This document provides detailed documentation of the heat transfer models implemented in the NeqSim fluid mechanics package.

Related Documentation:

Table of Contents

Overview

NeqSim implements comprehensive heat transfer models for:

The models are based on established correlations and are coupled with the rigorous thermodynamic calculations in NeqSim.

Theoretical Background

Energy Balance

The energy conservation equation for pipe flow:

\[\frac{\partial (\rho h)}{\partial t} + \frac{\partial (\rho v h)}{\partial z} = \dot{Q}_{wall} + \dot{Q}_{interphase}\]

Where:

Heat Transfer Mechanisms

Mechanism Equation Application
Conduction $q = -k \nabla T$ Through solid walls, stagnant fluids
Convection $q = h (T_w - T_f)$ Flowing fluids to walls
Radiation $q = \epsilon \sigma (T_1^4 - T_2^4)$ High temperature systems
Latent heat $\dot{Q} = \dot{m} \Delta H_{vap}$ Phase change

Single-Phase Heat Transfer

Dimensionless Numbers

Number Definition Physical Meaning
Nusselt (Nu) $h \cdot d / k$ Ratio of convective to conductive heat transfer
Prandtl (Pr) $\mu c_p / k$ Ratio of momentum to thermal diffusivity
Reynolds (Re) $\rho v d / \mu$ Ratio of inertial to viscous forces
Péclet (Pe) $Re \cdot Pr$ Ratio of advective to diffusive heat transport

Prandtl Number

The Prandtl number characterizes the relative thickness of thermal and velocity boundary layers:

// Calculated in FlowNode
double Pr = viscosity * heatCapacity / thermalConductivity;

Typical values:

Fluid Pr
Gases 0.7 - 1.0
Water 1.7 - 13
Light oils 10 - 1000
Heavy oils 100 - 100,000

Correlations

Laminar Flow (Re < 2300)

Constant wall temperature: \(Nu = 3.66\)

Constant heat flux: \(Nu = 4.36\)

Developing flow (Sieder-Tate): \(Nu = 1.86 \left(\frac{Re \cdot Pr \cdot d}{L}\right)^{1/3} \left(\frac{\mu}{\mu_w}\right)^{0.14}\)

Turbulent Flow (Re > 10,000)

Dittus-Boelter equation: \(Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{n}\)

Where:

Gnielinski correlation (more accurate): \(Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}\)

Valid for: $3000 < Re < 5 \times 10^6$, $0.5 < Pr < 2000$

Transition Region (2300 < Re < 10,000)

Gnielinski correlation or linear interpolation between laminar and turbulent.

Implementation

The equations above describe common correlations, not a literal implementation of InterphaseTransportCoefficientBaseClass: that base class returns zero for its heat-transfer methods. The single-phase InterphasePipeFlow implementation uses Nu = 3.66 below |Re| = 2000 and a friction-factor/Chilton-Colburn expression above that threshold. Other classes, including PipeBeggsAndBrills, have their own coefficient calculations. Select and inspect the actual model rather than assuming all models use the same transition rule.

Two-Phase Heat Transfer

Flow Pattern Effects

Heat transfer in two-phase flow depends strongly on the flow pattern:

Flow Pattern Dominant Mechanism Heat Transfer Characteristics
Stratified Convection in each phase Independent gas/liquid correlations
Annular Film evaporation/condensation High liquid-side coefficients
Slug Alternating mechanisms Time-averaged values
Bubble Enhanced liquid mixing Increased liquid-side coefficient
Mist Droplet evaporation Reduced wall wetting

Two-Phase Multiplier Approach

Some correlations use a two-phase multiplier:

\[h_{TP} = F \cdot h_{LO}\]

Where:

Flow Pattern-Specific Correlations

Stratified Flow

Gas and liquid are treated separately:

\(h_{gas} = \text{Single-phase correlation with } d_h = 4A_G/P_G\) \(h_{liquid} = \text{Single-phase correlation with } d_h = 4A_L/P_L\)

Annular Flow

Liquid film: \(h_L = 0.023 \cdot Re_f^{0.8} \cdot Pr_L^{0.4} \cdot \frac{k_L}{\delta}\)

Where $\delta$ is the film thickness and $Re_f = 4\Gamma/\mu_L$ is the film Reynolds number.

Gas core: \(h_G = 0.023 \cdot Re_G^{0.8} \cdot Pr_G^{0.4} \cdot \frac{k_G}{d - 2\delta}\)

Wall Heat Transfer

Overall Heat Transfer Coefficient

For heat transfer from fluid to surroundings through the pipe wall:

\[\frac{1}{U} = \frac{1}{h_i} + \frac{r_i \ln(r_o/r_i)}{k_{wall}} + \frac{r_i}{r_o \cdot h_o}\]

Where:

Insulation

For insulated pipes, add insulation resistance:

\[\frac{1}{U} = \frac{1}{h_i} + \frac{r_i \ln(r_o/r_i)}{k_{wall}} + \frac{r_i \ln(r_{ins}/r_o)}{k_{ins}} + \frac{r_i}{r_{ins} \cdot h_o}\]

Buried Pipelines

For buried pipelines, the outer resistance includes soil conduction:

\[R_{soil} = \frac{\ln(2z/r_o)}{2\pi k_{soil}}\]

Where $z$ is the burial depth.

Usage in NeqSim

The low-level geometry class is PipeData. This complete example calculates wall and outer-environment resistance, excluding the inner fluid film. The returned coefficient is referenced to the inner pipe area. It does not run a flow solver or determine an outlet temperature.

import org.apache.logging.log4j.LogManager;
import org.apache.logging.log4j.Logger;
import neqsim.fluidmechanics.geometrydefinitions.pipe.PipeData;

Logger logger = LogManager.getLogger("PipeWallExample");
PipeData pipe = new PipeData(0.30, 4.6e-5); // Inner diameter, roughness in m
pipe.setCarbonSteelWall(0.01); // 10 mm steel
pipe.addMineralWoolInsulation(0.05); // 50 mm insulation
pipe.setAirEnvironment(288.15, 2.0); // K, wind velocity m/s
double wallAndOuterCoefficient = pipe.calcOverallHeatTransferCoefficient();
logger.info("Wall plus outer coefficient: {} W/(m2 K)", wallAndOuterCoefficient);

For PipeBeggsAndBrills, an effective U-value is set with setHeatTransferCoefficient(double). To include computed inner convection, wall conduction, insulation, and outer convection, select DETAILED_U with the class-specific setters described in the pipeline guide.

Interphase Heat Transfer

Heat Transfer Between Phases

At the gas-liquid interface:

\[\dot{Q}_{GL} = h_{GL} \cdot a_i \cdot (T_G - T_L)\]

Where:

Interfacial Area

The interfacial area depends on the flow pattern:

Flow Pattern Interfacial Area $a_i$
Stratified $W/A_{pipe}$ (width / cross-section)
Annular $\pi(d - 2\delta)/A_{pipe}$
Bubble $6\epsilon_G/d_b$
Droplet $6\epsilon_L/d_d$

Chilton-Colburn Analogy

The heat and mass transfer coefficients are related:

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

Or in terms of j-factors:

\[j_H = j_D\]

Where: \(j_H = \frac{Nu}{Re \cdot Pr^{1/3}} = St \cdot Pr^{2/3}\) \(j_D = \frac{Sh}{Re \cdot Sc^{1/3}}\)

Heat-Mass Transfer Coupling

Latent Heat Effects

When mass transfer occurs, the associated enthalpy must be considered:

\[\dot{Q}_{total} = \dot{Q}_{sensible} + \dot{Q}_{latent}\] \[\dot{Q}_{latent} = \sum_j N_j \cdot \Delta H_{vap,j}\]

Interface Energy Balance

At the gas-liquid interface, the energy balance:

\[h_G (T_G - T_i) + \sum_j N_j H_j^G = h_L (T_i - T_L) + \sum_j N_j H_j^L\]

Rearranging:

\[h_G (T_G - T_i) - h_L (T_i - T_L) = \sum_j N_j (H_j^L - H_j^G) = -\sum_j N_j \Delta H_{vap,j}\]

Ackermann Correction

For high mass transfer rates, the sensible heat transfer is modified:

\[\dot{Q}_{sensible} = h \cdot \Phi \cdot (T_{bulk} - T_i)\]

Where the Ackermann correction factor:

\[\Phi = \frac{\phi}{e^\phi - 1}\]

And: \(\phi = \frac{\sum_j N_j c_{p,j}}{h}\)

Implementation

FluidBoundaryInterface exposes setHeatTransferCalc(boolean), solve(), and getInterphaseHeatFlux(int phase). Heat flux is read one phase at a time; there is no no-argument array getter. The film model couples interphase heat and mass transfer internally. The worked node example below starts at phase equilibrium and therefore checks a near-zero driving-force baseline; it is not an evaporation or condensation-rate experiment.

Implementation Classes

Class Hierarchy

FluidBoundary
├── heatTransferCoefficient[2]    // Gas, Liquid
├── heatTransferCorrection[2]     // Ackermann factors
├── prandtlNumber[2]
└── interphaseHeatFlux[2]

InterphaseTransportCoefficientBaseClass
├── calcWallHeatTransferCoefficient()
├── calcInterphaseHeatTransferCoefficient()
└── calcWallFrictionFactor()

Key Methods

Object Public method Result / input
InterphaseTransportCoefficientInterface calcWallHeatTransferCoefficient(int, double, FlowNodeInterface) Wall coefficient for a phase and Prandtl number
InterphaseTransportCoefficientInterface calcInterphaseHeatTransferCoefficient(int, double, FlowNodeInterface) Interphase coefficient
FluidBoundaryInterface setHeatTransferCalc(boolean) Enable/disable interphase heat calculation
FluidBoundaryInterface getInterphaseHeatFlux(int) Heat flux for the specified phase, W/m²

Usage Examples

Basic Heat Transfer in Pipe Flow

This self-contained example uses a process-level PipeBeggsAndBrills model with single-phase gas. It specifies an effective U-value and ambient boundary; SRK with the classic mixing rule supplies thermodynamic properties. As with the pipeline guide, put imports at class scope and statements in a method.

import org.apache.logging.log4j.LogManager;
import org.apache.logging.log4j.Logger;
import neqsim.process.equipment.pipeline.PipeBeggsAndBrills;
import neqsim.process.equipment.stream.Stream;
import neqsim.thermo.system.SystemSrkEos;

Logger logger = LogManager.getLogger("GasCoolingExample");
SystemSrkEos gas = new SystemSrkEos(373.15, 50.0); // 100 C, 50 bara
gas.addComponent("methane", 0.95);
gas.addComponent("ethane", 0.05);
gas.setMixingRule("classic");
Stream inlet = new Stream("Hot Gas", gas);
inlet.setFlowRate(10000.0, "kg/hr");
inlet.run();

PipeBeggsAndBrills pipeline = new PipeBeggsAndBrills("Cooling Pipeline", inlet);
pipeline.setDiameter(0.30); // m
pipeline.setLength(10000.0); // m
pipeline.setNumberOfIncrements(20);
pipeline.setConstantSurfaceTemperature(10.0, "C");
pipeline.setHeatTransferCoefficient(5.0); // W/(m2 K), SPECIFIED_U mode
pipeline.run();

double[] temperatureK = pipeline.getTemperatureProfile();
for (int i = 0; i < temperatureK.length; i++) {
    logger.info("x={} m; T={} C", pipeline.getLengthProfile().get(i),
        temperatureK[i] - 273.15);
}
logger.info("Outlet pressure: {} bara", pipeline.getOutletPressure("bara"));

Expect cooling from 100 °C toward the 10 °C surroundings and a positive pressure drop. The chosen U-value is an example input, not a prediction of a particular insulation system.

Two-Phase with Interphase Heat Transfer

This complete flow-node example starts with an equilibrium methane/decane mixture. Both phases initially have the same temperature, so the calculated interphase heat flux should be close to zero. A finite transfer-rate study requires non-equilibrium phase compositions or temperatures and a validated spatial/time integration.

import org.apache.logging.log4j.LogManager;
import org.apache.logging.log4j.Logger;
import neqsim.fluidmechanics.flownode.twophasenode.twophasepipeflownode.StratifiedFlowNode;
import neqsim.fluidmechanics.geometrydefinitions.pipe.PipeData;
import neqsim.thermo.system.SystemSrkEos;
import neqsim.thermodynamicoperations.ThermodynamicOperations;

Logger logger = LogManager.getLogger("InterphaseHeatExample");
SystemSrkEos fluid = new SystemSrkEos(298.15, 10.0);
fluid.addComponent("methane", 0.8);
fluid.addComponent("n-decane", 0.2);
fluid.setMixingRule("classic");
fluid.setTotalFlowRate(1.0, "kg/sec");
new ThermodynamicOperations(fluid).TPflash();
fluid.initProperties();

PipeData geometry = new PipeData(0.1);
StratifiedFlowNode node = new StratifiedFlowNode(fluid, geometry);
node.initFlowCalc();
node.init();
node.getFluidBoundary().setHeatTransferCalc(true);
node.getFluidBoundary().setMassTransferCalc(true);
node.getFluidBoundary().useThermodynamicCorrections(true, 0);
node.getFluidBoundary().useThermodynamicCorrections(true, 1);
node.getFluidBoundary().useFiniteFluxCorrection(true, 0);
node.getFluidBoundary().useFiniteFluxCorrection(true, 1);
node.getFluidBoundary().solve();
double gasHeatFlux = node.getFluidBoundary().getInterphaseHeatFlux(0);
double liquidHeatFlux = node.getFluidBoundary().getInterphaseHeatFlux(1);
logger.info("Gas-side: {} W/m2; liquid-side: {} W/m2", gasHeatFlux, liquidHeatFlux);

Condensation in Pipeline

The following equilibrium example checks whether cooling produces a liquid phase. Thermodynamic phase fraction and hydrodynamic holdup are different quantities: use outlet fluid phase volumes for equilibrium partitioning, and the pipeline’s holdup profile for in-situ liquid inventory. This model does not estimate finite-rate nucleation or condensation kinetics.

import org.apache.logging.log4j.LogManager;
import org.apache.logging.log4j.Logger;
import neqsim.process.equipment.pipeline.PipeBeggsAndBrills;
import neqsim.process.equipment.stream.Stream;
import neqsim.thermo.system.SystemSrkEos;

Logger logger = LogManager.getLogger("CondensationExample");
SystemSrkEos gas = new SystemSrkEos(320.0, 80.0);
gas.addComponent("methane", 0.80);
gas.addComponent("ethane", 0.10);
gas.addComponent("propane", 0.05);
gas.addComponent("n-butane", 0.03);
gas.addComponent("n-pentane", 0.02);
gas.setMixingRule("classic");
Stream inlet = new Stream("Warm Gas", gas);
inlet.setFlowRate(5000.0, "kg/hr");
inlet.run();

PipeBeggsAndBrills pipeline = new PipeBeggsAndBrills("Cold Flowline", inlet);
pipeline.setLength(5000.0);
pipeline.setDiameter(0.20);
pipeline.setNumberOfIncrements(20);
pipeline.setConstantSurfaceTemperature(4.0, "C");
pipeline.setHeatTransferCoefficient(25.0);
pipeline.run();

logger.info("Inlet phases: {}; outlet phases: {}", inlet.getFluid().getNumberOfPhases(),
    pipeline.getOutletStream().getFluid().getNumberOfPhases());
double[] holdup = pipeline.getLiquidHoldupProfile();
logger.info("Outlet: {} C; liquid holdup: {}", pipeline.getOutletTemperature("C"),
    holdup[holdup.length - 1]);

References

  1. Incropera, F.P., DeWitt, D.P., et al. (2007). Fundamentals of Heat and Mass Transfer. 6th ed. Wiley.

  2. Gnielinski, V. (1976). New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng., 16(2), 359-368.

  3. Dittus, F.W., Boelter, L.M.K. (1930). Heat transfer in automobile radiators of the tubular type. Univ. Calif. Publ. Eng., 2(13), 443-461.

  4. Chilton, T.H., Colburn, A.P. (1934). Mass transfer (absorption) coefficients: Prediction from data on heat transfer and fluid friction. Ind. Eng. Chem., 26(11), 1183-1187.

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

  6. Bird, R.B., Stewart, W.E., Lightfoot, E.N. (2002). Transport Phenomena. 2nd ed. Wiley.