HeatStructure2DRadiationCouplerRZ

This component couples two cylindrical, 2D heat structures via opaque, gray, diffuse radiation.

Usage

This component has the following restrictions:

  • The coupled heat structures must derive from HeatStructureCylindricalBase, for example, HeatStructureCylindrical.

  • Only one boundary name may be provided in each of the "primary_boundary" and "secondary_boundary" parameters.

  • The meshes along the coupled boundaries must be aligned. Each element on a boundary is paired with the nearest element on the coupled boundary. The alignment check requires that each element on a boundary has exactly one element from the coupled boundary paired to it.

  • The boundaries must be radial, i.e., on either the inner or outer cylindrical surfaces, not the flat surfaces.

Input Parameters

  • primary_boundaryThe boundary of the first heat structure to couple

    C++ Type:BoundaryName

    Unit:(no unit assumed)

    Controllable:No

    Description:The boundary of the first heat structure to couple

  • primary_emissivityEmissivity for the primary side

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Emissivity for the primary side

  • primary_heat_structureThe first heat structure to couple

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:The first heat structure to couple

  • secondary_boundaryThe boundary of the second heat structure to couple

    C++ Type:BoundaryName

    Unit:(no unit assumed)

    Controllable:No

    Description:The boundary of the second heat structure to couple

  • secondary_emissivityEmissivity for the secondary side

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Emissivity for the secondary side

  • secondary_heat_structureThe second heat structure to couple

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:The second heat structure to couple

Required Parameters

  • stefan_boltzmann_constant5.67037e-08Stefan Boltzmann constant [W/(m^2-K^4)]. This constant is provided as a parameter to allow different precisions.

    Default:5.67037e-08

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Stefan Boltzmann constant [W/(m^2-K^4)]. This constant is provided as a parameter to allow different precisions.

Optional Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Set the enabled status of the MooseObject.

Advanced Parameters

Formulation

The heat conduction equation is the following: ρcp\pdTt(kT)=q\eqc \rho c_p \pd{T}{t} - \nabla \cdot (k \nabla T) = q''' \eqc where

  • ρ\rho is density,

  • cpc_p is specific heat capacity,

  • kk is thermal conductivity,

  • TT is temperature, and

  • qq''' is a volumetric heat source.

Multiplying by a test function ϕi\phi_i and integrating by parts over the domain Ω\Omega gives \prρcp\pdTt,ϕiΩ+\prkT,ϕiΩkT,ϕinΩ=\prq,ϕiΩ\eqc \pr{\rho c_p \pd{T}{t}, \phi_i}_\Omega + \pr{k \nabla T, \nabla\phi_i}_\Omega - \left\langle k \nabla T, \phi_i\mathbf{n}\right\rangle_{\partial\Omega} = \pr{q''', \phi_i}_\Omega \eqc where Ω\partial\Omega is the boundary of the domain Ω\Omega.

For Neumann boundary conditions on the boundary Γ\Gamma, kTnk \nabla T \cdot \mathbf{n} is replaced with a known incoming heat flux function qbq_b:

kTn=qbxΓ\eqpk \nabla T \cdot \mathbf{n} = q_b \qquad \mathbf{x} \in \Gamma \eqp

This component computes and applies the boundary flux qbq_b for each boundary. For two opaque, gray, diffuse surfaces in an enclosure, the heat flux qiq_i to surface ii is the following (Incropera et al., 2002):

qi=σ(Tj4Ti4)Ri\eqcq_i = \frac{\sigma (T_j^4 - T_i^4)}{\mathcal{R}_i} \eqc

where Ri\mathcal{R}_i is sometimes described as a radiation resistance:

Ri=1ϵiϵi+1Fi,j+1ϵjϵjAiAj\eqc\mathcal{R}_i = \frac{1 - \epsilon_i}{\epsilon_i} + \frac{1}{F_{i,j}} + \frac{1 - \epsilon_j}{\epsilon_j}\frac{A_i}{A_j} \eqc

where

  • ϵi\epsilon_i is the emissivity of surface ii,

  • σ\sigma is the Stefan-Boltzmann constant,

  • Fi,jF_{i,j} is the view factor from surface ii to surface jj,

  • TiT_i is the temperature of surface ii, and

  • AiA_i is the area of surface ii.

The surface ii that is enclosed by the other surface has its view factor set to unity:

Fi,j=1\eqcF_{i,j} = 1 \eqc

whereas the other is computed using the reciprocity rule:

Fj,i=AiAj\eqpF_{j,i} = \frac{A_i}{A_j} \eqp

References

  1. Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine, and others. Fundamentals of Heat and Mass Transfer. Wiley New York, sixth edition, 2002.[BibTeX]