Mixing Rules in NeqSim

This guide provides comprehensive documentation on mixing rules available in NeqSim, including mathematical formulations, usage patterns, and recommendations for different applications.

Introduction to Mixing Rules
Setting Mixing Rules
Classic Mixing Rules
Huron-Vidal Mixing Rules
Wong-Sandler Mixing Rule
CPA Mixing Rules
Søreide-Whitson Mixing Rule
Binary Interaction Parameters (kij)
Complete Reference Table
Examples by Application

1. Introduction to Mixing Rules

Mixing rules determine how pure-component parameters from an equation of state are combined to calculate mixture properties. For cubic equations of state like SRK and PR, the key parameters are:

a: Energy parameter (attraction term)
b: Co-volume parameter (repulsion term)

The general form of cubic EoS mixing rules is:

\[a_{mix} = \sum_i \sum_j x_i x_j a_{ij}\] \[b_{mix} = \sum_i x_i b_i\]

Where the cross-parameter $a_{ij}$ is typically calculated using a combining rule:

\[a_{ij} = \sqrt{a_i a_j}(1 - k_{ij})\]

The binary interaction parameter $k_{ij}$ is crucial for accurate phase equilibrium predictions.

2. Setting Mixing Rules

NeqSim provides three ways to set mixing rules:

2.1 By Integer Value

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.8);
fluid.addComponent("CO2", 0.2);
fluid.setMixingRule(2);  // Classic mixing rule

2.2 By Name (String)

fluid.setMixingRule("classic");
fluid.setMixingRule("HV");
fluid.setMixingRule("WS");

2.3 By Enum Type

import neqsim.thermo.mixingrule.EosMixingRuleType;

fluid.setMixingRule(EosMixingRuleType.CLASSIC);
fluid.setMixingRule(EosMixingRuleType.byName("classic"));
fluid.setMixingRule(EosMixingRuleType.byValue(2));

2.4 With GE Model (for Huron-Vidal/Wong-Sandler)

// Huron-Vidal with UNIFAC
fluid.setMixingRule("HV", "UNIFAC_UMRPRU");

// Wong-Sandler with NRTL
fluid.setMixingRule("WS", "NRTL");

3. Classic Mixing Rules

3.1 NO (kij = 0) - Type 1

The simplest mixing rule with no binary interactions. All $k_{ij}$ values are set to zero.

\[a_{ij} = \sqrt{a_i a_j}\]

Use case: Quick calculations, ideal mixture approximations, or when no interaction parameters are available.

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.7);
fluid.addComponent("ethane", 0.3);
fluid.setMixingRule(1);  // or fluid.setMixingRule("NO");

3.2 CLASSIC - Type 2

The standard van der Waals one-fluid mixing rule with binary interaction parameters from the NeqSim database.

\[a_{mix} = \sum_i \sum_j x_i x_j \sqrt{a_i a_j}(1 - k_{ij})\] \[b_{mix} = \sum_i x_i b_i\]

Use case: General hydrocarbon systems, natural gas processing, most industrial applications.

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.85);
fluid.addComponent("CO2", 0.10);
fluid.addComponent("nitrogen", 0.05);
fluid.setMixingRule(2);  // or fluid.setMixingRule("classic");

3.3 CLASSIC_T - Type 8

Classic mixing rule with temperature-dependent binary interaction parameters:

\[k_{ij}(T) = k_{ij,0} + k_{ij,T} \cdot T\]

Where $k_{ij,0}$ is the reference value and $k_{ij,T}$ is the temperature coefficient.

Use case: Systems where kij varies significantly with temperature.

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.8);
fluid.addComponent("H2S", 0.2);
fluid.setMixingRule(8);  // Temperature-dependent classic

3.4 CLASSIC_T2 - Type 12

Alternative temperature-dependent formulation:

\[k_{ij}(T) = k_{ij,0} + \frac{k_{ij,T}}{T}\]

Use case: Systems requiring inverse temperature dependency.

fluid.setMixingRule(12);

4. Huron-Vidal Mixing Rules

Huron-Vidal (HV) mixing rules combine cubic EoS with activity coefficient models (like NRTL or UNIFAC) for improved liquid-phase behavior.

4.1 Mathematical Framework

The excess Gibbs energy from an activity coefficient model is incorporated into the EoS:

\[a_{mix} = b_{mix} \left( \sum_i x_i \frac{a_i}{b_i} - \frac{G^E}{\Lambda} \right)\]

Where:

$G^E$ is the excess Gibbs energy from the activity coefficient model
$\Lambda$ is the EoS-specific constant (≈ 0.693 for SRK)

4.2 CLASSIC_HV - Type 3

Basic Huron-Vidal mixing rule with NRTL parameters from the database.

SystemInterface fluid = new SystemSrkEos(350.0, 10.0);
fluid.addComponent("methanol", 0.3);
fluid.addComponent("water", 0.4);
fluid.addComponent("methane", 0.3);
fluid.setMixingRule(3);  // or fluid.setMixingRule("CLASSIC_HV");

4.3 HV - Type 4

Enhanced Huron-Vidal with temperature-dependent parameters (HVDijT):

\[D_{ij}(T) = D_{ij,0} + D_{ij,T} \cdot T\]

Use case: Polar/non-polar mixtures, alcohol-water-hydrocarbon systems.

SystemInterface fluid = new SystemPrEos(320.0, 20.0);
fluid.addComponent("ethanol", 0.2);
fluid.addComponent("water", 0.5);
fluid.addComponent("propane", 0.3);
fluid.setMixingRule(4);  // or fluid.setMixingRule("HV");

4.4 Huron-Vidal with UNIFAC

For predictive calculations without fitted parameters, use UNIFAC:

SystemInterface fluid = new SystemSrkEos(320.0, 15.0);
fluid.addComponent("ethanol", 0.3);
fluid.addComponent("n-hexane", 0.4);
fluid.addComponent("water", 0.3);

// HV with UNIFAC activity coefficients
fluid.setMixingRule("HV", "UNIFAC_UMRPRU");

Available GE models for HV:

"NRTL" - Non-Random Two-Liquid (default)
"UNIFAC_UMRPRU" - UNIFAC with UMR-PRU parameters
"UNIFAC" - Standard UNIFAC
"UNIFAC_PSRK" - UNIFAC with PSRK parameters

4.5 Accessing HV Parameters

// Get the mixing rule interface
HVMixingRulesInterface hvRule = (HVMixingRulesInterface) fluid.getPhase(0).getMixingRule();

// Get/Set HV parameters
double dij = hvRule.getHVDijParameter(0, 1);
hvRule.setHVDijParameter(0, 1, 500.0);  // Set Dij in K

double alpha = hvRule.getHValphaParameter(0, 1);
hvRule.setHValphaParameter(0, 1, 0.3);  // Set alpha (non-randomness)

5. Wong-Sandler Mixing Rule

5.1 Overview

The Wong-Sandler (WS) mixing rule provides theoretically correct behavior at both low and high densities by matching:

The second virial coefficient of the mixture
The excess Gibbs energy at infinite pressure

5.2 Mathematical Formulation

\[b_{mix} = \frac{\sum_i \sum_j x_i x_j \left( b - \frac{a}{RT} \right)_{ij}}{1 - \frac{A_\infty^E}{CRT} - \sum_i x_i \frac{a_i}{RTb_i}}\] \[a_{mix} = b_{mix} \left( \sum_i x_i \frac{a_i}{b_i} + \frac{A_\infty^E}{C} \right)\]

Where $C$ is an EoS-specific constant and $A_\infty^E$ is the excess Helmholtz energy at infinite pressure.

5.3 WS - Type 5

SystemInterface fluid = new SystemPrEos(350.0, 30.0);
fluid.addComponent("methanol", 0.2);
fluid.addComponent("water", 0.5);
fluid.addComponent("CO2", 0.3);
fluid.setMixingRule(5);  // or fluid.setMixingRule("WS");

5.4 Wong-Sandler with NRTL

SystemInterface fluid = new SystemSrkEos(320.0, 20.0);
fluid.addComponent("acetone", 0.3);
fluid.addComponent("water", 0.4);
fluid.addComponent("methane", 0.3);

// WS with NRTL activity coefficients
fluid.setMixingRule("WS", "NRTL");

5.5 Accessing WS Parameters

HVMixingRulesInterface wsRule = (HVMixingRulesInterface) fluid.getPhase(0).getMixingRule();

// Get/Set Wong-Sandler kij parameter
double kijWS = wsRule.getKijWongSandler(0, 1);
wsRule.setKijWongSandler(0, 1, 0.1);

Use case: High-pressure VLE, polar/non-polar mixtures, CO2 capture systems.

6. CPA Mixing Rules

For Cubic-Plus-Association (CPA) equations of state, specialized mixing rules handle both the cubic EoS part and the association term.

6.1 CPA_MIX - Type 7

Classic mixing rule with CPA-specific binary interaction parameters from the database.

SystemInterface fluid = new SystemSrkCPAstatoil(320.0, 50.0);
fluid.addComponent("water", 0.1);
fluid.addComponent("methane", 0.8);
fluid.addComponent("MEG", 0.1);
fluid.setMixingRule(7);  // CPA classic mixing

6.2 CLASSIC_T_CPA - Type 9

Temperature-dependent classic mixing rule for CPA systems:

fluid.setMixingRule(9);

6.3 CLASSIC_TX_CPA - Type 10

Temperature and composition dependent mixing rule for CPA. This is the recommended mixing rule for CPA systems as it:

Uses asymmetric kij parameters (kij ≠ kji)
Includes temperature dependency
Automatically selects the appropriate sub-type based on parameter symmetry

SystemInterface fluid = new SystemSrkCPAstatoil(330.0, 100.0);
fluid.addComponent("water", 0.15);
fluid.addComponent("methane", 0.70);
fluid.addComponent("MEG", 0.10);
fluid.addComponent("CO2", 0.05);
fluid.setMixingRule(10);  // Recommended for CPA

6.4 Cross-Association Parameters

CPA mixing rules also handle association site interactions between different molecules:

Association Scheme	Description	Examples
CR-1	Cross-association with single site	Water-MEG
ER	Elliott combining rule	General polar systems
4C	Four-site model	Water, glycols
2B	Two-site model	Alcohols

7. Søreide-Whitson Mixing Rule

7.1 Overview

The Søreide-Whitson mixing rule is specifically designed for systems containing:

Sour gases (CO2, H2S, N2)
Hydrocarbons
Aqueous brine solutions

It uses composition and salinity-dependent binary interaction parameters.

7.2 SOREIDE_WHITSON - Type 11

import neqsim.thermo.system.SystemSoreideWhitson;

SystemSoreideWhitson fluid = new SystemSoreideWhitson(350.0, 200.0);
fluid.addComponent("methane", 0.70);
fluid.addComponent("CO2", 0.15);
fluid.addComponent("H2S", 0.05);
fluid.addComponent("water", 0.10);

// Add salinity (important for sour gas solubility)
fluid.addSalinity(2.0, "mole/sec");

fluid.setMixingRule(11);  // Søreide-Whitson mixing rule

7.3 Salinity-Dependent kij

The mixing rule calculates kij for water-gas interactions based on salinity (S in mol/kg):

For CO2-water: $k_{ij} = -0.31092(1 + 0.156 S^{0.75}) + 0.236(1 + 0.178 S^{0.98}) T_r - 21.26 e^{-6.72^{T_r} - S}$

For N2-water: $k_{ij} = -1.702(1 + 0.026 S^{0.75}) + 0.443(1 + 0.081 S^{0.75}) T_r$

For hydrocarbons-water: $k_{ij} = (1 + a_0 S) A_0 + (1 + a_1 S) A_1 T_r + (1 + a_2 S) A_2 T_r^2$

Where $T_r$ is the reduced temperature and the $A$ and $a$ coefficients depend on the acentric factor.

Use case: Reservoir fluids with aqueous phases, sour gas processing, CCS with brine.

8. Binary Interaction Parameters (kij)

8.1 Default Database Values

NeqSim loads kij values from its internal database when you call setMixingRule(). These are stored in the inter table.

8.2 Setting Custom kij Values

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.9);
fluid.addComponent("CO2", 0.1);
fluid.setMixingRule("classic");

// Get the mixing rule interface
EosMixingRulesInterface mixRule = fluid.getPhase(0).getMixingRule();

// Set custom kij (symmetric: kij = kji)
mixRule.setBinaryInteractionParameter(0, 1, 0.12);

// Get current kij value
double kij = mixRule.getBinaryInteractionParameter(0, 1);
System.out.println("kij(CH4-CO2) = " + kij);

8.3 Setting Asymmetric kij (for CPA)

// For CPA systems with asymmetric parameters
mixRule.setBinaryInteractionParameterij(0, 1, 0.08);  // kij
mixRule.setBinaryInteractionParameterji(0, 1, 0.12);  // kji

8.4 Setting Temperature-Dependent kij

// For mixing rules 8, 9, 12
// kij(T) = kij0 + kijT * f(T)
mixRule.setBinaryInteractionParameter(0, 1, 0.10);    // kij0
mixRule.setBinaryInteractionParameterT1(0, 1, 0.001); // kijT

8.5 Displaying kij Matrix

double[][] kijMatrix = mixRule.getBinaryInteractionParameters();
for (int i = 0; i < fluid.getNumberOfComponents(); i++) {
    for (int j = 0; j < fluid.getNumberOfComponents(); j++) {
        System.out.printf("kij[%d][%d] = %.4f  ", i, j, kijMatrix[i][j]);
    }
    System.out.println();
}

9. Complete Reference Table

Type	Name	String Key	Description	Best For
1	`NO`	`"NO"`	All kij = 0	Quick estimates, ideal mixtures
2	`CLASSIC`	`"classic"`	Classic van der Waals with database kij	General hydrocarbons
3	`CLASSIC_HV`	`"CLASSIC_HV"`	Huron-Vidal with database parameters	Polar mixtures
4	`HV`	`"HV"`	Huron-Vidal with T-dependent parameters	Alcohol-water-HC
5	`WS`	`"WS"`	Wong-Sandler	High-pressure polar systems
7	`CPA_MIX`	`"CPA_MIX"`	Classic for CPA systems	Water-HC with CPA
8	`CLASSIC_T`	`"CLASSIC_T"`	T-dependent kij (linear)	Temperature-sensitive kij
9	`CLASSIC_T_CPA`	`"CLASSIC_T_CPA"`	T-dependent for CPA	CPA with T-dependency
10	`CLASSIC_TX_CPA`	`"CLASSIC_TX_CPA"`	T and x dependent for CPA	Recommended for CPA
11	`SOREIDE_WHITSON`	`"SOREIDE_WHITSON"`	Salinity-dependent	Sour gas, brine systems
12	`CLASSIC_T2`	`"CLASSIC_T2"`	T-dependent (inverse)	Alternative T-dependency

10. Examples by Application

10.1 Natural Gas Processing

SystemInterface gas = new SystemSrkEos(280.0, 70.0);
gas.addComponent("nitrogen", 0.02);
gas.addComponent("CO2", 0.03);
gas.addComponent("methane", 0.85);
gas.addComponent("ethane", 0.06);
gas.addComponent("propane", 0.04);
gas.setMixingRule("classic");  // Type 2

ThermodynamicOperations ops = new ThermodynamicOperations(gas);
ops.TPflash();

10.2 Glycol Dehydration (CPA)

SystemInterface fluid = new SystemSrkCPAstatoil(320.0, 50.0);
fluid.addComponent("methane", 0.80);
fluid.addComponent("water", 0.05);
fluid.addComponent("TEG", 0.15);  // Triethylene glycol
fluid.setMixingRule(10);  // CLASSIC_TX_CPA - recommended for CPA

ThermodynamicOperations ops = new ThermodynamicOperations(fluid);
ops.TPflash();

10.3 Amine Gas Treating

SystemInterface fluid = new SystemSrkEos(323.0, 20.0);
fluid.addComponent("CO2", 0.15);
fluid.addComponent("H2S", 0.05);
fluid.addComponent("methane", 0.60);
fluid.addComponent("MDEA", 0.10);  // Methyldiethanolamine
fluid.addComponent("water", 0.10);
fluid.setMixingRule("HV", "NRTL");  // Huron-Vidal with NRTL

ThermodynamicOperations ops = new ThermodynamicOperations(fluid);
ops.TPflash();

10.4 CO2 Capture and Storage

SystemInterface fluid = new SystemPrEos(350.0, 150.0);
fluid.addComponent("CO2", 0.90);
fluid.addComponent("nitrogen", 0.05);
fluid.addComponent("oxygen", 0.02);
fluid.addComponent("water", 0.03);
fluid.setMixingRule("WS");  // Wong-Sandler for high pressure

ThermodynamicOperations ops = new ThermodynamicOperations(fluid);
ops.TPflash();

10.5 Sour Gas with Brine

SystemSoreideWhitson fluid = new SystemSoreideWhitson(380.0, 250.0);
fluid.addComponent("methane", 0.65);
fluid.addComponent("CO2", 0.20);
fluid.addComponent("H2S", 0.05);
fluid.addComponent("water", 0.10);
fluid.addSalinity("NaCl", 3.0, "mole/sec");  // Add NaCl salinity
fluid.setMixingRule(11);  // Søreide-Whitson

ThermodynamicOperations ops = new ThermodynamicOperations(fluid);
ops.TPflash();

System.out.println("Number of phases: " + fluid.getNumberOfPhases());
fluid.prettyPrint();

10.6 Pharmaceutical/Chemical (UNIFAC Predictive)

SystemInterface fluid = new SystemSrkEos(298.0, 1.0);
fluid.addComponent("ethanol", 0.3);
fluid.addComponent("acetone", 0.3);
fluid.addComponent("water", 0.4);
fluid.setMixingRule("HV", "UNIFAC_UMRPRU");  // Predictive UNIFAC

ThermodynamicOperations ops = new ThermodynamicOperations(fluid);
ops.TPflash();