Conjugate Heat Transfer (CHT) in OpenFOAM: From Physics to Practical Implementation

Conjugate Heat Transfer (CHT) refers to thermal problems in which heat transfer occurs simultaneously in solid and fluid regions, with thermal coupling at their interfaces. In real engineering systems—such as heat exchangers, electronic cooling assemblies, turbine blades, and solar receivers—heat does not remain confined to a single domain. It conducts through solids, convects within fluids, and may radiate between surfaces. A proper numerical model must therefore account for all three modes of heat transfer in a coupled and physically consistent manner.

Conjugate heat transfer is not a separate mode of heat transfer — it is the coupling of conduction, convection, and radiation across different regions.

This article explains the physical meaning of conduction, convection, and radiation, presents their governing differential equations, and describes how each is modeled and implemented in OpenFOAM.

Conduction – Heat Transfer by Diffusion

Conduction is the transfer of thermal energy through a material due to microscopic molecular interactions. There is no bulk motion of matter involved. Energy flows from regions of higher temperature to lower temperature because of a temperature gradient. In solids, conduction is typically the dominant mode of heat transfer.

The physical basis of conduction is described by Fourier’s Law:

\mathbf{q} = -k \nabla T

where $\mathbf{q}$ is the heat flux vector, $k$ is the thermal conductivity, and $T$ is temperature. The negative sign indicates that heat flows in the direction of decreasing temperature.

Applying conservation of energy leads to the transient heat diffusion equation:

\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T)

Here, $\rho$ is density and $c_p$ is specific heat capacity. When thermal conductivity is constant, the equation simplifies to:

\frac{\partial T}{\partial t} = \alpha \nabla^2 T

where thermal diffusivity $\alpha$ is defined as:

\alpha = \frac{k}{\rho c_p}

Thermal diffusivity is a material property that determines how quickly temperature disturbances spread inside a solid.

For steady-state conduction without internal heat generation, the equation reduces to:

\nabla^2 T = 0

This is the Laplace equation.

In OpenFOAM, pure heat conduction in solids can be solved using the solver laplacianFoam . This solver directly handles diffusion-type equations and discretizes the Laplacian operator using the finite volume method. It is particularly useful for steady or transient solid conduction problems where fluid flow is not involved.

If your problem contains only diffusion and no velocity field, laplacianFoam is the most direct and physically consistent starting point.

Convection – Heat Transfer Due to Fluid Motion

Convection occurs when heat is transported by the bulk motion of fluid molecules. When a solid surface is heated, the adjacent fluid layer increases in temperature. If the fluid is free to move, temperature differences may lead to density variations, which generate motion and transport heat away from the surface.

Unlike conduction, convection cannot exist without a velocity field.

The governing energy equation for a moving fluid includes both diffusion and convection:

\rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{U} \cdot \nabla T \right) = \nabla \cdot (k \nabla T)

The term $\mathbf{U} \cdot \nabla T$ represents convective transport, where $\mathbf{U}$ is the velocity field.

The convective term is what makes thermal problems in fluids fundamentally different from solid conduction problems.

Buoyancy and the Boussinesq Approximation

In buoyancy-driven flows, temperature variations cause density changes, which in turn generate body forces. For small temperature differences, the density variation can be approximated using the Boussinesq approximation:

\rho = \rho_0 \left[ 1 – \beta (T – T_0) \right]

Here, $\beta$ is the thermal expansion coefficient. The key assumption is that density variations are small and only significant in the buoyancy term of the momentum equation. Everywhere else, density is treated as constant.

The Boussinesq approximation allows you to model natural convection without solving a fully compressible flow.

In OpenFOAM, buoyancy-driven incompressible flows are handled using the solver buoyantBoussinesqFoam. This solver couples the momentum and energy equations while applying the Boussinesq assumption.

When temperature differences are large and compressibility effects become important, the solver buoyantPimpleFoam is used. It solves the fully compressible Navier–Stokes equations together with the energy equation, making it suitable for high-temperature flows and significant density variations.

Choosing between buoyantBoussinesqFoam and buoyantPimpleFoam is primarily a physics decision — not just a numerical one.

Radiation – Heat Transfer Without a Medium

Radiation differs fundamentally from conduction and convection because it does not require a material medium. Energy is transferred through electromagnetic waves. A classic example is the transfer of heat from the Sun to the Earth.

Radiation is the only mode of heat transfer that can occur in a vacuum.

The radiative heat flux emitted by a surface is described by the Stefan–Boltzmann law:

q = \sigma \epsilon T^4

where $\sigma$ is the Stefan–Boltzmann constant and $\epsilon$ is surface emissivity. The fourth-power dependence on temperature explains why radiation becomes dominant at high temperatures.

The complete description of radiative heat transfer in participating media is given by the Radiative Transfer Equation (RTE), which accounts for absorption and emission along a radiation path.

In OpenFOAM, radiation effects can be modeled using methods such as fvDOM and the viewFactor radiation model.

The fvDOM method discretizes angular space into a finite number of directions and solves the radiative transfer equation for each direction. It is appropriate for participating media and high-temperature applications. The view factor model is typically used for surface-to-surface radiation in enclosures where the medium does not significantly absorb or emit radiation.

Radiation models introduce additional source terms in the energy equation, coupling thermal radiation with conduction and convection.

Conjugate Heat Transfer – Coupling Everything Together

In real systems, conduction, convection, and radiation rarely act independently. In a heat exchanger, heat conducts through the solid wall, convects into the fluid, and may also radiate between hot surfaces.

At the solid–fluid interface, two physical conditions must always be satisfied: temperature continuity and heat flux conservation. Mathematically, this means that temperature must be equal on both sides of the interface and that the normal heat flux leaving the solid equals the heat flux entering the fluid.

A CHT simulation enforces energy conservation across regions — not just within them.

In OpenFOAM, multi-region CHT simulations are performed by defining separate meshes for solid and fluid regions and coupling them through interface boundary conditions. Each region solves its own energy equation, and the solver ensures flux continuity at the interface automatically.

This capability allows realistic simulations of complex engineering systems where thermal interaction between materials and fluids is critical.

From Theory to Practice

Understanding the theory is essential, but mastery comes from implementation. That is why we have developed a comprehensive course that takes you from fundamental heat transfer principles to advanced thermal and buoyancy-driven flow simulations in OpenFOAM.

Practical Heat Transfer Simulation with OpenFOAM: A Hands-On Beginner Course

The course begins with heat conduction using the laplacianFoam solver, covering governing equations and boundary conditions. It then progresses to buoyancy-driven incompressible flows using the Boussinesq approximation, and further to fully compressible buoyant flows using buoyantPimpleFoam. You will learn how to set up multi-region meshes for conjugate heat transfer simulations, configure thermophysical models, implement radiation using fvDOM, apply source terms through FVOptions, and select appropriate discretization schemes to improve solution accuracy.

By the end of the course, you will not just know the equations — you will know how to implement them correctly in real OpenFOAM cases.

All lecture notes, case files, and step-by-step instructions are provided, enabling you to reproduce every simulation and build a strong foundation for your own research or engineering projects.

Conjugate Heat Transfer is about understanding how energy moves through different media and how those mechanisms interact. When properly configured, OpenFOAM becomes a powerful and flexible platform for solving advanced thermal problems with strong physical fidelity.