in turn the activities of the components areidentical in both the splitted phases. For liquid species, we can follow the same formalismas gaseous species and can write in the form of mole fraction and activity coefficient ( ).


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For any gas, if we chose the standard state as the pure gas at temperature and pressure ofinterest, then we can write the activity coefficient for the gas as the ratio of the fugacitycoefficient in the mixture ( i) .Let us consider an aqueous carbonic fluid with NaCl (three components) that washomogeneous and splits to two phases an aqueous-rich and a carbonic-rich and let the

fraction of the first be represented as and the other fraction is (1- ). For the equilibriumcondition, we can write

Where alpha and beta are the two phases (say the aqueous and the carbonic). Thefollowing conditions must satisfy

Representing bulk as (B), the following mass balance equation can be written:

Equations 1-6 suggest that in order to compute phase equilibria in the system, we knowthe bulk composition (mole fraction each component) and have computed the fugacity


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coefficients ( i) (discussed later). We have to make an initial guess of (or X B). We canhave estimates of mole fractions of components in phase beta (X i) using equation 6.This will permit us to calculate X i . We can check for the condition as stated in eq-2. Ifthe condition is not satisfied we can take another value of . However, to make thecalculation procedure more logical (for the next choice of ), we shall take a more

rational approach.

This algorithm will result in calculation of composition of the coexisting phases. The pre-requisite to this calculation is the computation of fugacity coefficients of species and thatcan reasonably done from PVTX relationship as detailed below.

Whenever we want to compute the excess free energy of mixing in a system (binary orternary) we have to take into account the activity coefficient term:


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Therefore, for computation of phase equilibria, one has to compute the fugacitycoefficients of the component fluid species and have to explore how fugacity or fugacity

coefficients could be computed from PVTX relationships.


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If P 0 is sufficiently low, P=f is a good approximation. Thus


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Thus given a bulk composition of a ternary fluid (say H 2O-CO 2-NaCl), on calculation ofthe molar volume (using procedure outlined before to calculate the a and b terms for themixture), the fugacity coefficient of each component i can be computed if the partialderivative terms can be evaluated for each species.

They can be approximated as follows:


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In order to compute the fugacity coefficient for species in the mixed aqueous-carbonicfluid, one has to compute the individual partial derivative terms for each species and inorder to do that, the a and b terms has to be evaluated.

The same can be achieved from the following empirical equations:

The above set of parameters will enable you to calculate the fugacity coefficients for eachspecies in the mixture at specified P, T and chosen value of W NaCl and X CO 2.

Taking W NaCl to be zero would imply working out the H 2O-CO 2 binary. Try reproducingthe solvus in the system at different temperatures (at fixed pressure) using the G Ex

expression. You will have to compute fugacity coefficient for the pure species ( i0) inorder to compute the activity coefficient. Reproduce the solvus at 1 kilo bar pressure.


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Modeling of Phase Equilibria

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