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13.3.7 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.



Gray-Diffuse Radiation


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 [ 248], 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 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 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


 q_{{\rm out},k} = \epsilon_k \sigma T_k^4 + \rho_k q_{{\rm in},k} (13.3-70)

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


 A_k q_{{\rm in},k} = \sum_{j=1}^N A_j q_{{\rm out},j} F_{jk} (13.3-71)

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


 A_j F_{jk} = A_k F_{kj} \mbox{ for } j=1,2,3, \ldots N (13.3-72)

so that


 q_{{\rm in},k} = \sum_{j=1}^N F_{kj} q_{{\rm out},j} (13.3-73)

Therefore,


 q_{{\rm out},k} = \epsilon_k \sigma T_k^4 + \rho_k \sum_{j=1}^N F_{kj} q_{{\rm out},j} (13.3-74)

which can be written as


 J_k = E_k + \rho_k \sum_{j=1}^N F_{kj} J_j (13.3-75)

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


 {\mbox{\boldmath$K$}} {\mbox{\boldmath$J$}} = {\mbox{\boldmath$E$}} (13.3-76)

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  13.3-76 is referred to as the radiosity matrix equation. The view factor between two finite surfaces $i$ and $j$ is given by


 F_{ij} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos \theta_i \cos \theta_j}{\pi r^2} \delta_{ij} dA_i dA_j (13.3-77)

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 are 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.

A new algorithm has been implemented for the creation of surface clusters which is faster and supports non-conformal interfaces, hanging nodes, or grid 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:


 T_{sc} = \left(\frac{\sum_f A_f T_f^4}{\sum A_f} \right) ^{1/4} (13.3-78)

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


 A_i F_{ij} = A_j F_{ji} (13.3-79)

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 [ 191] can be used to ensure that conservation is satisfied, i.e.,


 \sum F_{ij} = 1.0 (13.3-80)


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