Specifies radiative boundary conditions for the enclosure, p1_model and discrete_ordinate radiation models.

For the enclosure model, a radiation heat flux condition is applied to the surface. For the p1_model the condition specifies the emissivity of the surface used by the Marshak boundary condition. For the discrete_ordinate model, the condition specifies surface properties, such as emissivity and diffuse fraction. The emissivity is used to account for radiation emission from the surface. For example, $\epsilon \sigma {T}_{w}^{4}/\pi$ , where $\epsilon$ is the emissivity, $\sigma$ is Stefan-Boltzmann constant, and ${T}_{w}$ is the wall temperature.

AcuSolve Command

User-given name.

## Parameters

shape (enumerated) [no default]
Shape of the surfaces in this set.
three_node_triangle or tri3
Three-node triangle.
six_node_triangle or tri6
Six-node triangle.
element_set or elem_set (string) [no default]
User-given name of the parent element set.
surfaces (array) [no default]
List of element surfaces.
surface_sets (list) [={}]
List of surface set names (strings) to use in this command. When using this option, the connectivity, shape, and parent element of the surfaces are provided by the surface set container and it is unnecessary to specify the shape, element_set and surfaces parameters directly to the RADIATION_SURFACE command. This option is used in place of directly specifying these parameters. In the event that both of the surface_sets and surfaces parameters are provided, the full collection of surface elements is read and a warning message is issued. The surface_sets option is the preferred method to specify the surface elements. This option provides support for mixed element topologies and simplifies pre-processing and post-processing.
type (enumerated) [=wall]
Type of the boundary surface. For the enclosure and p1_model radiation models, the type can be either wall or opening. For the discrete_ordinate radiation model the type can be wall, opening or radiation_interface.
auto
Automatic radiation surface treatment to determine whether a surface is treated as type = wall or type = opening. Used with all radiation models.
wall
Wall. Requires an emissivity_model for all radiation models and agglomeration for enclosure.
opening
Opening. Requires emissivity_model and opening_temperature.
Radiation_interface. Enables the transmission and reflection of radiative intensity at an interface between two participating media. Available when radiation = discrete_ordinates.
external_emissivity_model (string) [=none]
User-given name of the external emissivity model of an exterior facing surface if radiation_interface_type = external.
external_temperature (real)>=0 [= 273.15]
Defines the temperature of the fluid surrounding the domain if radiation_interface_type = external.
external_temperature_multiplier_function (string) [=none]
User-given name of the multiplier function for scaling the external temperature. If none, no scaling is performed.
specular_ordinate_averaging (enumerated) [=one_ordinate]
The specular ordinate averaging parameter is used to determine the direction of the specular ordinate. Two averging methods are available: one_ordinate and three_ordinates. If specular_ordinate_averaging = one_ordinate, this method is to search for the closest specular ordinate direction, if specular_ordinate_averaging = three_ordinates, the specular ordinate direction is calculated by averaging the three closest ordinate directions. This parameter is used for specular interfaces when the diffusion fraction is less than 1.0. Requires radiation_interface. Available when radiation = discrete_ordinates.
emissivity_model (string) [no default]
User-given name of the emissivity model. Used with wall and opening types.
opening_temperature or temp (real) >=0 [=273.15]
Opening temperature. Used with opening type.
opening_temperature_multiplier_function (string) [=none]
User-given name of the multiplier function for scaling the opening temperature. If none, no scaling is performed. Used with opening type.
agglomeration (boolean) [=on]
Flag specifying whether to agglomerate the surface elements. Used with wall type. A value of on requires max_agglomeration_surfaces, max_agglomeration_angle and max_agglomeration_radius.
max_agglomeration_surfaces (integer) >=0 [=25]
Maximum number of surfaces in one agglomeration. Used with on agglomeration. If zero, this option is ignored.
max_agglomeration_angle or angle (real) >=0 <=180 [=10]
Maximum angle between surfaces allowed in any agglomeration. Used with on agglomeration.
Maximum radius of agglomeration. Used with on agglomeration. If zero, this option is ignored.
integrated_output_frequency or intg_freq (integer) >=0 [=1]
Time step frequency at which to output the integrated radiation heat flux. If zero, this option is ignored.
integrated_output_time_interval or intg_intv (real) >=0 [=0]
Time frequency at which to output the integrated radiation heat flux. If zero, this option is ignored.
nodal_output_frequency or nodal_freq (integer) >=0 [=0]
Time step frequency at which to output radiation heat flux at the nodes of the surface. If zero, this option is ignored.
nodal_output_time_interval or nodal_intv (real) >=0 [=0]
Time frequency at which to output radiation heat flux at the nodes of the surface. If zero, this option is ignored.
diffused_fraction (real) [=1.0]
Diffused fraction defines the proportion of reflected radiation intensity at a surface that is diffused.

## Description

This command specifies a radiation heat flux condition on a set of surfaces (element faces). This condition is coupled to all other radiation surfaces. The RADIATION command provides a detailed description of this coupling.

The surfaces of a radiation surface are defined with respect to the elements of an element set. For example,
ELEMENT_SET( "interior" ) {
shape                        = four_node_tet
elements                     = { 1, 8, 3, 4, 9 ;
3, 3, 4, 9, 5 ;
... }
...
}
RADIATION_SURFACE( "wall" ) {
type                         = wall
shape                        = three_node_triangle
element_set                  = "interior"
surfaces                     = { 1, 12, 9, 3, 4 ;
3, 52, 5, 3, 4 ; }
emissivity_model             = "emissivity"
integrated_output_frequency  = 2
}

specifies a radiation heat flux condition to be applied to two surfaces of the element set "interior" using the emissivity model "emissivity", and the integral of the radiation heat flux is to be output every two steps.

There are two main forms of this command. The legacy version (or single topology version) of the command relies on the use of the surfaces parameter to define the surfaces. When using this form of the command, all surfaces within a given set must have the same shape, and it is necessary to include both the element_set and shape parameters in the command. shape specifies the shape of the surface. This shape must be compatible with the shape of the "parent" element set whose user-given name is provided by element_set. The element set shape is specified by the shape parameter of the ELEMENT_SET command. The compatible shapes are:
Element Shape
Surface Shape
four_node_tet
three_node_triangle
five_node_pyramid
three_node_triangle
five_node_pyramid
six_node_wedge
three_node_triangle
six_node_wedge
eight_node_brick
ten_node_tet
six_node_triangle

The surfaces parameter contains the faces of the element set. This parameter is a multi-column array. The number of columns depends on the shape of the surface. For three_node_triangle, this parameter has five columns, corresponding to the element number, of the parent element set, a unique, within this set surface number, and the three nodes of the element face. For four_node_quad, surfaces has six columns, corresponding to the element number, a surface number, and the four nodes of the element face. For six_node_triangle, surfaces has eight columns, corresponding to the element number, a surface number, and the six nodes of the element face. One row per surface must be given. The three, four, or six nodes of the surface may be in any arbitrary order, since they are reordered internally based on the parent element definition.

The surfaces may be read from a file. For the above example, the surfaces may be placed in a file, such as wall.srf:
1 12 9 3 4
3 52 5 3 4
RADIATION_SURFACE ( "no-slip wall" ) {
shape        = three_node_triangle
element_set  = "interior"
surfaces     = Read( "wall.srf" )
...
}
The mixed topology form of the RADIATION_SURFACE command provides a more powerful and flexible mechanism for defining the surfaces. Using this form of the command, it is possible to define a collection of surfaces that contains different element shapes. This is accomplished through the use of the surface_sets parameter. The element faces are first created in the input file using the SURFACE_SET command, and are then referred to by the RADIATION_SURFACE command. For example, a collection of triangular and quadrilateral element faces can be defined using the following SURFACE_SET commands.
SURFACE_SET( "tri faces" ) {
surfaces       = { 1, 1, 1, 2, 4 ;
2, 2, 3, 4, 6 ;
3, 3, 5, 6, 8 ; }
shape          = three_node_triangle
volume_set     = "tetrahedra"
}
SURFACE_SET( "quad faces" ) {
surfaces       = { 1, 1, 1, 2, 4, 9 ;
2, 2, 3, 4, 6, 12 ;
3, 3, 5, 6, 8, 15 ; }
volume_set     = "prisms"
Then, a single RADIATION_SURFACE command is defined that contains the tri and quad faces as follows:
RADIATION_SURFACE ( "no-slip wall" ) {
surface_sets       = {"tri_faces", "quad_faces"}
...
}
The list of surface sets can also be placed in a file, such as surface_sets.srfst:
tri faces
quad faces
RADIATION_SURFACE ( "no-slip wall" ) {
...
}

The mixed topology version of the RADIATION_SURFACE command is preferred. This version provides support for multiple element topologies within a single instance of the command and simplifies pre-processing and post-processing. In the event that both the surface_sets and surfaces parameters are provided in the same instance of the command, the full collection of surface elements is read and a warning message is issued. Although the single and mixed topology formats of the commands can be combined, it is strongly recommended that they are not.

For the enclosure model, all data from all RADIATION_SURFACE commands are pre-processed to form view factors. Since it is usually too expensive to store and process radiation heat transfer between a pair of each element faces, the surface data are "agglomerated" to reduce the number of view factors. Several parameters are provided to help control the agglomeration. For example,
RADIATION_SURFACE( "wall" ) {
...
type                        = wall
emissivity_model            = "emissivity"
agglomeration               = on
max_agglomeration_surfaces  = 100
max_agglomeration_angle     = 20
diffused_fraction           = 0.9
}

specifies that each agglomeration contains no more than 100 surfaces, the angles between the outward normals of these surfaces are no more than 20 degrees, and the radius of the agglomerated surface no greater than 50 percent of the radius of the entire surface set. In addition to these constraints, each agglomeration must contain only one emissivity model. Several parameters must be the same across all radiation surfaces; these are given in the RADIATION command. When accuracy is more important than the cost of computing the view factors, for example, a small, hot surface, agglomeration should be set to off. In this case the radiation heat flux will be computed for each element face in the set.

The opening type provides a method of fully enclosing a fluid domain that is not be completely surrounded by walls. This type is appropriate for inlets, outlets, and surfaces that approximate infinity in external flows. The primary assumption is that the surface is at a single given temperature of opening_temperature. This assumption allows the entire set to be combined into one agglomerated facet, so agglomeration and associated parameters are ignored. An opening is typically modeled as a black body, with an emissivity of one. However, other emissivity models may be used with an opening.

The opening_temperature_multiplier_function parameter may be used to scale the opening temperature. For example,
RADIATION_SURFACE( "inlet" ) {
...
type                                    = opening
emissivity_model                        = "black body"
opening_temperature                     = 1
opening_temperature_multiplier_function = "inlet temperature"
}
MULTIPLIER_FUNCTION( "inlet temperature" ) {
type                                    = cubic_spline
curve_fit_values                        = { 0, 295 ;
12, 312 ;
24, 320 ; }
curve_fit_variable                      = time
}
EMISSIVITY_MODEL( "black body" ) {
type                                    = constant
emissivity                              = 1
}

If either integrated_output_frequency or integrated_output_time_interval is non-zero, the surface integral of the radiation heat flux will be output at the end of the run. If both are zero, no integrated radiation heat flux data is written to disk.

Similarly, if either nodal_output_frequency or nodal_output_time_interval is non-zero, the nodal values of the radiation heat flux will be output at the end of the run. If both are zero, no nodal radiation heat flux data is written to disk.

Run times may not coincide with integrated_output_time_interval or nodal_output_time_interval. In these cases, the corresponding data are output for every time step which passes through a multiple of output_time_interval or nodal_output_time_interval.

Once the surface quantities have been written to disk, they can be translated to other formats using the AcuTrans program and other post-processing modules; see the AcuSolve Programs Reference Manual for details.

## Reflection and Transmission for Specular Interfaces of Type Internal

For specular interfaces the diffused_fraction = 0.
Reflection at the interface is governed by the angle of incidence of radiative intensity to a surface and the refractive indices of the two media. The cosine of the incident angle for the incoming ordinate is given by(1)
$\mathrm{cos}{\theta }_{1}={\Omega }_{I}^{1}\cdot n$
where $n$ is the outward facing normal direction at the interface (towards the second medium) and ${\Omega }_{I}^{1}$ is the unit direction vector of incoming radiation intensity to the surface, given by(2)
${\Omega }_{I}^{1}={\Omega }_{R}^{1}-2\left({\Omega }_{R}^{1}\cdot n\right)n$

where ${\Omega }_{R}^{1}$ is the unit reflected ordinate direction vector and also represents the current ordinate direction being solved. The equivalent calculation can also be performed for medium two.

Radiative intensity that is transmitted into a second medium undergoes refraction governed by Snell’s law, ${n}_{1}\mathrm{sin}{\theta }_{1}={n}_{2}\mathrm{sin}{\theta }_{2}$ , or equivalently in vector form ${n}_{1}\left(n×{\Omega }_{I}^{1}\right)={n}_{2}\left(n×{\Omega }_{R}^{2}\right)$ , where ${n}_{1}$ and ${n}_{2}$ are the refractive indices of mediums one and two defined in MATERIAL_RADIATION_MODEL for the material_model of each ELEMENT_SET. ${\theta }_{1}$ and ${\theta }_{2}$ are the angles of incidence and refraction of radiative intensity relative to the interface normal, respectively. The incoming direction vector in medium two for a ray refracted from medium two to one is given by(3)
$\frac{{n}_{1}}{{n}_{2}}{\Omega }_{R}^{1}+\left(\frac{{n}_{1}}{{n}_{2}}\mathrm{cos}{\theta }_{1}-\sqrt{1-{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}\left(1-\mathrm{cos}{\theta }_{1}\right)}\right)n$

providing the expression under the radicand is greater than zero; otherwise total internal reflection occurs, which is discussed later.

The actual reflected and refracted directions differ slightly from the calculated direction since these directions will unlikely coincide with a discrete ordinate direction. Since the number of directions is governed by the order of radiation_quadrature in EQUATION (S2-S16), higher quadrature orders are more accurate for interface problems.

Depending on the refractive indices of the two media and the angle of incidence, ${\theta }_{1}$ , the proportion of radiation intensity that is reflected and transmitted will vary. If ${n}_{1}<{n}_{2}$ , then the radiative intensity in medium one will be partially reflected and partially transmitted into a cone defined by the critical angle, ${\theta }_{c}$ , which is given by:(4)

The extremity being an intensity ray that grazes the interface and is transmitted exactly at the critical angle into the other domain.

The reflected proportion from ${\Omega }_{I}^{1}\to {\Omega }_{R}^{1}$ , or reflectance, is given by (5)
${r}_{12}=\frac{1}{2}{\left(\frac{{n}_{1}\mathrm{cos}{\theta }_{1}-{n}_{2}\mathrm{cos}{\theta }_{2}}{{n}_{1}\mathrm{cos}{\theta }_{1}+{n}_{2}\mathrm{cos}{\theta }_{2}}\right)}^{2}+\frac{1}{2}{\left(\frac{{n}_{2}\mathrm{cos}{\theta }_{1}-{n}_{1}\mathrm{cos}{\theta }_{2}}{{n}_{2}\mathrm{cos}{\theta }_{1}+{n}_{1}\mathrm{cos}{\theta }_{2}}\right)}^{2}$

and the transmitted proportion from ${\Omega }_{I}^{2}\to {\Omega }_{R}^{1}$ , or transmittance, is given by ${\tau }_{21}=1-{r}_{12}$ .

In the second medium, for the current scenario where ${n}_{2}>{n}_{1}$ , if ${\theta }_{2}<{\theta }_{c}$ the radiative intensity is, as for medium one, partially reflected and partially transmitted. The reflection coefficient is as described above since ${r}_{21}={r}_{12}$ . If ${\theta }_{2}>{\theta }_{c}$ , then total internal reflection occurs and ${r}_{21}=1.0$ and ${\tau }_{12}=0.0$ , meaning no transmission of radiative intensity into the second medium or from the first medium. This is shown in the image above with the gray dashed lines.

In AcuSolve, the outgoing radiative intensity on side one of the interface for the current ordinate direction, ${\Omega }_{R}^{1}$ , is given by(6)
Where for medium one, the first term on the right-hand side represents the reflected intensity in medium one and the second term represents the transmitted intensity from medium two to one. For medium two, if the current ordinate direction is ${\Omega }_{R}^{2}$ then the intensity outgoing radiative intensity is given by(7)

For ${n}_{2}<{n}_{1}$ , the subscripts of the above analysis must be exchanged, and total internal reflection will now occur in medium one.

## Reflection and Transmission for Diffuse Interfaces of Type Internal

For diffuse interfaces the diffused_fraction = 1.

If the interface is diffused, the reflectivity of the interface is given by the hemispherically averaged reflectance:(8)
${r}_{D,12}=\frac{1}{2}+\frac{\left(n-1\right)\left(3n+1\right)}{6{\left(n+1\right)}^{2}}-\frac{2{n}^{3}\left({n}^{2}+2n-1\right)}{\left({n}^{4}-1\right)\left({n}^{2}+1\right)}+\frac{8{n}^{4}\left({n}^{4}+1\right)\mathrm{ln}n}{{\left({n}^{4}-1\right)}^{2}\left({n}^{2}+1\right)}+\frac{{n}^{2}{\left({n}^{2}-1\right)}^{2}}{{\left({n}^{2}+1\right)}^{3}}\mathrm{ln}\left(\frac{n-1}{n+1}\right)$

Where $n={n}_{1}/{n}_{2}$ is the ratio of refractive indices. ${n}_{1}$ always represents the medium with higher refractive index and ${n}_{2}$ the medium of lower refractive index.

The transmission from medium one to two is given by ${\tau }_{D,12}=1-{r}_{D,12}$ .

For the reverse direction the reflectance and transmittance are given by ${r}_{D,21}=1-\frac{1}{{n}^{2}}\left(1-{r}_{D,12}\right)$ and ${\tau }_{D,21}=\frac{1}{{n}^{2}}{\tau }_{D,12}$ , respectively.

The incoming radiative intensity to the interface is given by the hemispherically averaged intensity for medium one and medium two:(9)
where $n$ is the outward facing normal. From these fluxes, the outgoing intensity at the wall for the current ordinate direction, $\Omega$ , is given by(10)
$I\left(\Omega \right)={r}_{D,12}\frac{{Q}_{1}}{\pi }+{\tau }_{D,21}\frac{{Q}_{2}}{\pi }$

## Reflection and Transmission for Partially Specular and Partially Diffuse Interfaces of Type Internal

For partially specular and partially diffuse interfaces 0.0 < diffused_fraction < 1.0.

Interfaces between semi-transparent media are typically not 100 percent diffused or specular and the diffuse fraction lies somewhere between zero and one. In this range the outgoing radiative intensity is treated as a linear combination of the specular and diffuse components, for example(11)
$I\left(\Omega \right)=\left(1-\alpha \right){I}^{S}\left(\Omega \right)+\text{α}{I}^{D}\left(\Omega \right)$

where $\text{α}$ is the diffuse fraction, ${I}^{S}\left(\Omega \right)$ is the outgoing specular component of radiative intensity, and ${I}^{D}\left(\Omega \right)$ is the outgoing diffuse component of radiative intensity. For example, in medium one in the image above the components would be and ${I}^{D}\left(\Omega \right)={r}_{D,12}\frac{{Q}_{1}}{\pi }+{\tau }_{D,21}\frac{{Q}_{2}}{\pi }$ .

## Reflection and Transmission for Diffuse Interfaces of Type External

If the radiation_interface_type = external, then the medium surrounding the model is considered to participate in the transfer of radiation. For the surrounding medium no mesh is required, rather a mathematical model is used to determine the radiative intensity. The model assumes that the surrounding medium has uniform radiative intensity in all directions (the radiative flux is isotropic). The isotropic radiative intensity is given by the following blackbody source(12)
${I}_{EXT}=\frac{{\epsilon }_{EXT}\sigma {T}_{EXT}{}^{4}}{\pi }$

where ${\epsilon }_{EXT}$ is the exterior emissivity and is set to one in AcuSolve, $\sigma$ is the Stefan-Boltzmann constant, and ${T}_{EXT}$ is the temperature of the surrounding fluid. At the external interface ${I}_{EXT}$ is transferred into the medium. This condition can only be applied to boundaries as the interface is only modeled mathematically.

The following example demonstrates the AcuSolve input command used to simulate specular transmission between participating media.
RADIATION_SURFACE( " Lens-inner_Lens_Air"" ) {
...
}
External radiation is modeled as follows:
RADIATION_SURFACE( "Lens-outer" ) {
...
}
RADIATION_SURFACE( "Interface_fluid" ) {
}