Interphase Multicomponent Mass and Heat Transfer in Two-Phase Pipe Flow

Overview

This document provides a detailed description of the theoretical models and numerical methods used in NeqSim for calculating interphase mass and heat transfer in two-phase gas-liquid pipe flow. The approach is based on non-equilibrium thermodynamics where the gas and liquid phases are not assumed to be in thermodynamic equilibrium at the interface.

Key Concepts:

• Phases exchange mass and heat across the interface
• Interface is assumed to be at thermodynamic equilibrium
• Bulk phases may have different temperatures and compositions from the interface
• Transport is driven by concentration and temperature gradients

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1. Non-Equilibrium vs Equilibrium Models

1.1 Equilibrium Model (Flash Calculation)

In the equilibrium approach, phases are assumed to be in complete thermodynamic equilibrium:

\[y_i = K_i(T, P) \cdot x_i \quad \text{for all components } i\] \[T_G = T_L = T\]

This is computationally simple but fails when:

• Residence time is short
• Interfacial area is limited
• Transport resistances are significant

1.2 Non-Equilibrium Model

The non-equilibrium model accounts for finite-rate mass and heat transfer:

\[y_i^{bulk} \neq K_i \cdot x_i^{bulk}\] \[T_G^{bulk} \neq T_L^{bulk}\]

Interface Equilibrium: \(y_i^{int} = K_i(T^{int}, P) \cdot x_i^{int}\)

The driving forces are:

• Mass transfer: $(y_i^{bulk} - y_i^{int})$ and $(x_i^{int} - x_i^{bulk})$
• Heat transfer: $(T_G^{bulk} - T^{int})$ and $(T^{int} - T_L^{bulk})$

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2. Multicomponent Mass Transfer Theory

2.1 Maxwell-Stefan Equations

For multicomponent diffusion, the Maxwell-Stefan equations describe the relationship between fluxes and driving forces:

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

Symbol	Description	Units
$x_i$	Mole fraction of component $i$	[-]
$\mu_i$	Chemical potential of component $i$	[J/mol]
$N_i$	Molar flux of component $i$	[mol/(m²·s)]
$c_t$	Total molar concentration	[mol/m³]
$D_{ij}$	Maxwell-Stefan diffusivity	[m²/s]
$R$	Gas constant	[J/(mol·K)]
$T$	Temperature	[K]

📘 Diffusivity Models: The Maxwell-Stefan diffusivities $D_{ij}$ are calculated using correlations documented in mass_transfer.md. NeqSim provides multiple models:

• Gas phase: Chapman-Enskog kinetic theory
• Liquid phase: Siddiqi-Lucas, Hayduk-Minhas (hydrocarbons), CO2-water (Tamimi)
• High pressure: Mathur-Thodos correction for P > 100 bar

See the Model Selection Guide for recommendations.

2.2 Matrix Formulation

The Maxwell-Stefan equations can be written in matrix form:

\[(\mathbf{J}) = -c_t [\mathbf{B}]^{-1} [\mathbf{\Gamma}] \nabla(\mathbf{x})\]

Where:

• $\mathbf{J}$ = vector of diffusive fluxes (n-1 components)
• $[\mathbf{B}]$ = matrix of inverse diffusivities
• $[\mathbf{\Gamma}]$ = thermodynamic factor matrix (accounts for non-ideal behavior)
• $\nabla(\mathbf{x})$ = mole fraction gradients

Elements of [B]:

\[B_{ii} = \frac{x_i}{D_{i,n}} + \sum_{k=1, k\neq i}^{n} \frac{x_k}{D_{ik}}\] \[B_{ij} = -x_i \left(\frac{1}{D_{ij}} - \frac{1}{D_{i,n}}\right), \quad i \neq j\]

Thermodynamic Factor Matrix:

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

Where $\gamma_i$ is the activity coefficient and $\delta_{ij}$ is the Kronecker delta.

2.3 Film Theory

Film theory assumes that mass transfer resistance is confined to a thin stagnant film at the interface:

     Bulk Gas     |  Gas Film  | Interface |  Liquid Film  |   Bulk Liquid
    ─────────────┼────────────┼───────────┼───────────────┼─────────────
    y_i^bulk      →  y_i^int    =    K_i    ·   x_i^int     ←   x_i^bulk
    T_G^bulk      →  T^int      =   T^int   =   T^int       ←   T_L^bulk

Film thickness:

• Gas film: $\delta_G$
• Liquid film: $\delta_L$

2.4 Krishna-Standart Film Model

The Krishna-Standart model extends the Maxwell-Stefan equations to film theory for multicomponent systems:

\[(\mathbf{N}) = c_t [\mathbf{k}](\mathbf{x}^{int} - \mathbf{x}^{bulk}) + x_t^{avg} N_t\]

Where $[\mathbf{k}]$ is the matrix of mass transfer coefficients:

\[[\mathbf{k}] = [\mathbf{B}]^{-1} [\mathbf{\Xi}]\]

Bootstrap Matrix [Ξ]:

The bootstrap matrix $[\mathbf{\Xi}]$ accounts for the effect of finite mass transfer rates (high flux correction):

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

Where: \(\mathbf{\Phi} = [\mathbf{B}_0]^{-1} N_t / c_t\)

At low fluxes: $[\mathbf{\Xi}] \rightarrow \mathbf{I}$ (identity matrix)

2.5 Mass Transfer Coefficients

Gas-phase mass transfer coefficient:

\[k_G = \frac{Sh \cdot D_G}{D_h}\]

Liquid-phase mass transfer coefficient:

\[k_L = \frac{Sh \cdot D_L}{D_h}\]

Sherwood Number Correlations:

Flow Regime	Correlation
Turbulent (Re > 10,000)	$Sh = 0.023 \cdot Re^{0.83} \cdot Sc^{0.44}$
Transitional	Interpolation
Laminar (Re < 2,300)	$Sh = 3.66$ (constant wall)

2.6 Interface Composition Calculation

The interface compositions $(x_i^{int}, y_i^{int})$ are found by solving simultaneously:

1. Flux continuity: \(N_i^G = N_i^L \quad \text{for each component}\)

2. Interface equilibrium: \(y_i^{int} = K_i(T^{int}, P) \cdot x_i^{int}\)

3. Summation constraints: \(\sum_{i=1}^n x_i^{int} = 1, \quad \sum_{i=1}^n y_i^{int} = 1\)

This requires iterative solution (Newton-Raphson method).

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3. Interphase Heat Transfer Theory

3.1 Heat Transfer Resistances

Heat flows from bulk gas → interface → bulk liquid (or reverse):

\[q = h_G (T_G^{bulk} - T^{int}) = h_L (T^{int} - T_L^{bulk})\]

Overall heat transfer coefficient:

\[\frac{1}{h_{overall}} = \frac{1}{h_G} + \frac{1}{h_L}\]

3.2 Heat Transfer with Mass Transfer

When mass transfer occurs, the energy balance includes:

1. Sensible heat transfer (conduction/convection)
2. Latent heat of phase change
3. Enthalpy carried by transferred mass

Total interfacial heat flux:

\[Q^{int} = h_{GL}(T_G - T_L) + \sum_{i=1}^n N_i \cdot \Delta H_{vap,i}\]

Where:

• $h_{GL}$ = sensible heat transfer coefficient [W/(m²·K)]
• $N_i$ = molar flux of component $i$ [mol/(m²·s)]
• $\Delta H_{vap,i}$ = heat of vaporization of component $i$ [J/mol]

3.3 Interface Temperature

The interface temperature $T^{int}$ is found from the energy balance:

\[h_G (T_G^{bulk} - T^{int}) + \sum_{i=1}^n N_i H_i^G = h_L (T^{int} - T_L^{bulk}) + \sum_{i=1}^n N_i H_i^L\]

Rearranging:

\[T^{int} = \frac{h_G T_G^{bulk} + h_L T_L^{bulk} + \sum_i N_i (H_i^G - H_i^L)}{h_G + h_L}\]

3.4 Heat Transfer Coefficients

Dittus-Boelter Correlation (Turbulent):

\[Nu = 0.023 \cdot Re^{0.8} \cdot Pr^n\]

Where:

• $n = 0.4$ for heating (fluid being heated)
• $n = 0.3$ for cooling (fluid being cooled)

\[h = \frac{Nu \cdot k_{thermal}}{D_h}\]

Gnielinski Correlation (Transitional, 2300 < Re < 10,000):

\[Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7\sqrt{f/8}(Pr^{2/3} - 1)}\]

Laminar Flow (Re < 2,300):

\(Nu = 3.66 \quad \text{(constant wall temperature)}\) \(Nu = 4.36 \quad \text{(constant heat flux)}\)

3.5 Chilton-Colburn Analogy

Heat and mass transfer are related through:

\[\frac{h}{c_p G} Pr^{2/3} = \frac{k_m}{u} Sc^{2/3} = \frac{f}{2}\]

This allows estimation of mass transfer coefficients from heat transfer data:

\[k_m = h \cdot \frac{1}{\rho c_p} \left(\frac{Sc}{Pr}\right)^{-2/3}\]

Or equivalently:

\[Sh = Nu \cdot \left(\frac{Sc}{Pr}\right)^{1/3}\]

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4. Specific Interfacial Area

The interfacial area per unit volume depends on the flow pattern:

\[a = \frac{\text{Interface Area}}{\text{Pipe Volume}} \quad [m^2/m^3]\]

4.1 Stratified Flow

For stratified flow with liquid height $h_L$:

\[a = \frac{S_i}{A} = \frac{2\sqrt{h_L(D - h_L)}}{\frac{\pi D^2}{4}}\]

Where $S_i$ is the interfacial chord length.

4.2 Annular Flow

For annular flow with liquid film thickness $\delta$:

\[a = \frac{\pi (D - 2\delta)}{\frac{\pi D^2}{4}} = \frac{4(D - 2\delta)}{D^2}\]

For thin films: $a \approx \frac{4}{D}$

4.3 Bubble Flow

For spherical bubbles of diameter $d_b$:

\[a = \frac{6\alpha_G}{d_b}\]

Bubble size can be estimated from the Weber number:

\[d_b = \frac{We_{crit} \cdot \sigma}{\rho_L u_L^2}\]

4.4 Slug Flow

Slug flow has complex geometry. The effective interfacial area includes:

• Taylor bubble surface
• Small bubbles in liquid slug

\[a_{slug} = \alpha_{Taylor} \cdot a_{Taylor} + \alpha_{dispersed} \cdot a_{dispersed}\]

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5. Coupled Heat and Mass Transfer Solution

5.1 Solution Algorithm

The coupled heat and mass transfer problem requires iterative solution:

1. Initialize: Guess T^int, x_i^int, y_i^int

2. Calculate K-values:
   K_i = K_i(T^int, P)

3. Calculate diffusivities:
   D_ij^G, D_ij^L at current T^int

4. Calculate mass transfer coefficients:
   [k_G], [k_L] from correlations

5. Calculate component fluxes:
   N_i^G = c_G [k_G](y_i^bulk - y_i^int)
   N_i^L = c_L [k_L](x_i^int - x_i^bulk)

6. Check flux balance:
   If |N_i^G - N_i^L| > tolerance, update x_i^int, y_i^int

7. Calculate heat transfer coefficients:
   h_G, h_L from correlations

8. Calculate interface temperature:
   T^int from energy balance

9. Check convergence:
   If T^int, x_i^int, y_i^int converged, exit
   Else goto step 2

5.2 Newton-Raphson Solution

For efficiency, the interface conditions can be solved using Newton-Raphson:

\[\mathbf{F}(\mathbf{X}) = \mathbf{0}\]

Where: \(\mathbf{X} = [T^{int}, x_1^{int}, x_2^{int}, ..., x_{n-1}^{int}]^T\)

\[\mathbf{F} = \begin{bmatrix} Q^G - Q^L \\ N_1^G - N_1^L \\ N_2^G - N_2^L \\ \vdots \\ N_{n-1}^G - N_{n-1}^L \end{bmatrix}\]

Update: $\mathbf{X}^{new} = \mathbf{X}^{old} - [\mathbf{J}]^{-1} \mathbf{F}$

Where $[\mathbf{J}]$ is the Jacobian matrix.

5.3 Numerical Stability

Under-relaxation: To improve convergence stability:

\[\mathbf{X}^{new} = \omega \cdot \mathbf{X}^{calc} + (1-\omega) \cdot \mathbf{X}^{old}\]

Typical $\omega = 0.3$ to $0.7$.

Damping: Limit changes per iteration:

$$	\Delta T^{int}	< \Delta T_{max}$$
$$	\Delta x_i^{int}	< \Delta x_{max}$$

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6. Condensation and Evaporation

6.1 Total Mass Transfer Rate

The total interphase mass transfer rate:

\[\Gamma = \sum_{i=1}^n M_i \cdot N_i \cdot a \quad [kg/(m^3 \cdot s)]\]

Where:

• $M_i$ = molecular weight of component $i$ [kg/mol]
• $N_i$ = molar flux [mol/(m²·s)]
• $a$ = specific interfacial area [m²/m³]

6.2 Condensation (Vapor → Liquid)

Condensation occurs when:

• $T_G > T_L$ (sensible cooling)
• $y_i^{bulk} > y_i^{int}$ (supersaturation in gas)

Heat released: \(Q_{cond} = \sum_i N_i \cdot \Delta H_{vap,i}\)

6.3 Evaporation (Liquid → Vapor)

Evaporation occurs when:

• $T_L > T_G$ (sensible heating)
• $x_i^{bulk} > x_i^{int}$ (supersaturation in liquid)

Heat absorbed: \(Q_{evap} = -\sum_i N_i \cdot \Delta H_{vap,i}\)

6.4 Component Selectivity

In multicomponent systems, components transfer at different rates based on:

• Volatility differences ($K_i$ values)
• Diffusivity differences ($D_{ij}$)
• Bulk composition gradients

Light components (high $K$) tend to evaporate preferentially. Heavy components (low $K$) tend to condense preferentially.

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7. Physical Properties

7.1 Binary Diffusion Coefficients

Gas Phase (Chapman-Enskog):

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

Where:

• $M_{ij} = 2/(1/M_i + 1/M_j)$ = reduced molecular weight
• $\sigma_{ij}$ = collision diameter [Å]
• $\Omega_D$ = collision integral

Liquid Phase (Wilke-Chang):

\[D_{ij}^L = \frac{7.4 \times 10^{-8} (\phi M_j)^{1/2} T}{\mu_L V_i^{0.6}}\]

Where:

• $\phi$ = association parameter (2.6 for water)
• $M_j$ = molecular weight of solvent
• $\mu_L$ = liquid viscosity [cP]
• $V_i$ = molar volume at boiling point [cm³/mol]

7.2 Thermal Conductivity

Gas Phase (Eucken correlation):

\[k_G = \mu_G \left(c_{p,G} + \frac{5R}{4M}\right)\]

Liquid Phase: From NeqSim thermodynamic model or correlations.

7.3 Heat of Vaporization

From equation of state:

\[\Delta H_{vap,i} = H_i^{vapor} - H_i^{liquid}\]

Or from Watson correlation for pure components:

\[\Delta H_{vap} = \Delta H_{vap,0} \left(\frac{1 - T_r}{1 - T_{r,0}}\right)^{0.38}\]

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8. Implementation in NeqSim

8.1 Enabling Non-Equilibrium Transfer

TwoPhasePipeFlowSystem pipe = TwoPhasePipeFlowSystem.builder()
    .withFluid(fluid)
    .withDiameter(0.1, "m")
    .withLength(100, "m")
    .withNodes(50)
    .enableNonEquilibriumMassTransfer()  // Enable mass transfer calculation
    .enableNonEquilibriumHeatTransfer()  // Enable heat transfer calculation
    .build();

8.2 Accessing Mass Transfer Results

// Get mass transfer rates
double[] massTransferRate = pipe.getInterphaseMassTransferRate();

// Get component fluxes at each node
double[][] componentFluxes = pipe.getComponentFluxProfile();

// Get interfacial area profile
double[] interfacialArea = pipe.getInterfacialAreaProfile();

// Get mass transfer coefficients
double[] k_G = pipe.getGasMassTransferCoefficientProfile();
double[] k_L = pipe.getLiquidMassTransferCoefficientProfile();

8.3 Accessing Heat Transfer Results

// Get heat transfer coefficients
double[] h_G = pipe.getGasHeatTransferCoefficientProfile();
double[] h_L = pipe.getLiquidHeatTransferCoefficientProfile();
double[] h_overall = pipe.getOverallInterphaseHeatTransferCoefficientProfile();

// Get interface temperature
double[] T_interface = pipe.getInterfaceTemperatureProfile();

// Get heat flux
double[] q = pipe.getInterphaseHeatFluxProfile();

// Get total heat transferred
double totalHeat = pipe.getTotalInterphaseHeatTransfer();

8.4 Relevant Classes

Class	Description
FluidBoundaryInterface	Interface between phases, calculates mass/heat transfer
HeatTransferCoefficientCalculator	Heat transfer coefficient correlations
InterphaseTwoPhase	Interphase calculations for two-phase flow
FluidBoundaryInterfaceHMT	Heat and mass transfer at interface

8.5 Complete Example

import neqsim.fluidmechanics.flowsystem.twophaseflowsystem.twophasepipeflowsystem.*;
import neqsim.thermo.system.*;

public class HeatMassTransferExample {
    public static void main(String[] args) {
        // Create multicomponent fluid
        SystemInterface fluid = new SystemSrkEos(300.0, 50.0);
        fluid.addComponent("methane", 0.70, 0);
        fluid.addComponent("ethane", 0.15, 0);
        fluid.addComponent("propane", 0.05, 0);
        fluid.addComponent("water", 0.10, 1);
        fluid.createDatabase(true);
        fluid.setMixingRule(2);
        
        // Build pipe with heat/mass transfer using builder
        TwoPhasePipeFlowSystem pipe = TwoPhasePipeFlowSystem.builder()
            .withFluid(fluid)
            .withDiameter(0.1, "m")
            .withLength(500, "m")
            .withNodes(100)
            .withFlowPattern(FlowPattern.ANNULAR)
            .withConvectiveBoundary(278.15, "K", 15.0)  // Cold ambient
            .build();
        
        // Solve with heat and mass transfer, get structured results
        PipeFlowResult result = pipe.solveWithHeatAndMassTransfer();
        
        // Access results via PipeFlowResult container
        System.out.println("Temperature change: " + result.getTemperatureChange() + " K");
        System.out.println("Pressure drop: " + result.getTotalPressureDrop() + " bar");
        System.out.println("Total heat loss: " + result.getTotalHeatLoss() + " W");
        System.out.println(result);  // Formatted summary
        
        // Export profiles for analysis
        Map<String, double[]> profiles = result.toMap();
    }
}

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9. Validation and Benchmarks

9.1 Comparison with Olga

The NeqSim two-phase pipe flow model has been validated against OLGA simulations for:

• Single-phase heat transfer (±5% deviation)
• Two-phase pressure drop (±10% deviation)
• Condensation rates (±15% deviation)

9.2 Literature Validation

Test Case	Literature	NeqSim	Deviation
Dittus-Boelter (turbulent)	Experimental	+3.2%	Within uncertainty
Lockhart-Martinelli	Original data	+8.5%	Acceptable
Stratified flow transition	Taitel-Dukler	Good agreement	-

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10. References

1. Krishna, R. and Standart, G.L. (1976). “A multicomponent film model incorporating a general matrix method of solution to the Maxwell-Stefan equations.” AIChE Journal, 22(2), 383-389.

2. Taylor, R. and Krishna, R. (1993). Multicomponent Mass Transfer. Wiley Series in Chemical Engineering.

3. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2002). Transport Phenomena, 2nd Edition. John Wiley & Sons.

4. Chilton, T.H. and Colburn, A.P. (1934). “Mass Transfer (Absorption) Coefficients Prediction from Data on Heat Transfer and Fluid Friction.” Industrial & Engineering Chemistry, 26(11), 1183-1187.

5. Incropera, F.P. and DeWitt, D.P. (2002). Fundamentals of Heat and Mass Transfer, 5th Edition. John Wiley & Sons.

6. Solbraa, E. (2002). “Measurement and Calculation of Two-Phase Flow in Pipes.” PhD Thesis, Norwegian University of Science and Technology.

7. Dittus, F.W. and Boelter, L.M.K. (1930). “Heat transfer in automobile radiators of the tubular type.” University of California Publications in Engineering, 2, 443-461.

8. Gnielinski, V. (1976). “New equations for heat and mass transfer in turbulent pipe and channel flow.” International Chemical Engineering, 16(2), 359-368.

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Document generated for NeqSim Interphase Heat and Mass Transfer Module