Diffusivity Models

This guide documents the diffusion coefficient calculation methods available in NeqSim for gas and liquid systems.

Table of Contents

Overview

Diffusion coefficients describe the rate of molecular transport due to concentration gradients. They are essential for:

Units:

Available model names (for setDiffusionCoefficientModel):

Model Name Phase Class
"Chapman-Enskog" Gas Diffusivity
"Wilke Lee" Gas WilkeLeeDiffusivity
"Fuller-Schettler-Giddings" Gas FullerSchettlerGiddingsDiffusivity
"Siddiqi Lucas" Liquid SiddiqiLucasMethod
"Wilke-Chang" Liquid WilkeChangDiffusivity
"Tyn-Calus" Liquid TynCalusDiffusivity
"Hayduk-Minhas" Liquid HaydukMinhasDiffusivity
"CSP" Gas/Liquid CorrespondingStatesDiffusivity
"High Pressure" Liquid HighPressureDiffusivity
"Alkanol amine" Aqueous AmineDiffusivity

Setting a diffusivity model:

fluid.initPhysicalProperties();
fluid.getPhase("gas").getPhysicalProperties().setDiffusionCoefficientModel("Fuller-Schettler-Giddings");
fluid.getPhase("oil").getPhysicalProperties().setDiffusionCoefficientModel("Siddiqi Lucas");

Types of Diffusion Coefficients

Binary Diffusion Coefficient ($D_{ij}$)

The diffusion coefficient for species $i$ moving through species $j$ at infinite dilution.

Effective Diffusion Coefficient ($D_i^{eff}$)

The effective diffusivity of species $i$ in a multicomponent mixture (Wilke approximation):

\[D_i^{eff} = \frac{1 - x_i}{\sum_{j \neq i} \frac{x_j}{D_{ij}}}\]

Maxwell-Stefan Diffusion Coefficients

The fundamental diffusion coefficients describing molecular interactions, related to Fick diffusion through thermodynamic factors.

Gas-Phase Models

Chapman-Enskog (Default)

The default gas-phase model based on rigorous Chapman-Enskog kinetic theory.

Class: neqsim.physicalproperties.methods.gasphysicalproperties.diffusivity.Diffusivity

Equation:

\[D_{ij} = \frac{0.00266 \, T^{3/2}}{P \sqrt{M_{ij}} \, \sigma_{ij}^2 \, \Omega_D}\]

where:

Collision integral:

\[\Omega_D = \frac{1.06036}{(T^*)^{0.15610}} + \frac{0.19300}{\exp(0.47635 \, T^*)} + \frac{1.03587}{\exp(1.52996 \, T^*)} + \frac{1.76474}{\exp(3.89411 \, T^*)}\]

Key features:

Usage:

fluid.getPhase("gas").getPhysicalProperties().setDiffusionCoefficientModel("Chapman-Enskog");

Wilke-Lee

Modified Chapman-Enskog with the Wilke-Lee empirical correction factor for improved accuracy.

Class: WilkeLeeDiffusivity

Equation:

\[D_{ij} = \frac{(1.084 - 0.249\sqrt{1/M_i + 1/M_j}) \times 10^{-4} \, T^{3/2} \sqrt{1/M_i + 1/M_j}}{P \, \sigma_{ij}^2 \, \Omega_D}\]

The prefactor $(1.084 - 0.249\sqrt{1/M_i + 1/M_j})$ is an empirical correction that slightly improves accuracy compared to standard Chapman-Enskog, particularly for asymmetric molecular pairs.

Key features:

Usage:

fluid.getPhase("gas").getPhysicalProperties().setDiffusionCoefficientModel("Wilke Lee");

Fuller-Schettler-Giddings

A widely used empirical method that avoids Lennard-Jones parameters entirely, using atomic diffusion volumes instead.

Class: FullerSchettlerGiddingsDiffusivity

Equation:

\[D_{ij} = \frac{1.013 \times 10^{-3} \, T^{1.75} \sqrt{1/M_i + 1/M_j}}{P \left((\sum v_i)^{1/3} + (\sum v_j)^{1/3}\right)^2}\]

where:

Diffusion volume table (selected values):

Species $\sum v$
H₂ 7.07
N₂ 17.9
O₂ 16.6
CO₂ 26.9
H₂O 12.7
CH₄ 25.14
C₂H₆ 45.66
C₃H₈ 66.18

For components not in the table, the critical volume is used as fallback: $\sum v \approx 0.285 \, V_c$ where $V_c$ is in cm³/mol.

Key features:

Usage:

fluid.getPhase("gas").getPhysicalProperties()
    .setDiffusionCoefficientModel("Fuller-Schettler-Giddings");

Liquid-Phase Models

All liquid-phase models calculate infinite-dilution binary diffusion coefficients. The Vignes mixing rule is applied for finite concentrations:

\[D_{ij}^{mix} = (D_{ij}^0)^{x_j} (D_{ji}^0)^{x_i}\]

Siddiqi-Lucas

Correlates liquid diffusion with solvent viscosity and solute molar volume.

Class: SiddiqiLucasMethod

Aqueous systems (water as solvent):

\[D_{ij}^0 = 2.98 \times 10^{-7} \frac{T}{\eta_j^{1.026} \, V_i^{0.5473}}\]

Non-aqueous systems:

\[D_{ij}^0 = 9.89 \times 10^{-8} \frac{T}{\eta_j^{0.907} \, V_i^{0.45} \, V_j^{-0.265}}\]

where:

Key features:

Usage:

fluid.getPhase("aqueous").getPhysicalProperties()
    .setDiffusionCoefficientModel("Siddiqi Lucas");

Wilke-Chang

Classic correlation relating diffusion to solvent association and viscosity.

Class: WilkeChangDiffusivity

Equation:

\[D_{ij}^0 = \frac{7.4 \times 10^{-8} (\phi_j M_j)^{0.5} T}{\eta_j \, V_i^{0.6}}\]

where:

Key features:

Usage:

fluid.getPhase("aqueous").getPhysicalProperties()
    .setDiffusionCoefficientModel("Wilke-Chang");

Tyn-Calus

Uses parachor-based molar volumes for improved accuracy.

Class: TynCalusDiffusivity

Equation:

\[D_{ij}^0 = 8.93 \times 10^{-8} \frac{V_j^{0.267}}{V_i^{0.433}} \frac{T}{\eta_j}\]

where:

Key features:

Usage:

fluid.getPhase("oil").getPhysicalProperties()
    .setDiffusionCoefficientModel("Tyn-Calus");

Hayduk-Minhas

Specialized for paraffin solutions and aqueous systems with different correlations for each.

Class: HaydukMinhasDiffusivity

Paraffin solutions:

\[D_{ij}^0 = 13.3 \times 10^{-8} \frac{T^{1.47} \eta_j^{(\epsilon)}}{V_i^{0.71}}\]

where $\epsilon = 10.2/V_i - 0.791$.

Aqueous solutions:

\[D_{ij}^0 = 1.25 \times 10^{-8} (V_i^{-0.19} - 0.292) T^{1.52} \eta_j^{9.58/V_i - 1.12}\]

Key features:

Usage:

fluid.getPhase("oil").getPhysicalProperties()
    .setDiffusionCoefficientModel("Hayduk-Minhas");

General Models

Corresponding States (CSP)

A generalized corresponding states method for both gas and liquid diffusion.

Class: CorrespondingStatesDiffusivity

Uses reduced temperature and density to correlate diffusion coefficients, consistent with the thermodynamic model.

Usage:

fluid.getPhase("gas").getPhysicalProperties().setDiffusionCoefficientModel("CSP");

High Pressure

Pressure-corrected liquid diffusion model for conditions where the standard low-pressure correlations break down.

Class: HighPressureDiffusivity

Usage:

fluid.getPhase("oil").getPhysicalProperties().setDiffusionCoefficientModel("High Pressure");

Amine Diffusivity

Specialized correlations for amine solutions used in gas treating.

Class: AmineDiffusivity

Includes:

Usage:

fluid.getPhase("aqueous").getPhysicalProperties()
    .setDiffusionCoefficientModel("Alkanol amine");

Lennard-Jones Parameters

The Chapman-Enskog and Wilke-Lee models require Lennard-Jones parameters ($\sigma$, $\epsilon/k_B$) for each component. NeqSim uses a two-tier approach:

  1. Diffusion-specific parameters — A built-in lookup table with standard values from Poling, Prausnitz and O’Connell (2001) and Bird, Stewart and Lightfoot (2002), covering ~35 common components. These are optimized for transport property calculations.

  2. Database fallback — For components not in the diffusion lookup table, NeqSim uses the LJ parameters stored in the component database. These may be optimized for equation-of-state calculations and can give less accurate diffusion coefficients.

Components with diffusion-specific LJ parameters:

Component $\sigma$ (Å) $\epsilon/k_B$ (K)
Helium 2.551 10.22
Hydrogen 2.827 59.7
Nitrogen 3.798 71.4
Oxygen 3.467 106.7
CO₂ 3.941 195.2
H₂S 3.623 301.1
Methane 3.758 148.6
Ethane 4.443 215.7
Propane 5.118 237.1
Water 2.641 809.1
Benzene 5.349 412.3
Methanol 3.626 481.8

The full list includes noble gases, halogenated compounds, and hydrocarbons up to n-octane.

Validation Against Experimental Data

All models have been validated against published experimental data. The table below shows results at 298.15 K and 1 atm.

Gas-Phase Validation

System Experimental (cm²/s) Chapman-Enskog Fuller Wilke-Lee Source
CH₄-N₂ 0.220 0.219 (0.7%) 0.216 (2.0%) 0.231 (5.0%) Marrero and Mason (1972)
CO₂-N₂ 0.167 0.155 (7.4%) 0.162 (2.7%) 0.166 (0.3%) Poling (2001)

Gas model accuracy summary:

Liquid-Phase Validation (CO₂ in water at 25°C)

Model Calculated (10⁻⁹ m²/s) Error vs 1.92 × 10⁻⁹ m²/s
Siddiqi-Lucas 1.33 -31%
Wilke-Chang 1.73 -10%
Hayduk-Minhas 1.63 -15%

Liquid model accuracy summary:

Physical Consistency Tests

All models pass the following consistency checks:

Model Selection Guide

Application Recommended Model Model String Typical Accuracy
Gas phase (general) Fuller-Schettler-Giddings "Fuller-Schettler-Giddings" ±2-5%
Gas phase (light gases) Chapman-Enskog or Wilke-Lee "Chapman-Enskog" / "Wilke Lee" ±1-8%
Aqueous liquids Wilke-Chang "Wilke-Chang" ±10-15%
Organic liquids Tyn-Calus or Hayduk-Minhas "Tyn-Calus" / "Hayduk-Minhas" ±10-20%
Wide P-T range CSP "CSP" ±15-25%
Amine systems Alkanol amine "Alkanol amine" Specialized
High pressure liquids High Pressure "High Pressure" Pressure-corrected

Usage Examples

Accessing Binary Diffusion Coefficients

import neqsim.thermo.system.SystemSrkEos;
import neqsim.thermodynamicoperations.ThermodynamicOperations;

// Create and flash fluid
SystemInterface fluid = new SystemSrkEos(300.0, 10.0);
fluid.addComponent("methane", 0.9);
fluid.addComponent("ethane", 0.05);
fluid.addComponent("CO2", 0.05);
fluid.setMixingRule("classic");

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

// Initialize physical properties
fluid.initPhysicalProperties();

// Get binary diffusion coefficients
double[][] Dij = fluid.getPhase("gas").getPhysicalProperties()
    .getDiffusivityCalc().getBinaryDiffusionCoefficients();

// Print diffusion matrix
int n = fluid.getPhase("gas").getNumberOfComponents();
for (int i = 0; i < n; i++) {
    for (int j = 0; j < n; j++) {
        System.out.println("D[" + i + "][" + j + "] = " + Dij[i][j] + " m2/s");
    }
}

Switching Between Gas Models

SystemInterface fluid = new SystemSrkEos(298.15, 1.01325);
fluid.addComponent("methane", 0.5);
fluid.addComponent("nitrogen", 0.5);
fluid.setMixingRule("classic");

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

// Compare gas models
String[] models = {"Chapman-Enskog", "Wilke Lee", "Fuller-Schettler-Giddings"};
for (String model : models) {
    fluid.getPhase("gas").getPhysicalProperties().setDiffusionCoefficientModel(model);
    double D = fluid.getPhase("gas").getPhysicalProperties()
        .diffusivityCalc.calcBinaryDiffusionCoefficient(0, 1, 0);
    System.out.println(model + ": D(CH4-N2) = " + D + " m2/s");
}

Liquid-Phase Diffusion in Water

SystemInterface fluid = new SystemSrkEos(298.15, 1.01325);
fluid.addComponent("CO2", 0.01);
fluid.addComponent("water", 0.99);
fluid.createDatabase(true);
fluid.setMixingRule(2);

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

// Compare liquid models
String[] models = {"Siddiqi Lucas", "Wilke-Chang", "Hayduk-Minhas", "Tyn-Calus"};
for (String model : models) {
    fluid.getPhase("aqueous").getPhysicalProperties().setDiffusionCoefficientModel(model);
    double D = fluid.getPhase("aqueous").getPhysicalProperties()
        .diffusivityCalc.calcBinaryDiffusionCoefficient(0, 1, 0);
    System.out.println(model + ": D(CO2-water) = " + D + " m2/s");
}

Effective Diffusivity

// Get effective diffusion coefficient
double[] Deff = fluid.getPhase("gas").getPhysicalProperties()
    .getDiffusivityCalc().getEffectiveDiffusionCoefficient();

for (int i = 0; i < n; i++) {
    String name = fluid.getPhase("gas").getComponent(i).getName();
    System.out.println("D_eff[" + name + "] = " + Deff[i] + " m2/s");
}

Diffusivity in Amine Solutions

SystemInterface fluid = new SystemSrkCPAstatoil(313.15, 1.0);
fluid.addComponent("CO2", 0.1);
fluid.addComponent("water", 0.7);
fluid.addComponent("MDEA", 0.2);
fluid.setMixingRule(10);

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

// Use amine-specific diffusivity model
fluid.initPhysicalProperties("AMINE");

double[][] D = fluid.getPhase("aqueous").getPhysicalProperties()
    .getDiffusivityCalc().getBinaryDiffusionCoefficients();

System.out.println("D_CO2 in amine: " + D[0][1] + " m2/s");

Physical Background

Pressure Dependence

Gases: \(D \propto \frac{1}{P}\)

At constant temperature, gas diffusivity is inversely proportional to pressure.

Liquids: Weak pressure dependence; can often be neglected at moderate pressures. Use the "High Pressure" model for elevated pressures.

Temperature Dependence

Gases: \(D \propto T^{1.5} \text{ (Chapman-Enskog/Wilke-Lee)} \quad \text{or} \quad D \propto T^{1.75} \text{ (Fuller)}\)

Liquids: \(D \propto T / \eta\)

Since viscosity decreases with temperature, liquid diffusivity increases.

Typical Values

Phase Diffusivity Range
Gas (1 bar) $10^{-5}$ to $10^{-4}$ m²/s
Gas (100 bar) $10^{-7}$ to $10^{-6}$ m²/s
Liquid $10^{-10}$ to $10^{-9}$ m²/s
Supercritical $10^{-8}$ to $10^{-7}$ m²/s

Multicomponent Diffusion

For multicomponent systems, the flux of species $i$ depends on gradients of all components:

\[J_i = -c_t \sum_{j=1}^{n} D_{ij} \nabla x_j\]

NeqSim calculates:

  1. Binary Maxwell-Stefan coefficients from pure component properties
  2. Effective diffusivities using the Wilke approximation
  3. Fick diffusion matrix (optional) including thermodynamic factors

References

  1. Chapman, S., Cowling, T.G. (1970). The Mathematical Theory of Non-uniform Gases, 3rd Ed. Cambridge University Press.
  2. Wilke, C.R., Lee, C.Y. (1955). Estimation of Diffusion Coefficients for Gases and Vapors. Ind. Eng. Chem., 47(6), 1253-1257.
  3. Fuller, E.N., Schettler, P.D., Giddings, J.C. (1966). New Method for Prediction of Binary Gas-Phase Diffusion Coefficients. Ind. Eng. Chem., 58(5), 18-27.
  4. Siddiqi, M.A., Lucas, K. (1986). Correlations for Prediction of Diffusion in Liquids. Can. J. Chem. Eng., 64, 839-843.
  5. Wilke, C.R., Chang, P. (1955). Correlation of Diffusion Coefficients in Dilute Solutions. AIChE J., 1(2), 264-270.
  6. Tyn, M.T., Calus, W.F. (1975). Diffusion Coefficients in Dilute Binary Liquid Mixtures. J. Chem. Eng. Data, 20(1), 106-109.
  7. Hayduk, W., Minhas, B.S. (1982). Correlations for Prediction of Molecular Diffusivities in Liquids. Can. J. Chem. Eng., 60, 295-299.
  8. Poling, B.E., Prausnitz, J.M., O’Connell, J.P. (2001). The Properties of Gases and Liquids, 5th Ed. McGraw-Hill.
  9. Bird, R.B., Stewart, W.E., Lightfoot, E.N. (2002). Transport Phenomena, 2nd Ed. Wiley.
  10. Marrero, T.R., Mason, E.A. (1972). Gaseous Diffusion Coefficients. J. Phys. Chem. Ref. Data, 1(1), 3-118.