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The Dynamics of Thermal Heat Flux

Key Takeaways

  • Thermal heat flux is the rate of thermal energy flowing per unit area of the heat transfer surface per unit time.

  • The heat flux density at a given point can be easily described with the help of Fourier’s law. 

  • When modeling the heat transfer in a fluid flow system, heat flux can be calculated for conduction, radiation, resistance, and diffusion.

Thermal heat flux simulation

Heat plays a significant role in the design of engineering systems such as electronic devices, aerodynamic components, and cooling systems. Often, the calculation of heat transfer in such systems is done using CFD simulations to analyze factors such as thermal heat flux. However, this heat flux analysis can depend on mechanisms such as conduction, radiation, resistance, and diffusion.

In the study of systems such as aerodynamics, the knowledge of these thermal heat flux types facilitates solving airflow problems and cooling system designs. This article will discuss thermal heat flux, its associated equation, and the advantages of using CFD applications to solve heat flux problems

Understanding Thermal Heat Flux

Heat flux can be observed in every situation where heat transfer occurs. For instance, when hot water flows through a colder body temperature or when we touch ice, heat transfer occurs from the hot to the cold surface. Given this understanding, thermal heat flux can be defined as the  measure of the rate of thermal energy that flows per unit area of the heat transfer surface per unit time. By this definition, the thermal heat flux has the unit of W/m² i.e., watt per square meter. At a certain point in space, the thermal heat flux is measured by limiting the surface size to the smallest degree. As both the magnitude and direction can be defined, heat flux can also be considered a vector quantity.

The calculation of thermal heat flux can be explained by Fourier’s law for heat conduction, which can be expressed as:

Heat flux equation

Note that:

q is the heat flux 

k is the thermal conductivity

∇T is the thermal gradient

The above formula can be used to derive the relationship between thermal conductivity, resistivity, resistance, and diffusivity in the thermal heat flux. 

Thermal Heat Flux Modes

Fourier’s equation plays an important role in the concept of thermal heat flux. The equation covers the basic modes of heat transfer including conduction, resistivity, resistance, and diffusivity. 


Thermal conduction is simply the measure of heat conduction between a fluid and the material surface due to the molecular interaction. The conductive heat flux equation can then be derived from Fourier’s heat flux equation:

Formula for heat flux due to thermal conductivity

The conductive heat flux vector is directly proportional to the temperature gradient. If we consider that heat transfer occurs during the flow from two surfaces with temperatures T1 and T2, where, T2>T1, and the distances between the two surfaces is L, then the heat flux formula can be expressed as:

Heat flux formula between two surfaces with known temperatures


During heat transfer mechanisms, the resistivity defines the temperature difference with which the heat flow is resisted. It is simply the inverse of thermal conductance. For 1-D steady state heat transfers through a plane wall, heat flux can be defined as:

 Heat flux formula for thermal resistance

Note that:

A is the wall area 

L is the distance between the boundaries

Thus, the resistivity, R, can be expressed as:

Relation between resistivity and heat flux


Radiation heat flux is associated with heat transfer because of electromagentic radiation and may occur without the need for any medium. The radiative heat flux becomes important in heating/cooling analysis when the surface temperatures are relatively higher. Therefore, for heat transfer from surfaces 1 to 2, the heat transfer due to radiation can be expressed as:

Heat transfer due to radiation

Note that:

∊ is the surface emissivity 

σ is Stefan-Boltzmann’s constant

The radiative energy flux is E = σT4.


In heat transfer problems, diffusivity indicates the dispersion rate of heat through an object. While the thermal diffusivity can be difficult to analyze among fluids, it can be presented for solids as: 

Diffusivity formula

Note that:

 k is the thermal conductivity

p is the density

c is the specific heat capacity

The heat equation in terms of thermal diffusivity in a uniform object can be expressed as:

Heat transfer due to thermal diffusivity

Note that ▽² is the Laplace operator. 

Thermal Heat Flux Considerations in Engineering Designs

In applications such as aerodynamics, the analysis of heat transfer, especially at high speed, is a must given aerodynamic heating and pressure conditions. The calculation of heat fluxes in such applications for conduction, resistivity, resistance, and diffusivity problems can be done through CFD solver applications that support multi-dimensional systems analysis. With tools such as Omnis, complex thermal management problems can be solved through high-fidelity CFD simulation and modeling.  

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About the Author

With an industry-leading meshing approach and a robust host of solver and post-processing capabilities, Cadence Fidelity provides a comprehensive Computational Fluid Dynamics (CFD) workflow for applications including propulsion, aerodynamics, hydrodynamics, and combustion.

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