Pebble heat conduction simulation

Reflector pebble heat conduction

Reflector pebbles are modeled as homogeneous graphite spherical pebbles. We represent them with a 1D mesh in spherical coordinates.

[Mesh]
  coord_type = 'RSPHERICAL'
  type = GeneratedMesh
  dim = 1
  xmin = 0.0
  xmax = '${fparse pebble_diameter / 2}'
  nx = 20
[]

The heat conduction equation with no source is described with a time derivative kernel and a diffusion kernel, specialized as HeatConduction kernels in the heat conduction module.

[Kernels]
  # inactive = 'time'
  [time]
    type = ADHeatConductionTimeDerivative
    variable = T
    specific_heat = 'cp_s'
    density_name = 'rho_s'
  []
  [diffusion]
    type = ADHeatConduction
    variable = T
    thermal_conductivity = 'k_s'
  []
[]

The center of this spherical pebble is naturally a zero flux symmetry boundary condition. The temperature at the outer surface of the pebble is the temperature of the solid phase. This temperature is received from the main-app in a Receiver postprocessor. It is then used by a PostprocessorDirichletBC to set the boundary condition.

[Postprocessors]
  [pebble_surface_T]
    type = Receiver
  []
[]

[BCs]
  [right]
    type = PostprocessorDirichletBC
    variable = T
    postprocessor = pebble_surface_T
    boundary = 'right'
  []
[]

Fueled pebble heat conduction multiscale simulation

The fueled pebble is modeled using a similar approach, except the sphere is not uniform. Its center is a graphite core, surrounded by a fuel matrix then a graphite shell. We still represent the pebble as a 1D spherical system, with each zone defined as a different subdomain.

[Mesh]
  coord_type = 'RSPHERICAL'
  [mesh]
    type = CartesianMeshGenerator
    dim = 1
    dx = '${rcore} ${fparse rfuel_matrix - rcore} ${fparse pebble_diameter/2 -
                                                    rfuel_matrix} '
    ix = '15 20 10'
    subdomain_id = '0 1 2'
  []

  [left]
    type = SideSetsAroundSubdomainGenerator
    block = 1
    new_boundary = 'fm_left'
    fixed_normal = true
    normal = '-1 0 0'
    input = mesh
  []
  [right]
    type = SideSetsAroundSubdomainGenerator
    block = 1
    new_boundary = 'fm_right'
    fixed_normal = true
    normal = '1 0 0'
    input = left
  []
[]

We use a Heat Source Decomposition approach (Novak et al., 2021) to solve the multiscale heat conduction problem with sources. The heat source is decomposed in its mean and fluctuation (of zero average) terms, corresponding to the meso- and micro-scale. This approach is a decomposition of the heat conduction equations.

The mesoscale equation is written in terms of the mesoscale temperature, which is a slowly varying long-wavelength thermal solution of the problem with the average heat source and averaged (mixed) thermal properties.

The conduction equation on the microscale, the TRISO particles here, is similarly written,

In the microscale, we use the actual material properties of the materials involved, since there is no homogenization of the problem involved. The solid phase temperature is then the superposition of the mesoscale and microscale solutions.

where is the local coordinate and i is the index of the particles or microscale media.

In the fuel matrix, there is heat generation from the fission reactions. We solve the mesoscale conduction equation with the mean heat source.

[Kernels]
  [time]
    type = ADHeatConductionTimeDerivative
    variable = T_fuel_matrix
    specific_heat = 'cp_s'
    density_name = 'rho_s'
  []

  [diffusion]
    type = ADHeatConduction
    variable = T_fuel_matrix
    thermal_conductivity = 'k_s'
  []

  [heat_source]
    type = HeatSrc
    variable = T_fuel_matrix
    heat_source = fuel_matrix_heat_source
    scaling_factor = '${fparse 1.0/fuel_matrix_phase_fraction}'
  []
[]

Since there is no fuel in the core and the shell, and we do not model non-fission heating, there is no heat source in those subdomains.

[Kernels]
  [time2]
    type = ADHeatConductionTimeDerivative
    variable = T_shell
    specific_heat = 'cp_s'
    density_name = 'rho_s'
  []

  [diffusion2]
    type = ADHeatConduction
    variable = T_shell
    thermal_conductivity = 'k_s'
  []
[]

The fuel matrix temperature and the graphite temperatures match on their interfaces. We use an interface kernel to impose the equality conditions on both sides of the fuel matrix.

where is the outer boundary of the microscale domain, which will be detailed further in the next section. is obtained from the TRISO scale calculation.

[InterfaceKernels]
  [diffusion]
    type = HeatDiffusionInterface
    variable = T_fuel_matrix
    neighbor_var = T_shell
    boundary = 'fm_left fm_right'
    k = 'k_s'
    k_neighbor = 'k_s'
  []
[]

The material definitions, and the mixing of material properties from different components of a phase, is covered in the macroscale thermal-hydraulics simulation input.

We solve the microscale equation in the fuel matrix with a separate simulation, using the MultiApp system. We pass the heat source to the microscale simulation, and receive the outer surface temperature of the microscale particle. The maximum and average temperatures reached in the microscale simulations are also passed for postprocessing purposes.

[MultiApps]
  [particle]
    type = TransientMultiApp
    execute_on = 'TIMESTEP_BEGIN'
    input_files = 'ss5_fuel_matrix.i'
    cli_args = 'Outputs/console=false'
  []
[]
[Transfers]
  [particle_heat_source]
    type = MultiAppVariableValueSampleTransfer
    to_multi_app = particle
    source_variable = fuel_matrix_heat_source
    variable = fuel_matrix_heat_source
  []

  [particle_surface_temp]
    type = MultiAppPostprocessorTransfer
    from_multi_app = particle
    from_postprocessor = surface_T
    to_postprocessor = particle_surface_T
    reduction_type = average
  []
[]

We cover the microscale equation treatment in the fuel matrix microscale simulation.

References