Radiation

For bodies with varying temperatures, the energy exchange can occur in three modes: conduction, convection and radiation. Conduction is the heat transfer by direct contact, convection is the heat transfer by motion of a material and radiation is the transfer of energy with the help of electromagnetic waves. In this article, we take a deeper look into the radiative heat transfer.

Electromagnetic radiation is emitted by heated surfaces. This means all materials with an absolute temperature above zero degrees Kelvin, radiate thermal energy. The hotter the object, the more it will radiate to its colder surroundings and the more important it becomes to take radiative heat transfer into account in your simulations. Radiation distinguishes itself from conduction and convection as it does not necessarily need a medium to carry it. Radiation can thus occur through any transparent medium, either a fluid or a solid, or a vacuum.

In radiative heat transfer, the Stefan-Boltzmann law is a well-known formula as it relates the heat flow rate emitted or absorbed from an object to the fourth power of its absolute surface temperature. \[q_{R} = \epsilon\sigma T^{4}A\] In the formula above, \(\epsilon \) represents the emissivity, \(\sigma \) the Stefan-Boltzmann constant, \(T\) the temperature expressed in Kelvin and \(A\) the surface of the emitting body. Besides the temperature of the surface, the rate at which a body radiates thermal radiation also depends on the surface properties. A surface with an emissivity of one is called a black body. This black body is an idealized object that would absorb all wavelengths of thermal radiation it would receive, without reflecting or transmitting any energy. Hence, it would totally absorb light and appear black at low temperatures. However, black bodies don’t exist in real life, in reality, all objects are gray bodies. This means incident radiation on a surface is partly reflected, absorbed and transmitted according to the surface properties. For gray surfaces, the following condition holds: \[\epsilon + \tau + \rho = 1\] Where \(\epsilon \) denotes the emissivity,\(\tau \) the transmissivity and \(\rho \) the reflectivity of the surface. Furthermore, Kirchoff discovered that a good emitter is also a good absorber so the emissivity (\(\epsilon \)) of a surface can be linked to the absorptivity of a surface (\(\alpha \)) as follows: \[\epsilon = \alpha \]

If a medium absorbs, emits or scatters a thermal ray as it passes through, it is called a participating medium. The ray can lose energy through absorption and scattering away from the ray. However, energy can also be gained from light sources in the medium through emission and scattering towards the ray. If the medium does not absorb, emit or scatter a thermal ray, the medium is called a non-participating medium.

The energy balance of the ray over an infinite layer of the medium, results in a differential equation, i.e. the radiative heat transfer equation (RTE). Within our software, the RTE is solved with the Finite Volume Discrete Ordinates Method or FvDOM model in short. This means the RTE is solved for a discrete number of finite solid angles. This model is the most comprehensive radiation model, which can be used for (semi-) transparent media and can account for scattering and wavelength dependent transmission. Our model can handle both participating and non-participating media. In the first case, the RTE is coupled with the general energy equation for fluids or solids and a source term is added in the energy equation to account for the radiative heat transfer, in the latter the source term is zero.

If large temperature differences are to be expected with respect to the surroundings, radiative heat transfer will become important as it relates to the fourth power of the temperature. In this case, radiation should be enabled. When the radiative heat transfer is expected to be small with regards to the convective heat transfer, radiation should not be enabled; unless this specifically needs to be modelled.

When radiative heat transfer is to be included in a specific region and the medium of the region is considered to be participating, (i.e. the medium absorbs, emits or scatters a thermal ray as it passes through) need to enable radiation in the concering region by checking the radiation box. In addition, the following parameters need to be set in the boundary settings of the boundaries related to the concerning region. If the concerning region is a solid type, also some material parameters related to radiation can be set:

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On the boundaries, an emissivity and absorptivity should be specified. The emissivity depends on the material of the surface and measures the capability to emit radiation. It is a representation of the ratio of energy radiated by the material to the energy radiated by a black body at the same temperature. The emissivity is considered constant for every material. In reality, the emissivity depends on the wavelength of the radiation that is being emitted. However, in CFD, rather the total emissivity is specified which is an integrated emissivity over all wavelengths. The emissivity of a surface is bound between 0 and 1. A perfect mirror for example, would reflect all energy, hence the emissivity and by Kirchhoff’s law also the absorptivity are zero. Within our software, the default value is set to 0.9.

The absorptivity is a dimensionless value between 0 and 1. If nothing is specified, the same value as the emissivity is taken by default.

When dealing with participating media in the simulation, some additional parameters need to be set. Since part of the radiative heat transfer can be absorbed in this case, leading to a change in radiation intensity, an absorption length should also be specified which represents the attenuation of the radiation intensity.