Equations for the eNRTL method

For the equilibrium calculations, this method uses the approach from Song and Chen[4] and has the following characteristics:

• Considers both the long-range and short-range interactions for the calculation of the phase equilibrium behavior in electrolyte mixtures. It uses the Pitzer-Debye-Huckel[1] model to represent the contribution of the long-range ion-ion interactions and the non-random two-liquid theory to represent the short-range interactions.

• Develops the excess Gibbs free energy expression as a generalized expression without consideration to the fused-salt reference state. Instead, the expressions for the binary interaction parameters take the fused-salt reference state into account along with the like-ion repulsion and local electroneutrality hypotheses. This simplification eliminates the ionic change fraction quantities from the excess Gibbs free energy expression and the subsequent derivations for activity coefficients.

• Normalizes the activity coefficients by choosing the reference states for the ionic and molecular species, that is, a symmetric or unsymmetric reference state. The reference state for the molecular solvents is pure liquid. We use the unsymmetric reference state for the ionic species, which is the infinite dilution state in aqueous solution. We use an asterisk (*) to represent the unsymmetric model in the activity coefficient.

We use the following equation to express the Gibbs energy of the system (gE) described by the local composition model:

We can also express this equation in terms of activity coefficients:

where

i is the molecular component (m), the cationic species (c), or the anionic species (a)

ln(gi*PDH) is the influence of long-range interactions (that is, the Debye-Huckel term)

ln(gi*lc) is the influence of the short-range interactions

DBornln(gi*) is the Born term, a correction to the Debye-Huckel term due to a difference between the dielectric constant of water and the dielectric constant of the solvent mixture

Short-range interactions

The NRTL activity coefficient term is different for molecules and ions. In these equations, we use the following abbreviation for composition (xi) and ion charge number (Ci):

where

zi is the number of charges on component i (this number is zero except for the cation and anion species)

The following equation gives the activity coefficient for molecules (gmlc) at the symmetric reference state (that is, pure liquid):

The following equations give the activity coefficients for the ionic species at the unsymmetric reference state (that is, infinite dilution in aqueous solution):

Long-range interactions

We use a term that describes the Debye-Huckel theory to account for the influence of the long-range interactions. We use a modification from Pitzer[1] to calculate the Debye-Huckel term.

For the solvent component (m), we use the following equation to obtain the activity coefficient:

For the cationic (i = c) and anionic (i = a) species in the unsymmetric reference state, we use the following equations to obtain the activity coefficient:

where

NA is Avogadro's number (6.0221×1023 mole-1)

v is the molar volume

Qe is the electronic charge (1.60206×10-19 Coulomb)

e0 is the permittivity of a vacuum (8.854×10-12 Coulomb2/N-m2)

e is the dielectric constant

kB is the Boltzmann constant (1.38065×10-23 J/K)

We use the following mixing rules for mixed solvents:

where

Ms is the molecular weight of pure solvent s

We express the dielectric constant of the pure solvent as a temperature-dependent function:

where

Ai is a coefficient from the thermodynamic data bank

Bi is a coefficient from the thermodynamic data bank

Tref is 298.15 K

When the solvent is pure water, we use the following temperature-dependent correlation from Chen et al.[2] to calculate the Debye-Huckel parameter (AØ):

where

Mw is the molecular weight of the pure water in kg/kmol

Born term

The equation for the Debye-Huckel term uses a reference state of electrolyte components in infinitely diluted solution in pure water. Therefore, we use the Born term to correct the Debye-Huckel term so that we can use the equations for mixed solvents instead of only pure water. The Born term accounts for the difference between the dielectric constants for water and for the solvent mixture.

We can use the following expression to calculate the Born term in terms of the activity coefficient (gi*) for the cation and anion species:

where

ri is the ionic radius

esolv is the dielectric constant of the solvent mixture

eH2O is the dielectric constant of pure water

If there are no better values of ri available, we recommend that you use a value of 3×10-10 m.

The Born correction to the activity coefficient for solvent components is zero. When the solvent mixture is pure water, DBornln(gi*) = 0 for ions as well.

Interaction parameters

There are three types of model-adjustable binary parameters: molecule-molecule binary parameters, molecule-electrolyte binary parameters, and electrolyte-electrolyte binary parameters. Here, "electrolyte" represents a cation (c) and anion (a) pair. The symmetric non-randomness parameter (aij) and the unsymmetric binary interaction energy parameter (tij) are two kinds of parameters that we need to model the eNRTL method. Typically, we set the NRTL non-randomness parameters to a uniform value of aij = 0.3 for molecule-molecule pairs and aij = 0.2 for molecule-electrolyte and electrolyte-electrolyte pairs.

For molecule-molecule, molecule-electrolyte, and electrolyte-electrolyte interactions, we define the interaction energy parameters (tij) as a function of the temperature as we do for the conventional NRTL equations:

where

T is in Kelvin

Aij, Bij, Cij, Dij, Eij, and Fij are the NRTL binary interaction parameters and are specific to an electrolyte, not to an ion

We use the following equation to obtain the non-adjustable molecule-ion and ion-ion interaction parameters that appear in the equations for the activity coefficients:

We use equations and mixing rules from Song and Chen[4] to calculate the aij and Gij values.

After we establish the interaction parameters, we can then calculate the activity coefficients.

Liquid specific heat capacity and liquid enthalpy for ions

For ions in the standard state (that is, infinite dilute aqueous solution), the heat capacity changes with temperature. We use the following three-parameter correlation from Thomsen[5] (Correlation 96 in AVEVA Thermodynamic Data Manager) to calculate the standard-state heat capacity for ion i (C¥p,i):

where

C1, C2, C3, and C4 are ion-specific parameters

According to Thomsen[5], C4 is a constant value of 200 K for all components.

If data for an ion is not available in Thomsen[5], we use Correlation 1 (the polynomial equation) instead of Correlation 96 to calculate Cp for that ion. See Equation forms for temperature-dependent properties in AVEVA Thermodynamic Data Manager for more information.

We obtain the liquid enthalpy for ions (H¥i) from the heat capacity according to the following equation:

where

DfH¥i is the heat of formation of ion i at infinite dilution

Vapor-liquid equilibrium for Henry's solutes

For molecular solutes (Henry's solutes), we use Henry's Law to model the equilibrium between the gaseous solute and the dissolved gas in the liquid phase:

where

Hi is the Henry's constant for component i in the mixed solvent

gi¥ is the aqueous infinite dilution (xw → 1) activity coefficient of molecular solute i

For some Henry's solutes, we can assume ideal behavior and set the activity coefficient (gi*) for that Henry's solute to one. This assumption simplifies the equilibrium equation between the gaseous solute and the dissolved gas to the following equation:

The Include activity coefficient in liquid fugacity for Henry's solutes (select for solutes in Component List) checkbox in the Equilibrium Options section of the Fluid Editor determines whether to include the activity coefficient in the vapor-liquid equilibrium (VLE) calculations for selected Henry's solutes.

When you clear the Include activity coefficient in liquid fugacity for Henry's solutes (select for solutes in Component List) checkbox, we set the activity coefficients for all Henry's solutes to one, and we use the simplified equilibrium equation for all Henry's solutes.

When you select the Include activity coefficient in liquid fugacity for Henry's solutes (select for solutes in Component List) checkbox, the Henry Activity column appears in the table in the Component List section of the Fluid Editor. You use the checkboxes in the Henry Activity column to choose which Henry's solutes use activity coefficients in their equilibrium equations.

If you clear the Henry Activity checkbox for a Henry's solute, we set its activity coefficient to one, and we use the simplified equilibrium equation for that Henry's solute.

If you select the Henry Activity checkbox for a Henry's solute, its equilibrium equation includes the activity coefficient.

The Henry Activity checkbox is selected for some Henry's solutes by default. You should review the Henry Activity checkbox selections for all the Henry's solutes in your Fluid Type to ensure that the VLE calculations meet your expectations.

You use the Include rigorous mixing with activity coefficient and critical volumes for Henry's Law checkbox in the Equilibrium Options section of the Fluid Editor to determine which mixing rule to use to calculate the Henry's constant in the mixed solvent (Hi). When you clear this checkbox, we use a simple additive mixing rule to calculate Hi:

where

A is the set of solvent components in the mixed solvent

XA is the mole fraction of solvent component A on a solute-free basis

HiA is the Henry's constant for component i in pure solvent A

When you select the Include rigorous mixing with activity coefficient and critical volumes for Henry's Law checkbox, we use the rigorous mixing rule to calculate Hi, which uses the activity coefficient and critical volume to account for the non-ideality of the system:

where

wA is a weighting factor

We calculate the weighting factor from the critical volumes of the pure solvents:

where

VcA and VcB are the critical volumes of solvent A and solvent B, respectively

You can override the critical volume data (Vc) for components on the Constants tab in the Component Data section of the Fluid Editor. Refer to Override constant property data for more information.

You can include a pressure correction in the calculation of HiA. You use the Apply Henry's Law Pressure Correction using Brelvi O'Connell Model checkbox in the Equilibrium Options section of the Fluid Editor to turn on or turn off the pressure correction. When you select this checkbox, we use the following equation to calculate HiA:

where

PAsat is the saturation pressure of solvent A at the current temperature

viA¥ is the partial molar volume of molecular solute i at infinite dilution in pure solvent A

We use the Brelvi-O'Connell method[6] to calculate viA¥ as a function of characteristic volumes:

where

vCi is the characteristic volume from Brelvi-O'Connell[6] of component i

vCA is the characteristic volume from Brelvi-O'Connell[6] of solvent A

v0A is the liquid molar volume of pure solvent A calculated from the temperature-dependent property correlation for liquid density for pure solvent A

The temperature-dependent property correlations for liquid density are defined by the pure component (PURECOMP) data bank that the Fluid Type uses and by the local thermodynamic data overrides specified on the Temperature Dependent tab in the Component Data section of the Fluid Editor. Refer to Override temperature-dependent property data for more information.

You can also provide the characteristic volume data for components as temperature-dependent property data on the Temperature Dependent tab. If data is not available for a component, we fill the characteristic volume for the component (vCi) with the critical volume (Vci) data.

Changes to the Liquid Density method-override option in the Fluid Editor do not affect these calculations. See Effects of specifying thermodynamic method overrides for more information.

For Henry's solutes that participate in the fluid reactions, the activity coefficients in the reaction equilibrium calculations are based on infinite dilution in aqueous solution while the Henry's constants in the phase equilibrium calculations are based on infinite dilution in the mixed solvent. For Henry's solutes that do not have any associated ternary interaction data, we do not include any activity coefficient contributions to the liquid phase fugacity.