# RADIATION

Specifies global radiation algorithms and parameters.

## Type

AcuSolve Command

## Syntax

RADIATION {parameters}

## Qualifier

This command has no qualifier.

## Parameters

- view_factor_type (enumerated) [=hemicube]
- Calculation method for the view factor.
- hemicube
- Hemicube. Requires num_hemicube_bins and max_surface_subdivision.

- num_hemicube_bins or bins (integer) >0 [=200]
- Number of bins for the view factor calculation. Used with hemicube view factor type.
- max_surface_subdivision (integer) >=1 [=1]
- Maximum number of segments to split a surface facet when proximity condition is violated. Used with hemicube view factor type.
- smoothing_type (enumerated) [=least_squares]
- Type of view factor smoothing algorithm.
- least_squares
- Least squares smoothing.

- stefan_boltzmann_constant (real) >0 [=5.67e-8]
- Stefan-Boltzmann constant. Default is in MKS units.
- num_symmetry_planes (integer) >=0 <=3 [=0]
- Number of orthogonal symmetry planes in geometrical model.
- symmetry_center or center (array) [={0,0,0}]
- Any point that lies at the intersection of all symmetry planes. Used with num_symmetry_planes>0.
- symmetry_direction_1 (array) [={1,0,0}]
- Normal direction to first symmetry plane. Used with num_symmetry_planes>0.
- symmetry_direction_2 (array) [={0,1,0}]
- Normal direction to second symmetry plane. Used with num_symmetry_planes>1.
- symmetry_direction_3 (array) [={0,0,1}]
- Normal direction to third symmetry plane. Used with num_symmetry_planes=3.

## Description

This command specifies the global parameters for enclosure radiation heat transfer as defined by the RADIATION_SURFACE and EMISSIVITY_MODEL commands. Radiation commands with default values are automatically created for the P1 radiation method to model radiative heat transfer in a grey, optically thick participating medium. These parameters do not apply to radiation boundary conditions defined by ELEMENT_BOUNDARY_CONDITION commands or to heat fluxes defined by SOLAR_RADIATION and related commands.

```
EQUATION {
...
radiation = enclosure
absolute_temperature_offset = 273.14
}
RADIATION {
view_factor_type = hemicube
num_hemicube_bins = 200
max_surface_subdivision = 4
smoothing_type = least_squares
stefan_boltzmann_constant = 5.67e-08
num_symmetry_planes = 1
symmetry_center = { 0, 0, 0 }
symmetry_direction_1 = { 1, 0, 0 }
}
```

specifies that the enclosure radiation equation is to be solved, the hemicube algorithm with 200 bins per side is used to calculate the view factor, each radiation facet may be split into a maximum of 4 sub-surfaces to reduce proximity and visibility errors, the least squares method is used to smooth the view factor, the Stefan-Boltzmann constant is 5.67x10-, and one symmetry plane is modeled.

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.

Symmetry planes may be modeled with num_symmetry_planes>0. This allows geometrical models that are a half, quarter, or eighth of the corresponding "ull"models. All planes must be mutually orthogonal. The radiation facets are reflected across each symmetry plane to create the full model. The view factors are computed on this model before being rescaled to the original model. A given surface facet may then be able to see itself. Normally no RADIATION_SURFACE command is used on a surface lying in any of the symmetry planes, except if (partial) shielding is modeled. Specifying symmetry planes in this command only affects the calculation of view factors and has no effect on any other aspect of the simulation model.