represents a thermal radiation boundary condition for PDEs with predicate pred indicating where it applies, with model variables vars and global parameters pars.

represents a thermal radiation boundary condition with local parameters specified in pars[lkey].

# Details

• HeatRadiationValue specifies a boundary condition for HeatTransferPDEComponent and is used as part of the modeling equation:
• HeatRadiationValue is typically used to model heating or cooling through radiation on some part of the boundary. Common examples include an electrical radiator or a fireplace.
• HeatRadiationValue models heating or cooling through radiation with dependent variable in , independent variables in and time variable in .
• Stationary variables vars are vars={Θ[x1,,xn],{x1,,xn}}.
• Time-dependent variables vars are vars={Θ[t,x1,,xn],t,{x1,,xn}}.
• The non-conservative time-dependent heat transfer model HeatTransferPDEComponent is based on a convection-diffusion model with mass density , specific heat capacity , thermal conductivity , convection velocity vector and heat source :
• The thermal radiation value HeatRadiationValue with the dimensionless emissivity, the Boltzmann constant, an ambient temperature and a reference temperature and boundary unit normal models:
• The emissivity is the effectiveness of a material emitting heat and can have a value in the range of .
• Model parameters pars as specified for HeatTransferPDEComponent.
• The following additional model parameters pars can be given:
•  parameter default symbol "AmbientTemperature" 0 , ambient temperature in "BoltzmannConstant" , Boltzmann constant in "Emissivity" 1 "ReferenceTemperature" 0 , reference temperature in
• The Boltzmann constant has units and the temperatures of the PDE model need to be specified in Kelvin.
• The "BoltzmannConstant" parameter can only be specified in pars, not with lkey.
• The default reference temperature is 0 Kelvin, but other units can be used after a conversion.
• To localize model parameters, a key lkey can be specified, and values from association pars[lkey] are used for model parameters.
• All model parameters may depend on any of , and , as well as other dependent variables.
• HeatRadiationValue is a special case of a HeatFluxValue.
• HeatRadiationValue evaluates to a generalized NeumannValue.
• The boundary predicate pred can be specified as in NeumannValue.
• If the HeatRadiationValue depends on parameters that are specified in the association pars as ,keypi,pivi,], the parameters are replaced with .

# Examples

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## Basic Examples(2)

Set up a thermal radiation boundary condition:

Model a temperature field and a thermal radiation boundary with:

Set up the heat transfer model variables :

Set up a region :

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

Specify boundary condition parameters with a constant ambient temperature of 25 °C and a surface emissivity of :

Specify the equation:

Set up initial conditions:

Solve the PDE:

Visualize the solution:

## Scope(6)

Define model variables vars for a transient acoustic pressure field with model parameters pars and a specific boundary condition parameter:

Define model variables vars for a transient acoustic pressure field with model parameters pars and multiple specific parameter boundary conditions:

Set up a reference temperature of absolute zero in degrees Celsius:

Set up a thermal radiation boundary condition with a reference and ambient temperature in Celsius:

If no value for emissivity is specified, then an emissivity of 1 is assumed:

Set up a thermal radiation boundary condition with ambient temperature emissivity :

### 2D(1)

Model a ceramic strip that is embedded in a high-thermal-conductive material. The side boundaries of the strip are maintained at a constant temperature . The top surface of the strip is losing heat via both heat convection and heat radiation to the ambient environment at . The bottom boundary, however, is assumed to be thermally insulated:

Model a temperature field and the thermal radiation and thermal transfer with:

Set up the heat transfer model variables :

Set up a rectangular domain with a width of and a height of :

Specify thermal conductivity :

Set up temperature surface boundary conditions at the left and right boundaries:

Set up a heat transfer boundary condition on the top surface:

Also set up a thermal radiation boundary condition on the top surface:

Set up the equation:

Solve the PDE:

Visualize the solution:

## Applications(1)

Model the temperature field and a thermal radiation boundary with:

Set up the heat transfer model variables :

Set up a region region:

Specify heat transfer model parameters density , specific heat capacity and thermal conductivity :

Specify boundary condition parameters with a constant ambient temperature of and a surface emissivity of :

Specify the equation:

Solve the PDE:

Visualize the solution:

#### CMS

Wolfram Language. 2020. "HeatRadiationValue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HeatRadiationValue.html.