5.3.7 Surface-to-Surface (S2S) Radiation Model Theory

The surface-to-surface radiation model can be used to account for the radiation exchange in an enclosure of gray-diffuse surfaces. The energy exchange between two surfaces depends in part on their size, separation distance, and orientation. These parameters are accounted for by a geometric function called a "view factor''.

The main assumption of the S2S model is that any absorption, emission, or scattering of radiation can be ignored; therefore, only "surface-to-surface'' radiation need be considered for analysis.

For information about setting up the model, see this section in the separate User's Guide.

Gray-Diffuse Radiation

ANSYS FLUENT's S2S radiation model assumes the surfaces to be gray and diffuse. Emissivity and absorptivity of a gray surface are independent of the wavelength. Also, by Kirchoff's law [ 234], the emissivity equals the absorptivity ( $\epsilon = \alpha$). For a diffuse surface, the reflectivity is independent of the outgoing (or incoming) directions.

The gray-diffuse model is what is used in ANSYS FLUENT. Also, as stated earlier, for applications of interest, the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Thus, according to the gray-body model, if a certain amount of radiant energy ( $E$) is incident on a surface, a fraction ( $\rho E$) is reflected, a fraction ( $\alpha E$) is absorbed, and a fraction ( $\tau E$) is transmitted. Since for most applications the surfaces in question are opaque to thermal radiation (in the infrared spectrum), the surfaces can be considered opaque. The transmissivity, therefore, can be neglected. It follows, from the conservation of energy, that $\alpha + \rho = 1$, since $\alpha = \epsilon$ (emissivity), and $\rho = 1-\epsilon$.

The S2S Model Equations

The energy flux leaving a given surface is composed of directly emitted and reflected energy. The reflected energy flux is dependent on the incident energy flux from the surroundings, which then can be expressed in terms of the energy flux leaving all other surfaces. The energy reflected from surface $k$ is

where $q_{{\rm out},k}$ is the energy flux leaving the surface, $\epsilon_k$ is the emissivity, $\sigma$ is Boltzmann's constant, and $q_{{\rm in},k}$ is the energy flux incident on the surface from the surroundings.

The amount of incident energy upon a surface from another surface is a direct function of the surface-to-surface "view factor,'' $F_{jk}$. The view factor $F_{jk}$ is the fraction of energy leaving surface $k$ that is incident on surface $j$. The incident energy flux $q_{{\rm in},k}$ can be expressed in terms of the energy flux leaving all other surfaces as

where $A_k$ is the area of surface $k$ and $F_{jk}$ is the view factor between surface $k$ and surface $j$. For $N$ surfaces, using the view factor reciprocity relationship gives

so that

Therefore,

which can be written as

where $J_k$ represents the energy that is given off (or radiosity) of surface $k$, and $E_k$ represents the emissive power of surface $k$. This represents $N$ equations, which can be recast into matrix form as

where ${\mbox{\boldmath$K$}}$ is an $N\times N$ matrix, ${\mbox{\boldmath$J$}}$ is the radiosity vector, and ${\mbox{\boldmath$E$}}$ is the emissive power vector.

Equation  5.3-76 is referred to as the radiosity matrix equation. The view factor between two finite surfaces $i$ and $j$ is given by

where $\delta_{ij}$ is determined by the visibility of $dA_j$ to $dA_i$. $\delta_{ij}$ = 1 if $dA_j$ is visible to $dA_i$ and 0 otherwise.

Clustering

The S2S radiation model is computationally very expensive when there is a large number of radiating surfaces. To reduce the computational time as well as the storage requirement, the number of radiating surfaces is reduced by creating surface "clusters''. The surface clusters are made by starting from a face and adding its neighbors and their neighbors until a specified number of faces per surface cluster is collected.

An algorithm has been implemented for the creation of surface clusters which is faster and supports non-conformal interfaces, hanging nodes, or mesh adaption. This algorithm is now the default. If you wish to use the old algorithm, you may use the TUI command but adaption and non-conformal interfaces will not be supported.

The radiosity, $J$, is calculated for the surface clusters. These values are then distributed to the faces in the clusters to calculate the wall temperatures. Since the radiation source terms are highly non-linear (proportional to the fourth power of temperature), care must be taken to calculate the average temperature of the surface clusters and distribute the flux and source terms appropriately among the faces forming the clusters.

The surface cluster temperature is obtained by area averaging as shown in the following equation:

where $T_{sc}$ is the temperature of the surface cluster, and $A_f$ and $T_f$ are the area and temperature of face $f$. The summation is carried over all faces of a surface cluster.

Smoothing

Smoothing can be performed on the view factor matrix to enforce the reciprocity relationship and conservation.

The reciprocity relationship is represented by

where $A_i$ is the area of surface $i$, $F_{ij}$ is the view factor between surfaces $i$ and $j$, and $F_{ji}$ is the view factor between surfaces $j$ and $i$.

Once the reciprocity relationship has been enforced, a least-squares smoothing method [ 175] can be used to ensure that conservation is satisfied, i.e.,