EMISSIVITY_MODEL

Specifies an emissivity model for the radiation equation.

Type

AcuSolve Command

Syntax

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

Qualifier

User-given name.

Parameters

type (enumerated) [=constant]
Type of the emissivity.
constant or const
Constant emissivity. Requires emissivity.
piecewise_linear or linear
Piecewise linear curve fit. Requires curve_fit_values and curve_fit_variable.
cubic_spline or spline
Cubic spline curve fit. Requires curve_fit_values and curve_fit_variable.
user_function or user
User-defined function. Requires user_function, user_values and user_strings.
emissivity or emis (real) >0 <=1 [=1]
Constant value of the emissivity. Used with constant type.
diffused_fraction (real) [= 1.0]
To enable specular reflection a RADIATION_SURFACE must be defined and the diffused fraction, a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ , and emissivity, ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379D@ , must both be less than one.
curve_fit_values or curve_values (array) [={0,0}]
A two-column array of independent-variable/emissivity data values. Used with piecewise_linear and cubic_spline types.
curve_fit_variable or curve_var (enumerated) [=temperature]
Independent variable of the curve fit. Used with piecewise_linear and cubic_spline types.
temperature or temp
Temperature.
user_function or user (string) [no default]
Name of the user-defined function. Used with user_function type.
user_values (array) [={}]
Array of values to be passed to the user-defined function. Used with user_function type.
user_strings (list) [={}]
Array of strings to be passed to the user-defined function. Used with user_function type.
multiplier_function (string) [=none]
User-given name of the multiplier function for scaling the viscosity. If none, no scaling is performed.

Description

This command specifies an ideal grey-surface emissivity model for the radiation equation. This model is only applicable to radiation surfaces, and is not used for the radiation_heat_flux variable in the ELEMENT_BOUNDARY_CONDITION command.

EMISSIVITY_MODEL commands are referenced by RADIATION_SURFACE commands:
EMISSIVITY_MODEL( "my emissivity model" ) {
   type                                = constant 
   emissivity                          = 0.5
}
RADIATION_SURFACE( "hot wall" ) {
	emmissivity_mdoel               = "my emmissivity model"
	...
}
The emissivity is the factor ε in the Stefan-Boltzmann law for the total emissive power of an ideal grey surface:(1)

where σ is the Stefan-Boltzmann constant, given by the Stefan_boltzmann_constant parameter of the RADIATION command; T is the temperature; and Toff is the offset to convert to an absolute temperature, given by the absolute_temperature_offset parameter of the EQUATION command.

A constant emissivity model applies a spatially constant emissivity, as in the above example.

Emissivity models of types piecewise_linear and cubic_spline may be used to define emissivity as a function of a single independent variable. For example,
EMISSIVITY_MODEL( "curve fit emissivity model" ) {
   type               = piecewise_linear
   curve_fit_values   = { 273, 0.2 ; 323, 0.2 ;    373, 0.3 ;    423, 0.4 ; }
   curve_fit_variable = temperature
}

defines emissivity as a function of temperature. In general, the problem must contain the variable defined by curve_fit_variable; this is not an issue here since radiation problems always contain temperature. The curve_fit_values parameter is a two-column array corresponding to the independent variable and the emissivity values. The independent variable values must be in ascending order. The limit point values of the curve fit are used when curve_fit_variable falls outside of the curve fit limits.

The curve_fit_values data may be read from a file. For the above example, the curve fit values may be placed in a file, such as emissivity.fit:
273      0.2
323      0.2
373      0.3
423      0.3
and read by:
EMISSIVITY_MODEL( "curve fit emissivity model" ) {
   type               = piecewise_linear
   curve_fit_values   = Read( "emissivity.fit" )
   curve_fit_variable = temperature 
}

An emissivity of type user_function may be used to model more complex behaviors; see the AcuSolve User- Defined Functions Manual for a detailed description of user-defined functions.

For example, consider an emissivity that is equal to 0.5 between temperatures of 300 and 400, and 0.3 otherwise. The input command may be given by:
EMISSIVITY_MODEL( "UDF emissivity model" ) {
   type          = user_function 
   user_function = "usrEmissivityExample"
   user_values   = { 300,    # lower temp. limit of band    
                     400,    # upper temp. limit of band    
                       0.3,  # default emissivity    
                       0.5 }   # emissivity in temperature band
}
where the user-defined function "usrEmissivityExample" may be implemented as follows:
#include "acusim.h"
#include "udf.h"
UDF_PROTOTYPE( usrEmissivityExample ) ;  /* function prototype */
Void usrEmissivityExample (
    UdfHd    udfHd,                      /* Opaque handle for accessing data */
	Real*    outVec,                     /* Output vector */
	Integer  nItems,                     /* Number of elements */
	Integer  vecDim                      /* = 1 */
) {
    Integer  elem ;                      /* an element counter */
    Real     temp0 ;                     /* lower temp. limit of band */
    Real     temp1 ;                     /* upper temp. limit of band */
    Real     eDef ;                      /* default emissivity */
    Real     eBand ;                     /* emissivity in temperature band */
    Real*    temp ;                      /* temperature */
    Real*    usrVals ;                   /* user values */
udfCheckNumUsrVals( udfHd, 4 ) ;         /* check for error */
    usrVals  = udfGetUsrVals( udfHd ) ;  /* get the user vals */
    temp0    = usrVals[0] ;              /* lower temp  */
    temp1    = usrVals[1] ;              /* upper temp */
    eDef     = usrVals[2] ;              /* default emissivity */
    eBand    = usrVals[3] ;              /* band emissivity */
temp            = udfGetRsfData( udfHd, UDF_RSF_TEMPERATURE ) ;  /* get the temp. */
for ( elem = 0 ; elem < nItems ; elem++ ) {
    if ( temp[elem] >= temp0 && temp[elem] <= temp1 ) {
       outVec[elem] = eBand ;
    } else {
       outVec[elem] = eDef ;
    }
}
} /* end of usrEmissivityExample() */

The dimension of the returned emissivity vector, outVec, is the number of elements.

The multiplier_function parameter may be used to uniformly scale the emissivity values. The value of this parameter refers to the user-given name of a MULTIPLIER_FUNCTION command in the input file. For example, a ramped emissivity may be specified by:
EMISSIVITY_MODEL( "ramped emissivity model" ) {
   type                = constant 
   emissivity          = 0.4 
   multiplier_function = "ramped" 
   }
   MULTIPLIER_FUNCTION( "ramped" ) {
   type               = piecewise_linear 
   curve_fit_values   = { 1, 0.1 ; 10, 1 }
   curve_fit_variable = time_step
}
The diffused fraction defines the proportion of reflected radiation intensity at a surface which is diffused, i.e., the reflection may also have a specular component. If the radiation intensity reflection coefficient at the surface is defined by(2)
ρ = ρ S + ρ D = 1 ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHbpGCcqGH9aqpcqaHbpGCpaWaaWbaaSqabeaapeGaam4uaaaa kiabgUcaRiabeg8aY9aadaahaaWcbeqaa8qacaWGebaaaOGaeyypa0 JaaGymaiabgkHiTiabew7aLbaa@43E0@
then the diffused reflection coefficient, ρ D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaW baaSqabeaaqaaaaaaaaaWdbiaadseaaaaaaa@38CC@ , is defined in terms of the diffused fraction ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@ ) and the emissivity of the surface by(3)
ρ D =α(1ε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaW baaSqabeaaqaaaaaaaaaWdbiaadseaaaGccqGH9aqpcqaHXoqycaGG OaGaaGymaiabgkHiTiabew7aLjaacMcaaaa@4023@
and the specular reflection coefficient by(4)
ρ S =(1α)(1ε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaW baaSqabeaaqaaaaaaaaaWdbiaadofaaaGccqGH9aqpcaGGOaGaaGym aiabgkHiTiabeg7aHjaacMcacaGGOaGaaGymaiabgkHiTiabew7aLj aacMcaaaa@4333@
If α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGymaaaa@3956@ , then the reflection at the surface is completely diffused. If α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGimaaaa@3955@ then the reflection is specular. The outgoing radiation intensity, I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaaaa@36C4@ , at the surface in terms of the above two reflection coefficients is given by(5)
I( r w , Ω i )=ε( r w ) I B ( r w )+ ρ D ( r w ) π n Ω j <0 N ω j I( r w , Ω j )|n Ω j |+ ρ S ( r w )I( r w , Ω S )n Ω i >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbGaaiikaiaadkhapaWaaSbaaSqaa8qacaWG3baapaqabaGc peGaaiilaiabfM6ax9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qaca GGPaGaeyypa0JaeqyTduMaaiikaiaadkhapaWaaSbaaSqaa8qacaWG 3baapaqabaGcpeGaaiykaiaadMeapaWaaSbaaSqaa8qacaWGcbaapa qabaGcpeGaaiikaiaadkhapaWaaSbaaSqaa8qacaWG3baapaqabaGc peGaaiykaiabgUcaRmaalaaabaGaeqyWdi3damaaCaaaleqabaWdbi aadseaaaGccaGGOaGaamOCa8aadaWgaaWcbaWdbiaadEhaa8aabeaa k8qacaGGPaaabaGaeqiWdahaamaaqahabaaaleaacaWGUbGaeyyXIC TaeuyQdC1damaaBaaameaapeGaamOAaaWdaeqaaSWdbiabgYda8iaa icdaaeaacaWGobaaniabggHiLdGcpaGaeqyYdC3aaSbaaSqaa8qaca WGQbaapaqabaGcpeGaamysaiaacIcacaWGYbWdamaaBaaaleaapeGa am4DaaWdaeqaaOWdbiaacYcacqqHPoWvpaWaaSbaaSqaa8qacaWGQb aapaqabaGcpeGaaiykaiaacYhacaWGUbGaeyyXICTaeuyQdC1damaa BaaaleaapeGaamOAaaWdaeqaaOWdbiaacYhacqGHRaWkcqaHbpGCpa WaaWbaaSqabeaapeGaam4uaaaakiaacIcacaWGYbWdamaaBaaaleaa peGaam4DaaWdaeqaaOWdbiaacMcacaWGjbGaaiikaiaadkhapaWaaS baaSqaa8qacaWG3baapaqabaGcpeGaaiilaiabfM6ax9aadaWgaaWc baWdbiaadofaa8aabeaak8qacaGGPaGaamOBaiabgwSixlabfM6ax9 aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH+aGpcaaIWaaaaa@89F9@
Michael Modest, Radiative Heat Transfer, Third Edition (2013)
where the first terms represent emission from the surface, the second term the diffused component incoming radiation heat flux and the third the specular component. The diffused component represents a sum over all radiation intensities along ordinates that are incident to the surface (i.e. a hemisphere of incoming radiation to the surface); n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ is the normal into the domain and Ω j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHPoWvpaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa@38ED@ the jth ordinate direction. The ordinate direction ( Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCfaaa@3784@ ), the total number of ordinate directions ( N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36C9@ ) and the weights ( ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C3@ ) are automatically defined by the order of the radiation_quadrature (S2, S4, S6, S8 & S10). The specular ordinate direction ( Ω S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHPoWvpaWaaSbaaSqaa8qacaWGtbaapaqabaaaaa@38D6@ ) is the direction that the radiation intensity must strike the surface to reflect in a specular fashion along the outgoing ordinate direction, Ω i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHPoWvpaWaaSbaaSqaa8qacaWGPbaapaqabaaaaa@38EC@ , and is given by(6)
Ω S = Ω i 2( Ω i n)n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHPoWvpaWaaSbaaSqaa8qacaWGtbaapaqabaGccqGH9aqppeGa euyQdC1damaaBaaaleaapeGaamyAaaWdaeqaaOGaeyOeI0IaaGOmai aacIcapeGaeuyQdC1damaaBaaaleaapeGaamyAaaWdaeqaaOGaeyyX ICTaamOBaiaacMcacaWGUbaaaa@46F8@

which means the angle that incident radiation intensity strikes the surface equals the angle of reflection.

To enable specular reflection a RADIATION_SURFACE must be defined and the diffused fraction, α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@ , and emissivity, ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379D@ , must both be less than one.

For example,
RADIATION_SURFACE( "wall" ) {
	…
	emissivity_model = "wall_emissivity" ;
	diffused_fraction = 0.9 ;
}
with a typical emissivity model:
EMISSIVITY_MODEL( "wall_emissivity" ) {
	type = constant ;
	emissivity = 0.8 ;
}