# AcuSolve Enclosure Radiation

## Enclosure Radiation Methodology

- View factor computation: In radiative heat transfer view factor is the proportion of radiation incident on one surface due to another surface. The view factors for each facet defining the radiation enclosure are computed using the hemicube algorithm and smoothed using least squares method as a pre-processing step. The view factors are not recomputed during the simulation.
- Heat flux addition: The radiative heat flux, based on view factors, computed using the Stefan-Boltzmann law and the total radiosity, based on Kirchoff's law, is added to the enthalpy transport equation during the solver run.

The enclosure radiation model is supported only on fluid mediums, that is, the fluid side of the fluid/solid surface.

View factor computation is an important point when dealing with moving mesh simulation. Since the view factors are not recomputed during the solver run the boundary elements forming the enclosure should not deform.

The total emissive power of a grey surface is given by the Stefan-Boltzmann law:

where
${\u03f5}_{i}$
is the emissivity of surface
i, given by an
EMISSIVITY_MODEL command;
$\sigma $
is the Stefan-Boltzmann constant, given by
stefan_boltzmann_constant;
T_{i} is the temperature of surface
i; and
T_{off} is the offset to convert to
an absolute temperature, given by the
absolute_temperature_offset parameter of the
EQUATION command. In addition, each surface receives part of
the total radiosity from each of the radiation surfaces:

where G_{i} is the total irradiation of
the surface i, F_{ij} is the view factor
from the surface i to surface
j and
J_{j} is the total radiosity from
surface j. The total radiosity of surface
i is

where ${\alpha}_{i}$ is the absorptivity and $1-{\alpha}_{i}$ is the reflectivity. From Kirchhoff' law and the grey surface assumption, $\alpha =\u03f5$ . The net radiation heat flux is thus ${q}_{i}={G}_{i}-{J}_{i}={\alpha}_{i}{G}_{i}-{W}_{i}$ and is added to the temperature equation. The radiation equation solves for all the radiation heat fluxes coupled together with the temperature equation.

View factors ${F}_{ij}$ are purely geometric entities defined as

where A_{i} and
A_{j} are the areas of surfaces
i and j,
respectively;
${\theta}_{i}$
and
${\theta}_{j}$
are the angles between the line connecting
$d{A}_{i}$
to
$d{A}_{j}$
and the normals to surfaces
$d{A}_{i}$
to
$d{A}_{j}$
; respectively; r
is the distance from
$d{A}_{i}$
to
$d{A}_{j}$
; and
${\delta}_{ij}$
is the visibility function, it is equal to one if
$d{A}_{i}$
to
$d{A}_{j}$
see each other, otherwise it is zero.

## View Factor Nomenclature

The view factors are computed using the hemicube algorithm. In a nutshell, in this
algorithm a hemicube is placed on the centroid of each radiation facet
i. The hemicube is discretized into
3n^{2} pixels; where
n is given by
num_hemicube_bins. All surfaces (facing surface
i) are then projected onto this hemicube. The
Z-buffering algorithm is used to compute the visibility. Once all surfaces are
projected, the pixels weighted by their view factor increments are added up to
determine a row of
F_{i}={F_{ij}}.

## Hemicube Discretization

The hemicube algorithm has three major assumptions, and hence sources of errors:
aliasing - the true projection of each visible face onto the hemicube can be
accurately accounted for by using a finite resolution hemicube; proximity - the
distance between faces is great compared to the effective diameter of the faces; and
visibility - the visibility between any two faces does not depend on the position
within either face. The aliasing error may be reduced through the use of larger
num_hemicube_bins values, at the expense of computational
cost. The computational cost is proportional to
n^{2} for sufficiently large values
of n. In addition, a random jittering algorithm
and a non-uniform hemicube pixelization are employed to minimize the aliasing error.
The proximity and visibility errors occur when two facing facets are too close to
each other. In this case, the geometry of one or both facets cannot be sufficiently
well-represented by their centroids. These errors can be reduced by splitting each
poorly-represented facet into smaller segments (sub-surfaces), computing the view
factors separately for each sub-surface and then adding them up. The computational
cost is proportional to the number of sub-surface divisions. The
max_surface_subdivision parameter may be used to trade-off
accuracy against cost.

A least-squares smoothing is automatically applied to the computed view factor matrix to enforce reciprocity and conservation:

${A}_{i}{F}_{ij}={A}_{j}{F}_{ji}$ (reciprocity)

${\sum}_{j}^{}{F}_{ij}=1$ (conservation)

The default for the Stefan-Boltzmann constant is given in MKS units: $\sigma =5.670\times {10}^{-8}\text{\hspace{0.17em}}W/\left({m}^{2}\xb0{K}^{4}\right)$ .

In British units it is: $\sigma =4.756\times {10}^{-11}\text{\hspace{0.17em}}Btu/\left(s\text{\hspace{0.17em}}f{t}^{2}\xb0{R}^{4}\right)$ .

In general, this constant must be converted to be consistent with the units chosen for use in the problem.