RADIATION_SURFACE

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, ε σ T w 4 / π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH1oqzcqaHdpWCcaWGubWdamaaDaaaleaapeGaam4DaaWdaeaa peGaaGinaaaakiaac+cacqaHapaCaaa@3EF8@ , where ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379D@ is the emissivity, σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37B9@ is Stefan-Boltzmann constant, and T w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaam4DaaWdaeqaaaaa@3845@ is the wall temperature.

Type

AcuSolve Command

Syntax

RADIATION_SURFACE("name") {parameters...}

Qualifier

User-given name.

Parameters

shape (enumerated) [no default]
Shape of the surfaces in this set.
three_node_triangle or tri3
Three-node triangle.
four_node_quad or quad4
Four-node quadrilateral
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
Radiation_interface. Enables the transmission and reflection of radiative intensity at an interface between two participating media. Available when radiation = discrete_ordinates.
radiation_interface_type (enumerated) [=internal]
The type of radiation_interface is specified through radiation_interface_type. The parameter has two values: internal and external. If radiation_interface_type = internal, there must be a participating media on each side of an interface, that is, the material for each adjacent ELEMENT_SET has a radiation_model. For this condition, the transfer of radiative intensity is both transmitted and reflected at the interface. If radiation_interface_type = external, there is a participating media on one side of an interface. The radiative intensity in the medium surrounding the model is calculated using the mathematical model. The external radiation interface is used with external_emissivity_model and external_temperature. The radiation_interface_type requires radiation_interface.
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.
max_agglomeration_radius or radius (real) >=0 [=0.25]
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
four_node_quad
six_node_wedge
three_node_triangle
six_node_wedge
four_node_quad
eight_node_brick
four_node_quad
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
and read by:
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 ; }
   shape          = four_node_quad
   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
and read using:
RADIATION_SURFACE ( "no-slip wall" ) { 
   surface_sets       = Read("surface_sets.srfst")
   ...
}

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
    max_agglomeration_radius    = 0.5
    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.

For the discete_ordinate radiation model, exchange of radiative intensity occurs at the interface between participating media. For this case the RADIATION_SURFACE type is set to radiation_interface and the radiation_interface_type can be either internal or external. For the internal radiation_interface_type, the condition is applied between two participating media (two media both with a MATERIAL_RADIATION_MODEL). For the external radiation_interface_type, the RADIATION_SURFACE is applied to the boundary of a semi-transparent media (has a MATERIAL_RADIATION_MODEL) and models radiative exchange with the surrounding environment. The exchange of radiative intensity depends on the diffused_fraction of the interface and the refractive indices of the media. The different scenarios are described below.

Reflection and Transmission for Specular Interfaces of Type Internal

For specular interfaces the diffused_fraction = 0.


Figure 1. . Reflected and refracted directions at the interface between two participating media of different refractive indices (n1 < n2). For the medium of higher refractive index if the incoming direction is greater than the critical angle total internal reflection occurs (no transmission occurs across the interface). This is represented by the gray dashed lines in the image.
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)
cos θ 1 = Ω I 1 n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaci4yaiaac+gacaGGZbGaeqiUde3damaaBaaaleaapeGaaGymaaWd aeqaaOWdbiabg2da9iaahM6apaWaa0baaSqaaGqad8qacaWFjbaapa qaa8qacaaIXaaaaOWdaiabgwSix=qacaWHUbaaaa@4365@
where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCOBaaaa@3703@ is the outward facing normal direction at the interface (towards the second medium) and Ω I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaacbmWdbiaa=Leaa8aabaWdbiaaigdaaaaa aa@393D@ is the unit direction vector of incoming radiation intensity to the surface, given by(2)
Ω I 1 = Ω R 1 2 ( Ω R 1 n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaacbmWdbiaa=Leaa8aabaWdbiaaigdaaaGc cqGH9aqpcaWHPoWdamaaDaaaleaapeGaa8NuaaWdaeaapeGaaGymaa aakiabgkHiTiaaikdadaqadaWdaeaapeGaaCyQd8aadaqhaaWcbaWd biaa=jfaa8aabaWdbiaaigdaaaGcpaGaeyyXIC9dbiaah6gaaiaawI cacaGLPaaacaWHUbaaaa@4865@

where Ω R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaacbmWdbiaa=jfaa8aabaWdbiaaigdaaaaa aa@3946@ 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 sin θ 1 = n 2 sin θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qaciGGZbGaaiyA aiaac6gacqaH4oqCpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey ypa0JaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qaciGGZbGa aiyAaiaac6gacqaH4oqCpaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@46B8@ , or equivalently in vector form n 1 ( n × Ω I 1 ) = n 2 ( n × Ω R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qadaqadaWdaeaa peGaaCOBaiabgEna0kaahM6apaWaa0baaSqaaGqad8qacaWFjbaapa qaa8qacaaIXaaaaaGccaGLOaGaayzkaaGaeyypa0JaamOBa8aadaWg aaWcbaWdbiaaikdaa8aabeaak8qadaqadaWdaeaapeGaaCOBaiabgE na0kaahM6apaWaa0baaSqaa8qacaWFsbaapaqaa8qacaaIYaaaaaGc caGLOaGaayzkaaaaaa@4B37@ , where n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3814@ and n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@3815@ are the refractive indices of mediums one and two defined in MATERIAL_RADIATION_MODEL for the material_model of each ELEMENT_SET. θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38D7@ and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@38D8@ 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)
n 1 n 2 Ω R 1 + ( n 1 n 2 cos θ 1 1 ( n 1 n 2 ) 2 ( 1 cos θ 1 ) ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaaIXaaapaqabaaa keaapeGaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaGcpeGaaC yQd8aadaqhaaWcbaWdbiaadkfaa8aabaWdbiaaigdaaaGccqGHRaWk daqadaWdaeaapeWaaSaaa8aabaWdbiaad6gapaWaaSbaaSqaa8qaca aIXaaapaqabaaakeaapeGaamOBa8aadaWgaaWcbaWdbiaaikdaa8aa beaaaaGcpeGaci4yaiaac+gacaGGZbGaeqiUde3damaaBaaaleaape GaaGymaaWdaeqaaOWdbiabgkHiTmaakaaapaqaa8qacaaIXaGaeyOe I0YaaeWaa8aabaWdbmaalaaapaqaa8qacaWGUbWdamaaBaaaleaape GaaGymaaWdaeqaaaGcbaWdbiaad6gapaWaaSbaaSqaa8qacaaIYaaa paqabaaaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYa aaaOGaaiikaiaaigdacqGHsislciGGJbGaai4BaiaacohacqaH4oqC paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiykaaWcbeaaaOGaay jkaiaawMcaaGqadiaa=5gaaaa@5E2D@

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, θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaaGymaaWdaeqaaaaa@38D7@ , the proportion of radiation intensity that is reflected and transmitted will vary. If n 1 < n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH8aapcaWG UbWdamaaBaaaleaapeGaaGOmaaWdaeqaaaaa@3B3B@ , then the radiative intensity in medium one will be partially reflected and partially transmitted into a cone defined by the critical angle, θ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaam4yaaWdaeqaaaaa@3904@ , which is given by:(4)
θ c = sin 1 n 1 n 2 ,   n 1 < n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaam4yaaWdaeqaaOWdbiabg2da9iGa cohacaGGPbGaaiOBa8aadaahaaWcbeqaa8qacqGHsislcaaIXaaaaO WaaSaaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaaIXaaapaqabaaa keaapeGaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaGcpeGaai ilaiaacckacqGHuhY2caWGUbWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiabgYda8iaad6gapaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@4CCD@

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 Ω I 1 Ω R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaWdbiaadMeaa8aabaWdbiaaigdaaaGccqGH sgIRcaWHPoWdamaaDaaaleaapeGaamOuaaWdaeaapeGaaGymaaaaaa a@3E5E@ , or reflectance, is given by (5)
r 12 = 1 2 ( n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 ) 2 + 1 2 ( n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaigdacaaIYaaapaqabaGcpeGaeyyp a0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaWaaeWaa8aaba Wdbmaalaaapaqaa8qacaWGUbWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiGacogacaGGVbGaai4CaiabeI7aX9aadaWgaaWcbaWdbiaaig daa8aabeaak8qacqGHsislcaWGUbWdamaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiGacogacaGGVbGaai4CaiabeI7aX9aadaWgaaWcbaWdbi aaikdaa8aabeaaaOqaa8qacaWGUbWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiGacogacaGGVbGaai4CaiabeI7aX9aadaWgaaWcbaWdbi aaigdaa8aabeaak8qacqGHRaWkcaWGUbWdamaaBaaaleaapeGaaGOm aaWdaeqaaOWdbiGacogacaGGVbGaai4CaiabeI7aX9aadaWgaaWcba Wdbiaaikdaa8aabeaaaaaak8qacaGLOaGaayzkaaWdamaaCaaaleqa baWdbiaaikdaaaGccqGHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaape GaaGOmaaaadaqadaWdaeaapeWaaSaaa8aabaWdbiaad6gapaWaaSba aSqaa8qacaaIYaaapaqabaGcpeGaci4yaiaac+gacaGGZbGaeqiUde 3damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTiaad6gapaWa aSbaaSqaa8qacaaIXaaapaqabaGcpeGaci4yaiaac+gacaGGZbGaeq iUde3damaaBaaaleaapeGaaGOmaaWdaeqaaaGcbaWdbiaad6gapaWa aSbaaSqaa8qacaaIYaaapaqabaGcpeGaci4yaiaac+gacaGGZbGaeq iUde3damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgUcaRiaad6ga paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaci4yaiaac+gacaGGZb GaeqiUde3damaaBaaaleaapeGaaGOmaaWdaeqaaaaaaOWdbiaawIca caGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaaaaa@869E@

and the transmitted proportion from Ω I 2 Ω R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaWdbiaadMeaa8aabaWdbiaaikdaaaGccqGH sgIRcaWHPoWdamaaDaaaleaapeGaamOuaaWdaeaapeGaaGymaaaaaa a@3E5F@ , or transmittance, is given by τ 21 = 1 r 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaaGOmaiaaigdaa8aabeaak8qacqGH 9aqpcaaIXaGaeyOeI0IaamOCa8aadaWgaaWcbaWdbiaaigdacaaIYa aapaqabaaaaa@3F32@ .

In the second medium, for the current scenario where n 2 > n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH+aGpcaWG UbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3B3F@ , if θ 2 < θ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabgYda8iab eI7aX9aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3CEE@ 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaikdacaaIXaaapaqabaGcpeGaeyyp a0JaamOCa8aadaWgaaWcbaWdbiaaigdacaaIYaaapaqabaaaaa@3CBC@ . If θ 2 > θ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg6da+iab eI7aX9aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3CF2@ , then total internal reflection occurs and r 21 = 1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaikdacaaIXaaapaqabaGcpeGaeyyp a0JaaGymaiaac6cacaaIWaaaaa@3C1B@ and τ 12 = 0.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaaGymaiaaikdaa8aabeaak8qacqGH 9aqpcaaIWaGaaiOlaiaaicdaaaa@3CE8@ , 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, Ω R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaWdbiaadkfaa8aabaWdbiaaigdaaaaaaa@393E@ , is given by(6)
I ( Ω R 1 ) = r 12 I ( Ω I 1 ) +   τ 21 ( n 1 n 2 ) 2 I ( Ω I 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysamaabmaapaqaa8qacaWHPoWdamaaDaaaleaapeGaamOuaaWd aeaapeGaaGymaaaaaOGaayjkaiaawMcaaiabg2da9iaadkhapaWaaS baaSqaa8qacaaIXaGaaGOmaaWdaeqaaOWdbiaadMeadaqadaWdaeaa peGaaCyQd8aadaqhaaWcbaWdbiaadMeaa8aabaWdbiaaigdaaaaaki aawIcacaGLPaaacqGHRaWkcaGGGcGaeqiXdq3damaaBaaaleaapeGa aGOmaiaaigdaa8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbi aad6gapaWaaSbaaSqaa8qacaaIXaaapaqabaaakeaapeGaamOBa8aa daWgaaWcbaWdbiaaikdaa8aabeaaaaaak8qacaGLOaGaayzkaaWdam aaCaaaleqabaWdbiaaikdaaaGccaWGjbWaaeWaa8aabaWdbiaahM6a paWaa0baaSqaa8qacaWGjbaapaqaa8qacaaIYaaaaaGccaGLOaGaay zkaaaaaa@57DD@
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 Ω R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQd8aadaqhaaWcbaWdbiaadkfaa8aabaWdbiaaikdaaaaaaa@393F@ then the intensity outgoing radiative intensity is given by(7)
I ( Ω R 2 ) = r 21 I ( Ω I 2 ) +   τ 12 ( n 2 n 1 ) 2 I ( Ω I 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysamaabmaapaqaa8qacaWHPoWdamaaDaaaleaapeGaamOuaaWd aeaapeGaaGOmaaaaaOGaayjkaiaawMcaaiabg2da9iaadkhapaWaaS baaSqaa8qacaaIYaGaaGymaaWdaeqaaOWdbiaadMeadaqadaWdaeaa peGaaCyQd8aadaqhaaWcbaWdbiaadMeaa8aabaWdbiaaikdaaaaaki aawIcacaGLPaaacqGHRaWkcaGGGcGaeqiXdq3damaaBaaaleaapeGa aGymaiaaikdaa8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbi aad6gapaWaaSbaaSqaa8qacaaIYaaapaqabaaakeaapeGaamOBa8aa daWgaaWcbaWdbiaaigdaa8aabeaaaaaak8qacaGLOaGaayzkaaWdam aaCaaaleqabaWdbiaaikdaaaGccaWGjbWaaeWaa8aabaWdbiaahM6a paWaa0baaSqaa8qacaWGjbaapaqaa8qacaaIXaaaaaGccaGLOaGaay zkaaaaaa@57DE@

For n 2 < n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH8aapcaWG UbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3B3B@ , 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 = 1 2 + ( n 1 ) ( 3 n + 1 ) 6 ( n + 1 ) 2 2 n 3 ( n 2 + 2 n 1 ) ( n 4 1 ) ( n 2 + 1 ) + 8 n 4 ( n 4 + 1 ) ln n ( n 4 1 ) 2 ( n 2 + 1 ) + n 2 ( n 2 1 ) 2 ( n 2 + 1 ) 3 ln ( n 1 n + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaadseacaGGSaGaaGymaiaaikdaa8aa beaak8qacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaa aacqGHRaWkdaWcaaWdaeaapeWaaeWaa8aabaWdbiaad6gacqGHsisl caaIXaaacaGLOaGaayzkaaWaaeWaa8aabaWdbiaaiodacaWGUbGaey 4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOnamaabmaapaqa a8qacaWGUbGaey4kaSIaaGymaaGaayjkaiaawMcaa8aadaahaaWcbe qaa8qacaaIYaaaaaaakiabgkHiTmaalaaapaqaa8qacaaIYaGaamOB a8aadaahaaWcbeqaa8qacaaIZaaaaOWaaeWaa8aabaWdbiaad6gapa WaaWbaaSqabeaapeGaaGOmaaaakiabgUcaRiaaikdacaWGUbGaeyOe I0IaaGymaaGaayjkaiaawMcaaaWdaeaapeWaaeWaa8aabaWdbiaad6 gapaWaaWbaaSqabeaapeGaaGinaaaakiabgkHiTiaaigdaaiaawIca caGLPaaadaqadaWdaeaapeGaamOBa8aadaahaaWcbeqaa8qacaaIYa aaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaaacqGHRaWkdaWcaaWd aeaapeGaaGioaiaad6gapaWaaWbaaSqabeaapeGaaGinaaaakmaabm aapaqaa8qacaWGUbWdamaaCaaaleqabaWdbiaaisdaaaGccqGHRaWk caaIXaaacaGLOaGaayzkaaGaciiBaiaac6gacaWGUbaapaqaa8qada qadaWdaeaapeGaamOBa8aadaahaaWcbeqaa8qacaaI0aaaaOGaeyOe I0IaaGymaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaO WaaeWaa8aabaWdbiaad6gapaWaaWbaaSqabeaapeGaaGOmaaaakiab gUcaRiaaigdaaiaawIcacaGLPaaaaaGaey4kaSYaaSaaa8aabaWdbi aad6gapaWaaWbaaSqabeaapeGaaGOmaaaakmaabmaapaqaa8qacaWG UbWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaacaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaacIca caWGUbWdamaaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaaIXaGaai yka8aadaahaaWcbeqaa8qacaaIZaaaaaaakiGacYgacaGGUbWaaeWa a8aabaWdbmaalaaapaqaa8qacaWGUbGaeyOeI0IaaGymaaWdaeaape GaamOBaiabgUcaRiaaigdaaaaacaGLOaGaayzkaaaaaa@94A6@

Where n = n 1 / n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaiabg2da9iaad6gapaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaai4laiaad6gapaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@3CE3@ is the ratio of refractive indices. n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3814@ always represents the medium with higher refractive index and n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@3815@ the medium of lower refractive index.

The transmission from medium one to two is given by τ D , 12 = 1 r D , 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamiraiaacYcacaaIXaGaaGOmaaWd aeqaaOWdbiabg2da9iaaigdacqGHsislcaWGYbWdamaaBaaaleaape GaamiraiaacYcacaaIXaGaaGOmaaWdaeqaaaaa@4224@ .

For the reverse direction the reflectance and transmittance are given by r D , 21 = 1 1 n 2 ( 1 r D , 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaadseacaGGSaGaaGOmaiaaigdaa8aa beaak8qacqGH9aqpcaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8 aabaWdbiaad6gapaWaaWbaaSqabeaapeGaaGOmaaaaaaGcdaqadaWd aeaapeGaaGymaiabgkHiTiaadkhapaWaaSbaaSqaa8qacaWGebGaai ilaiaaigdacaaIYaaapaqabaaak8qacaGLOaGaayzkaaaaaa@47CE@ and τ D , 21 = 1 n 2 τ D , 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamiraiaacYcacaaIYaGaaGymaaWd aeqaaOWdbiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWGUb WdamaaCaaaleqabaWdbiaaikdaaaaaaOGaeqiXdq3damaaBaaaleaa peGaamiraiaacYcacaaIXaGaaGOmaaWdaeqaaaaa@4458@ , respectively.

The incoming radiative intensity to the interface is given by the hemispherically averaged intensity for medium one and medium two:(9)
Q 1 = j = 1 N w j I j | n Ω j |             n Ω j > 0                     Q 2 = j = 1 N w j I j | n Ω j |             n Ω j 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyua8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpdaGf WbqabSWdaeaapeGaamOAaiabg2da9iaaigdaa8aabaWdbiaad6eaa0 WdaeaapeGaeyyeIuoaaOGaam4Da8aadaWgaaWcbaWdbiaadQgaa8aa beaak8qacaWGjbWdamaaBaaaleaapeGaamOAaaWdaeqaaOWdbmaaem aapaqaa8qacaWHUbGaeyyXICTaaCyQd8aadaWgaaWcbaacbmWdbiaa =Pgaa8aabeaaaOWdbiaawEa7caGLiWoacaWFGcGaa8hOaiaa=bkaca WFGcGaa8hOaiaa=bkacaWHUbGaeyyXICTaaCyQd8aadaWgaaWcbaWd biaa=Pgaa8aabeaak8qacqGH+aGpcaaIWaGaa8hOaiaa=bkacaWFGc Gaa8hOaiaa=bkacaWFGcGaa8hOaiaa=bkacaWFGcGaa8hOaiaadgfa paWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyypa0ZaaybCaeqal8 aabaWdbiaadQgacqGH9aqpcaaIXaaapaqaa8qacaWGobaan8aabaWd biabggHiLdaakiaadEhapaWaaSbaaSqaa8qacaWGQbaapaqabaGcpe Gaamysa8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qadaabdaWdaeaa peGaaCOBaiabgwSixlaahM6apaWaaSbaaSqaa8qacaWFQbaapaqaba aak8qacaGLhWUaayjcSdGaa8hOaiaa=bkacaWFGcGaa8hOaiaa=bka caWFGcGaaCOBaiabgwSixlaahM6apaWaaSbaaSqaa8qacaWFQbaapa qabaGcpeGaeyizImQaaGimaaaa@8C94@
where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCOBaaaa@3703@ is the outward facing normal. From these fluxes, the outgoing intensity at the wall for the current ordinate direction, Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyQdaaa@3741@ , is given by(10)
I ( Ω ) = r D , 12 Q 1 π + τ D , 21 Q 2 π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysamaabmaapaqaa8qacaWHPoaacaGLOaGaayzkaaGaeyypa0Ja amOCa8aadaWgaaWcbaWdbiaadseacaGGSaGaaGymaiaaikdaa8aabe aak8qadaWcaaWdaeaapeGaamyua8aadaWgaaWcbaWdbiaaigdaa8aa beaaaOqaa8qacqaHapaCaaGaey4kaSIaeqiXdq3damaaBaaaleaape GaamiraiaacYcacaaIYaGaaGymaaWdaeqaaOWdbmaalaaapaqaa8qa caWGrbWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcbaWdbiabec8aWb aaaaa@4D06@

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 ( Ω ) = ( 1 α ) I S ( Ω ) + α I D ( Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysamaabmaapaqaa8qacaWHPoaacaGLOaGaayzkaaGaeyypa0Za aeWaa8aabaWdbiaaigdacqGHsislcqaHXoqyaiaawIcacaGLPaaaca WGjbWdamaaCaaaleqabaWdbiaadofaaaGcdaqadaWdaeaapeGaaCyQ daGaayjkaiaawMcaaiabgUcaRiaabg7acaWGjbWdamaaCaaaleqaba WdbiaadseaaaGcdaqadaWdaeaapeGaaCyQdaGaayjkaiaawMcaaaaa @4B68@

where α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeySdaaa@3743@ is the diffuse fraction, I S ( Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaahaaWcbeqaa8qacaWGtbaaaOWaaeWaa8aabaWdbiaa hM6aaiaawIcacaGLPaaaaaa@3AE5@ is the outgoing specular component of radiative intensity, and I D ( Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaahaaWcbeqaa8qacaWGebaaaOWaaeWaa8aabaWdbiaa hM6aaiaawIcacaGLPaaaaaa@3AD6@ is the outgoing diffuse component of radiative intensity. For example, in medium one in the image above the components would be I S ( Ω ) = I ( Ω R 1 ) = r 12 I ( Ω I 1 ) +   τ 21 ( n 1 n 2 ) 2 I ( Ω I 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaahaaWcbeqaa8qacaWGtbaaaOWaaeWaa8aabaWdbiaa hM6aaiaawIcacaGLPaaacqGH9aqpcaWGjbWaaeWaa8aabaWdbiaahM 6apaWaa0baaSqaa8qacaWGsbaapaqaa8qacaaIXaaaaaGccaGLOaGa ayzkaaGaeyypa0JaamOCa8aadaWgaaWcbaWdbiaaigdacaaIYaaapa qabaGcpeGaamysamaabmaapaqaa8qacaWHPoWdamaaDaaaleaapeGa amysaaWdaeaapeGaaGymaaaaaOGaayjkaiaawMcaaiabgUcaRiaacc kacqaHepaDpaWaaSbaaSqaa8qacaaIYaGaaGymaaWdaeqaaOWdbmaa bmaapaqaa8qadaWcaaWdaeaapeGaamOBa8aadaWgaaWcbaWdbiaaig daa8aabeaaaOqaa8qacaWGUbWdamaaBaaaleaapeGaaGOmaaWdaeqa aaaaaOWdbiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaaki aadMeadaqadaWdaeaapeGaaCyQd8aadaqhaaWcbaWdbiaadMeaa8aa baWdbiaaikdaaaaakiaawIcacaGLPaaaaaa@5DBC@ and I D ( Ω ) = r D , 12 Q 1 π + τ D , 21 Q 2 π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaahaaWcbeqaa8qacaWGebaaaOWaaeWaa8aabaWdbiaa hM6aaiaawIcacaGLPaaacqGH9aqpcaWGYbWdamaaBaaaleaapeGaam iraiaacYcacaaIXaGaaGOmaaWdaeqaaOWdbmaalaaapaqaa8qacaWG rbWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbiabec8aWbaacq GHRaWkcqaHepaDpaWaaSbaaSqaa8qacaWGebGaaiilaiaaikdacaaI XaaapaqabaGcpeWaaSaaa8aabaWdbiaadgfapaWaaSbaaSqaa8qaca aIYaaapaqabaaakeaapeGaeqiWdahaaaaa@4E25@ .

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 E X T = ε E X T σ T E X T 4 π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadweacaWGybGaamivaaWdaeqaaOWd biabg2da9maalaaapaqaa8qacqaH1oqzpaWaaSbaaSqaa8qacaWGfb Gaamiwaiaadsfaa8aabeaak8qacqaHdpWCcaWGubWdamaaBaaaleaa peGaamyraiaadIfacaWGubaapaqabaGcdaahaaWcbeqaa8qacaaI0a aaaaGcpaqaa8qacqaHapaCaaaaaa@47FF@

where ε E X T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaamyraiaadIfacaWGubaapaqabaaa aa@3A8D@ is the exterior emissivity and is set to one in AcuSolve, σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdmhaaa@37CF@ is the Stefan-Boltzmann constant, and T E X T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadweacaWGybGaamivaaWdaeqaaaaa @39BF@ is the temperature of the surrounding fluid. At the external interface I E X T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadweacaWGybGaamivaaWdaeqaaaaa @39B4@ 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"" ) {
    ...
    type                                    = radiation_interface
    radiation_interface_type                = internal
}
External radiation is modeled as follows:
RADIATION_SURFACE( "Lens-outer" ) {
    ...
    type                                     = radiation_interface
    radiation_interface_type                 = external
    external_emissivity_model                = "my emissivity_model"
    external_temperature                     = 313.0
}
When radiative intensity is transmitted from one medium to another, the direction in which the ray is reflected is unlikely to match a specific ordinate. There are two methods for determining the direction of transmission: 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. Below is an example of specular_ordinate_averaging.
RADIATION_SURFACE( "Interface_fluid" ) {     
     ...
     type                                = radiation_interface 
     diffused_fraction                   = 0.0      
     specular_ordinate_averaging         = three_ordinates
}