This chapter introduces reduced-order heat transfer modeling for complex systems whose geometric detail cannot be resolved explicitly. Porous media models and heat exchanger models are presented as volume-averaged representations that replace detailed solid–fluid interfaces with equivalent momentum and energy source terms. The chapter explains the Representative Elementary Volume (REV) concept, local thermal equilibrium assumptions, one- and two-equation energy models, and practical CFD implementations for porous materials and compact heat exchangers.

Why This Chapter Exists

Many real engineering systems:

Contain complex internal geometries
Span multiple length scales
Are impossible to mesh explicitly

Examples:

Packed beds
Filters
Catalytic converters
Radiators
Compact heat exchangers

Instead of resolving geometry:

Its effect is modeled
Using momentum losses and heat exchange source terms

This chapter formalizes that abstraction.

Porous Media: Conceptual Foundation

3.1 What Is a Porous Medium?

A porous medium consists of:

A solid matrix
An interconnected pore space filled with fluid

Key features:

Geometry is highly complex
Pore scale is much smaller than system scale
Transport occurs in both phases simultaneously

3.2 Characteristic Variables

Important descriptors:

Porosity: fraction of volume available to fluid
- Global vs effective porosity
Specific surface area: fluid–solid interfacial area per volume
Permeability: resistance to flow through the matrix

These properties encode geometry into lumped parameters.

Representative Elementary Volume (REV)

The REV concept enables upscaling.

Key idea:

There exists a volume large enough to be statistically representative
But small enough to preserve spatial variation

At the REV scale:

Governing equations are volume-averaged
Microscopic detail is replaced by effective properties

This is the foundation of porous-media CFD models.

Momentum Modeling in Porous Media

In porous zones:

Continuity remains unchanged
Momentum equations gain a sink term

Physical meaning:

The porous matrix resists flow
Resistance depends on velocity magnitude

Two contributions:

Viscous resistance (low velocities)
Inertial resistance (high velocities)

This framework generalizes Darcy’s law.

Velocity Definitions

Two velocity concepts exist:

Superficial velocity
- Based on total cross-sectional area
- Used by default in Fluent
Physical (pore) velocity
- Accounts for porosity
- More accurate for heat and mass transfer

Choosing between them affects:

Convective heat transfer
Energy source terms

Heat Transfer in Porous Media

7.1 Local Thermal Equilibrium (LTE)

The one-equation model assumes:

Fluid and solid matrix have the same temperature locally
Heat exchange between phases is fast

Implications:

Single energy equation
Effective thermal conductivity

Valid when:

Pore sizes are small
Heat exchange is strong

7.2 Effective Thermal Conductivity

Effective conductivity represents:

Conduction in solid
Conduction in fluid
Thermal dispersion due to velocity fluctuations

It depends on:

Porosity
Material properties
Flow regime
Matrix geometry

This is a closure problem, often empirical.

Non-Equilibrium Heat Transfer

8.1 Two-Equation Model

The non-equilibrium model relaxes the LTE assumption:

Fluid and solid have separate temperatures
Two coupled energy equations are solved

Coupling occurs through:

Interfacial heat transfer coefficient

Used when:

Large pores
High heat fluxes
Rapid transients
Strong phase-to-phase imbalance

8.2 Modeling Implications

Increased computational cost
Additional closure parameters required
More sensitive to mesh and material data

Often implemented using UDFs for flexibility.

Heat Exchangers as Porous Systems

9.1 Why Heat Exchangers Are Special

Heat exchangers:

Are porous to the primary flow
Contain an auxiliary flow (coolant)
Exchange heat without mixing fluids

Explicitly meshing fins and tubes is impractical in system-level CFD.

Heat Exchanger Modeling Philosophy

Two goals:

Predict pressure loss in the primary flow
Predict heat transfer to/from the auxiliary fluid

This is achieved by:

Treating the exchanger core as a porous zone
Embedding heat exchange models inside it

Macro Heat Exchanger Models

11.1 Core Idea

Auxiliary flow is treated as 1D
Primary flow is fully 3D
Heat exchanger core is divided into macroscopic cells (“macros”)

Each macro:

Applies an energy balance
Exchanges heat locally with the primary flow

11.2 Effectiveness-Based Model

Uses a global effectiveness:

Interpolated from performance curves
Applied locally across the core

Assumptions:

Auxiliary fluid hotter than primary
Primary heat capacity rate is limiting

Simple and robust, but restrictive.

11.3 NTU-Based Model

More general:

Works regardless of which fluid is hotter
Handles reverse flow
Allows variable properties

Effectiveness is computed from NTU and capacity ratios.

Preferred for:

General engineering simulations
Automotive and HVAC applications

Dual-Cell Heat Exchanger Model

For highly non-uniform auxiliary flow:

Two overlapping meshes are used
One for primary flow
One for auxiliary flow

Coupled only through heat transfer.

Advantages:

Captures coolant maldistribution
Allows variable density and temperature fields

Limitations:

Higher cost
More setup complexity

Heat Exchanger Groups

Heat exchangers can be:

Connected in series
Connected in parallel

Useful for:

Radiator stacks
Multi-stage cooling systems

Grouping allows system-level modeling without geometric detail.

Engineering Intuition

Porous models trade geometry for parameters
Accuracy depends more on closure quality than mesh density
One-equation models are usually sufficient
Two-equation models are reserved for strong thermal imbalance
Heat exchangers are structured porous media with an extra energy pathway

Rule of thumb:

If geometry matters more than flow structure, use a porous model.

Study Priorities

If time is limited, the most important concepts to look into:

REV concept and upscaling logic
Porosity, permeability, and resistance terms
One- vs two-equation energy models
Effective thermal conductivity meaning
Macro vs dual-cell heat exchanger models
When simplification is justified

Key Takeaways

Porous media models replace geometry with averaged physics.
Momentum losses encode flow resistance.
Heat transfer can be modeled under equilibrium or non-equilibrium assumptions.
Effective properties are the central modeling challenge.
Heat exchangers are porous media with auxiliary energy transport.
Reduced-order models enable system-scale CFD.

Heat transfer Chapter 5: Porous media and heat exchangers