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:

Setting a diffusivity model:

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

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

Maxwell-Stefan Diffusion Coefficients

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

Available Models

Wilke-Lee (Gases)

The Wilke-Lee method is based on Chapman-Enskog kinetic theory for gases.

Class: WilkeLeeDiffusivity

Equation: \(D_{ij} = \frac{(1.084 - 0.249\sqrt{1/M_i + 1/M_j}) \times 10^{-4} T^{1.5} \sqrt{1/M_i + 1/M_j}}{P \sigma_{ij}^2 \Omega_D}\)

where:

Collision diameter combining rule: \(\sigma_{ij} = \frac{\sigma_i + \sigma_j}{2}\)

Collision integral approximation: \(\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp(DT^*)} + \frac{E}{\exp(FT^*)} + \frac{G}{\exp(HT^*)}\)

where $T^* = k_B T / \epsilon_{ij}$.

Applicable phases: Gas

Best for:

Usage:

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

Siddiqi-Lucas (Liquids)

The Siddiqi-Lucas method is designed for liquid-phase binary diffusion.

Class: SiddiqiLucasMethod

Aqueous systems: \(D_{ij} = 2.98 \times 10^{-7} \frac{T}{\eta_j^{1.026} V_i^{0.5473}}\)

Non-aqueous systems: \(D_{ij} = 9.89 \times 10^{-8} \frac{T}{\eta_j^{0.907} V_i^{0.45} V_j^{-0.265}}\)

where:

Applicable phases: Liquid (aqueous and organic)

Best for:

Usage:

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

Corresponding States (CSP)

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

Class: CorrespondingStatesDiffusivity

Principle: Uses reduced temperature and density to correlate diffusion:

\[D^* = D \cdot \frac{\sigma^2}{(M/N_A) \sqrt{k_B T / M}}\]

The reduced diffusivity is correlated against reduced temperature and density.

Applicable phases: Gas, Liquid

Best for:

Usage:

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

Amine Diffusivity

Specialized correlations for amine solutions used in gas treating.

Class: AmineDiffusivity

Includes:

Applicable phases: Aqueous amine solutions

Best for:

Usage:

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

Model Selection Guide

Application Recommended Model Notes
Gas phase Wilke Lee Based on kinetic theory
Aqueous liquids Siddiqi Lucas Validated for water
Organic liquids Siddiqi Lucas Use non-aqueous correlation
Wide P-T range CSP Corresponding states
Amine systems Alkanol amine Specialized for gas treating
CO₂ in water CO2water model Specific correlation

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");
    }
}

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");
}

Comparing Gas and Liquid Diffusivity

SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("CO2", 0.1);
fluid.addComponent("n-octane", 0.9);
fluid.setMixingRule("classic");
fluid.setMultiPhaseCheck(true);

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

// Compare gas and liquid diffusivities
if (fluid.hasPhaseType("gas")) {
    double[][] Dgas = fluid.getPhase("gas").getPhysicalProperties()
        .getDiffusivityCalc().getBinaryDiffusionCoefficients();
    System.out.println("Gas D_CO2-octane: " + Dgas[0][1] + " m²/s");
}

if (fluid.hasPhaseType("oil")) {
    double[][] Dliq = fluid.getPhase("oil").getPhysicalProperties()
        .getDiffusivityCalc().getBinaryDiffusionCoefficients();
    System.out.println("Liquid D_CO2-octane: " + Dliq[0][1] + " m²/s");
}
// Gas diffusivity is typically 10,000x larger than liquid

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.

Temperature Dependence

Gases: \(D \propto T^{1.5}\)

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. Wilke, C.R., Lee, C.Y. (1955). Estimation of Diffusion Coefficients for Gases and Vapors. I&EC.
  2. Siddiqi, M.A., Lucas, K. (1986). Correlations for Prediction of Diffusion in Liquids. Can. J. Chem. Eng.
  3. Fuller, E.N., et al. (1966). New Method for Prediction of Binary Gas-Phase Diffusion Coefficients. I&EC.
  4. Poling, B.E., et al. (2001). The Properties of Gases and Liquids, 5th Ed.