Mass Transfer Modeling in NeqSim

This document provides detailed documentation of the mass transfer models implemented in the NeqSim fluid mechanics package, focusing on diffusivity correlations and reactive mass transfer.

Related Documentation:

Table of Contents

Overview

NeqSim implements rigorous multicomponent mass transfer models based on non-equilibrium thermodynamics. The models are suitable for:

The theoretical foundation is described in:

Solbraa, E. (2002). Equilibrium and Non-Equilibrium Thermodynamics of Natural Gas Processing. Dr.ing. thesis, NTNU. Available at NVA

Theoretical Background

Fick’s Law vs Maxwell-Stefan

For binary systems, Fick’s law is adequate:

\[J_A = -D_{AB} \cdot c_t \cdot \nabla x_A\]

For multicomponent systems, NeqSim uses the Maxwell-Stefan equations:

\[-\frac{c_t}{RT} \nabla \mu_i = \sum_{j=1, j \neq i}^{n} \frac{x_i N_j - x_j N_i}{c_t D_{ij}}\]

Where:

Film Theory

The film theory assumes mass transfer occurs across a stagnant film of thickness $\delta$:

\[N_i = k_i \cdot c_t \cdot (x_{i,bulk} - x_{i,interface})\]

Where $k_i = D_i / \delta$ is the mass transfer coefficient.

Penetration Theory

For unsteady-state mass transfer (short contact times):

\[k_L = 2 \sqrt{\frac{D}{\pi t_c}}\]

Where $t_c$ is the contact time.

Single-Phase Mass Transfer

Wall Mass Transfer

Mass transfer from a flowing fluid to a solid wall (pipe wall, packing surface):

Dimensionless Numbers

Number Definition Physical Meaning
Sherwood (Sh) $k_c \cdot d / D$ Ratio of convective to diffusive mass transfer
Schmidt (Sc) $\nu / D$ Ratio of momentum to mass diffusivity
Reynolds (Re) $\rho v d / \mu$ Ratio of inertial to viscous forces

Correlations

Laminar Flow (Re < 2300): \(Sh = 3.66\)

Turbulent Flow (Re > 10000): \(Sh = 0.023 \cdot Re^{0.83} \cdot Sc^{0.33}\)

Transition Region (2300 < Re < 4000): Linear interpolation between laminar and turbulent values.

Diffusion Coefficients

Gas Phase

Chapman-Enskog theory for binary diffusion:

\[D_{AB} = \frac{0.00266 \cdot T^{3/2}}{P \cdot M_{AB}^{1/2} \cdot \sigma_{AB}^2 \cdot \Omega_D}\]

Where:

Liquid Phase

NeqSim provides multiple liquid-phase diffusivity models, each optimized for different applications:

1. Wilke-Chang Correlation (Default)
\[D_{AB} = 7.4 \times 10^{-8} \cdot \frac{(\phi M_B)^{0.5} \cdot T}{\mu_B \cdot V_A^{0.6}}\]

Where:

2. Siddiqi-Lucas Method

Uses group contribution based on molecular weight and solvent viscosity:

Best for: General aqueous and organic liquid systems at low to moderate pressures.

3. Hayduk-Minhas Method

Optimized for hydrocarbon systems (Hayduk & Minhas, 1982):

Best for: Hydrocarbon-hydrocarbon diffusion in oil/gas applications.

// Example: Using Hayduk-Minhas for oil system
PhysicalProperties physProps = system.getPhase(1).getPhysicalProperties();
Diffusivity diffModel = new HaydukMinhasDiffusivity(physProps);
diffModel.calcDiffusionCoefficients(0, 0);  // binaryMethod, multicomponentMethod
double Dij = diffModel.getMaxwellStefanBinaryDiffusionCoefficient(0, 1);
4. CO2-Water (Tamimi Correlation)

Specialized for CO2 diffusion in water, validated against experimental data:

\[D_{CO_2} = 2.35 \times 10^{-6} \cdot \exp\left(\frac{-2119}{T}\right)\]

Best for: Carbon capture applications, CO2 absorption/desorption studies.

5. High-Pressure Correction

For reservoir and deep-water conditions (>100 bar), apply Mathur-Thodos correction:

\[D_P = D_0 \cdot f(\rho_r)\]

The correction factor accounts for increased molecular crowding at high pressures and can reduce diffusivity by 10× at 400 bar.

// Example: High-pressure diffusivity
PhysicalProperties physProps = system.getPhase(1).getPhysicalProperties();
HighPressureDiffusivity hpModel = new HighPressureDiffusivity(physProps);
hpModel.calcDiffusionCoefficients(0, 0);  // applies HP correction automatically
double correctionFactor = hpModel.getPressureCorrectionFactor();

Model Selection Guide

Application Recommended Model Notes
General aqueous Siddiqi-Lucas (aqueous) Well-validated for dilute solutions
General organic Siddiqi-Lucas (non-aqueous) Good for organic solvents
Oil/gas hydrocarbons Hayduk-Minhas (paraffin) 2-3× higher than Siddiqi-Lucas, physically appropriate for oils
CO2 in water CO2-water (Tamimi) Best accuracy (±11% of literature)
Reservoir conditions High-pressure + Hayduk-Minhas Critical for P > 100 bar

Model Comparison Results

Based on validation testing at 300 K, 1 atm for CO2 in water (literature: 1.9×10⁻⁹ m²/s):

Model Predicted (m²/s) Error
Hayduk-Minhas (aqueous) 1.71×10⁻⁹ -10%
CO2-water (Tamimi) 2.12×10⁻⁹ +11%
Siddiqi-Lucas 1.39×10⁻⁹ -27%

For hydrocarbon systems, Hayduk-Minhas produces values 2-3.5× higher than Siddiqi-Lucas, which is consistent with the different physical basis of the correlations.

Automatic Model Selection

The DiffusivityModelSelector class can automatically choose the optimal model:

// Automatic model selection based on composition and conditions
PhaseInterface phase = system.getPhase(1);
PhysicalProperties physProps = phase.getPhysicalProperties();
DiffusivityModelSelector.DiffusivityModelType modelType = 
    DiffusivityModelSelector.selectOptimalModel(phase);
Diffusivity model = DiffusivityModelSelector.createModel(physProps, modelType);

// Or use auto-selection directly:
Diffusivity autoModel = DiffusivityModelSelector.createAutoSelectedModel(physProps);

Selection criteria:

Future Development Possibilities

  1. Concentration-dependent diffusivity (Vignes mixing rule): \(D_{AB,mix} = D_{AB}^{x_B} \cdot D_{BA}^{x_A}\) Currently implemented but may cause numerical issues with very different diffusivities.

  2. Binary interaction parameters: Allow user-tuning for specific component pairs.

  3. Additional correlations:
    • Tyn-Calus for associated liquids
    • Scheibel for high-viscosity systems
    • He-Yu for supercritical fluids
  4. Temperature extrapolation warnings: Alert users when operating outside correlation validity ranges (typically 273-400 K).

Two-Phase Mass Transfer

Interphase Mass Transfer

At the gas-liquid interface, mass transfer occurs from both sides:

    Gas Bulk     |  Interface  |    Liquid Bulk
                 |             |
    x_i,G,bulk --|-- x_i,I ----|-- x_i,L,bulk
                 |             |
      k_G        |  Equilibrium|      k_L
                 |   K_i       |

Two-Resistance Model

The overall mass transfer coefficient combines gas and liquid resistances:

\[\frac{1}{K_{OG}} = \frac{1}{k_G} + \frac{m}{k_L}\] \[\frac{1}{K_{OL}} = \frac{1}{k_L} + \frac{1}{m \cdot k_G}\]

Where $m = dy/dx$ is the slope of the equilibrium line.

Flow Regime Dependence

The mass transfer coefficients depend strongly on the flow regime:

Flow Regime Interfacial Area Gas-side $k_G$ Liquid-side $k_L$
Stratified $A_i = W \cdot L$ (flat interface) Smooth surface correlation Penetration theory
Annular $A_i = \pi d L$ (film on wall) Core flow correlation Film flow correlation
Droplet/Mist $A_i = 6\epsilon/d_p$ (droplet surface) External mass transfer Internal circulation
Bubble $A_i = 6\epsilon/d_b$ (bubble surface) External mass transfer Higbie penetration
Slug Combined film + slug Varies with position Varies with position

Stratified Flow

For stratified gas-liquid flow in pipes:

Gas-side (smooth interface): \(Sh_G = 0.023 \cdot Re_G^{0.83} \cdot Sc_G^{0.33}\)

Liquid-side (penetration theory): \(k_L = 2 \sqrt{\frac{D_L \cdot v_L}{\pi \cdot L}}\)

Annular Flow

For annular flow with liquid film:

Gas core: \(Sh_G = 0.023 \cdot Re_G^{0.8} \cdot Sc_G^{0.33} \cdot \left(1 + 0.1 \cdot (d/\delta)^{0.5}\right)\)

Liquid film: \(k_L = \frac{D_L}{\delta} \cdot f(Re_{film})\)

Multicomponent Mass Transfer

Krishna-Standart Model

NeqSim implements the Krishna-Standart multicomponent mass transfer model. For a system with $n$ components:

Binary Mass Transfer Coefficients

First, calculate binary coefficients from correlations:

for (int i = 0; i < nComponents; i++) {
    for (int j = 0; j < nComponents; j++) {
        // Schmidt number
        Sc[i][j] = kinematicViscosity / D[i][j];
        
        // Binary mass transfer coefficient
        k_binary[i][j] = calcInterphaseMassTransferCoefficient(phase, Sc[i][j], node);
    }
}

Mass Transfer Coefficient Matrix

Build the $(n-1) \times (n-1)$ matrix $[\mathbf{k}]$:

\[k_{ii} = \sum_{j \neq i} \frac{x_j}{k_{ij}} + \frac{x_i}{k_{in}}\] \[k_{ij} = -x_i \left(\frac{1}{k_{ij}} - \frac{1}{k_{in}}\right) \quad (i \neq j)\]

Where component $n$ is the reference (typically the most abundant).

Flux Calculation

The molar flux vector:

\[\mathbf{N} = c_t [\mathbf{k}]^{-1} (\mathbf{x}_{bulk} - \mathbf{x}_{interface})\]

Corrections

Thermodynamic Correction

For non-ideal solutions, the driving force includes activity coefficient gradients:

\[[\Gamma] = [\delta_{ij} + x_i \frac{\partial \ln \gamma_i}{\partial x_j}]\]

The corrected flux: \(\mathbf{N} = c_t [\mathbf{k}][\Gamma]^{-1} (\mathbf{x}_{bulk} - \mathbf{x}_{interface})\)

Finite Flux Correction (Stefan Flow)

For high mass transfer rates, the film theory correction:

\[[\Xi] = [\Phi](e^{[\Phi]} - I)^{-1}\]

Where $[\Phi]$ is the rate factor matrix.

Reactive Mass Transfer

Enhancement Factor

Chemical reactions in the liquid phase enhance mass transfer:

\[N_A = E \cdot k_L \cdot (C_{A,i} - C_{A,bulk})\]

Where $E \geq 1$ is the enhancement factor.

Hatta Number

The Hatta number characterizes the reaction regime:

\[Ha = \frac{\sqrt{k_{rxn} \cdot D_A}}{k_L}\]
Ha Range Regime Location of Reaction
Ha < 0.3 Slow Bulk liquid
0.3 < Ha < 3 Intermediate Film and bulk
Ha > 3 Fast Within film
Ha » 3 Instantaneous At interface

Enhancement Factor Models

Pseudo-First Order (Fast Reaction)

\[E = \frac{Ha}{\tanh(Ha)}\]

Instantaneous Reaction

\[E_{\infty} = 1 + \frac{D_B \cdot C_{B,bulk}}{\nu_B \cdot D_A \cdot C_{A,i}}\]

Where $\nu_B$ is the stoichiometric coefficient.

General Case (Danckwerts)

\[E = \frac{\sqrt{1 + Ha^2 \cdot (E_{\infty} - 1)/E_{\infty}}}{1 + (E_{\infty} - 1)^{-1}}\]

CO₂-Amine Systems

NeqSim includes specific models for CO₂ absorption:

Reaction Mechanism (MDEA)

CO₂ + MDEA + H₂O ⇌ MDEAH⁺ + HCO₃⁻  (slow, base-catalyzed)
CO₂ + OH⁻ ⇌ HCO₃⁻                   (parallel)

Reaction Kinetics

\[r_{CO2} = k_2 \cdot [CO_2] \cdot [MDEA]\]

With Arrhenius temperature dependence:

\[k_2 = A \cdot \exp\left(-\frac{E_a}{RT}\right)\]
Amine A (m³/mol·s) Eₐ (kJ/mol) Source
MDEA 4.01×10⁸ 42.0 Rinker et al. (1995)
MEA 4.4×10¹¹ 50.5 Hikita et al. (1977)
DEA 1.3×10¹⁰ 47.5 Blauwhoff et al. (1984)

High-Pressure Effects

From the thesis work, high-pressure effects on CO₂ absorption include:

  1. Reduced CO₂ capacity at high total pressure (up to 40% reduction at 200 bar)
  2. Thermodynamic non-ideality must be modeled consistently
  3. Reaction kinetics relatively unaffected by pressure (with N₂ as inert)

Implementation Classes

Class Hierarchy

FluidBoundary (abstract)
├── EquilibriumFluidBoundary
└── NonEquilibriumFluidBoundary (abstract)
    └── KrishnaStandartFilmModel
        └── ReactiveKrishnaStandartFilmModel
            └── ReactiveFluidBoundary

Key Classes

Class Location Purpose
FluidBoundary flownode.fluidboundary.heatmasstransfercalc Base class for interphase calculations
NonEquilibriumFluidBoundary ...nonequilibriumfluidboundary Non-equilibrium base
KrishnaStandartFilmModel ...filmmodelboundary Multicomponent film model
ReactiveKrishnaStandartFilmModel ...reactivefilmmodel With chemical reactions
EnhancementFactor ...enhancementfactor Enhancement factor calculations

Key Methods

// FluidBoundary
public abstract void solve();
public double[] getMolarFlux();
public double[] getHeatFlux();

// KrishnaStandartFilmModel
public double calcBinarySchmidtNumbers(int phase);
public double calcBinaryMassTransferCoefficients(int phase);
public double calcMassTransferCoefficients(int phase);
public void calcPhiMatrix(int phase);  // Finite flux correction

Usage Examples

Basic Two-Phase Mass Transfer

import neqsim.fluidmechanics.flownode.twophasenode.twophasepipeflownode.AnnularFlow;
import neqsim.fluidmechanics.geometrydefinitions.pipe.PipeData;
import neqsim.thermo.system.SystemSrkEos;

// Create two-phase system
SystemSrkEos fluid = new SystemSrkEos(300.0, 50.0);
fluid.addComponent("methane", 0.90);
fluid.addComponent("CO2", 0.05);
fluid.addComponent("water", 0.05);
fluid.setMixingRule("classic");

// Create pipe geometry
PipeData pipe = new PipeData(0.1);  // 0.1 m diameter

// Create flow node
AnnularFlow node = new AnnularFlow(fluid, pipe);
node.init();
node.initFlowCalc();

// Enable mass transfer
node.getFluidBoundary().setMassTransferCalc(true);
node.getFluidBoundary().solve();

// Get results
double[] molarFlux = node.getFluidBoundary().getMolarFlux();
System.out.println("CO2 flux: " + molarFlux[1] + " mol/m²·s");

With Thermodynamic Corrections

// Enable activity coefficient corrections
node.getFluidBoundary().setThermodynamicCorrections(0, true);  // Gas
node.getFluidBoundary().setThermodynamicCorrections(1, true);  // Liquid

// Enable Stefan flow correction
node.getFluidBoundary().setFiniteFluxCorrection(0, true);
node.getFluidBoundary().setFiniteFluxCorrection(1, true);

node.getFluidBoundary().solve();

CO₂ Absorption with Reaction

import neqsim.thermo.system.SystemSrkCPAstatoil;

// Create system with amine
SystemSrkCPAstatoil fluid = new SystemSrkCPAstatoil(313.15, 30.0);
fluid.addComponent("nitrogen", 0.85);
fluid.addComponent("CO2", 0.10);
fluid.addComponent("water", 0.04);
fluid.addComponent("MDEA", 0.01);
fluid.setMixingRule(10);  // CPA mixing rule

// Use reactive film model
// The enhancement factor is calculated automatically
// based on reaction kinetics and Hatta number

References

  1. Solbraa, E. (2002). Equilibrium and Non-Equilibrium Thermodynamics of Natural Gas Processing. Dr.ing. thesis, NTNU. NVA

  2. Krishna, R., Standart, G.L. (1976). Mass and energy transfer in multicomponent systems. Chem. Eng. Commun., 3(4-5), 201-275.

  3. Taylor, R., Krishna, R. (1993). Multicomponent Mass Transfer. Wiley.

  4. Danckwerts, P.V. (1970). Gas-Liquid Reactions. McGraw-Hill.

  5. Poling, B.E., Prausnitz, J.M., O’Connell, J.P. (2001). The Properties of Gases and Liquids. 5th ed. McGraw-Hill.

  6. Rinker, E.B., Ashour, S.S., Sandall, O.C. (1995). Kinetics and modelling of carbon dioxide absorption into aqueous solutions of N-methyldiethanolamine. Chem. Eng. Sci., 50(5), 755-768.