Multiphysics Legacy Model

Contact: Nicholas Stauff, nstauff.at.anl.gov

Mesh File

note

The mesh for this model is hosted on LFS. Please refer to LFS instructions to download it.

Mesh generation was performed with Cubit toolkit and the mesh file is used in BISON, and in the MultiApp (coupling BISON and Sockeye). A simplified 1/6 core was generated for preliminary assessment (Figure below). This mesh does not contain the helium gaps and stainless steel envelopes for moderators and heat pipes, both of which will be included in the later version of full core model. The mesh density in radial direction is high, as multiple small features (fuel rods, moderators, heat pipes and control drums) are involved.

BISON Model

The present BISON simulation utilized the heat conduction module in MOOSE, supported with materials models in BISON including the thermal properties of TRISO fuel, thermal and mechanical properties of SS316 (cladding material for YH2). Convective boundary conditions were defined at the top and bottom of the model with an external temperature of 800K and h=100 W/m²-K.

################################################################################
## NEAMS Micro-Reactor Application Driver                                     ##
## BISON Main Application input file                                          ##
## Thermal (Heat Conduction) model only                                       ##
## Constant and uniform power profile in fuel region                          ##
## POC: Nicholas Stauff, nstauff at anl.gov                                   ##
################################################################################
# If using or referring to this model, please cite as explained in
# https://mooseframework.inl.gov/virtual_test_bed/citing.html

[GlobalParams]
  flux_conversion_factor = 1
[]

# Note: the mesh is stored using git large file system (LFS)
[Mesh]
  file = ../mesh/mrad_mesh.e
[]

[Variables]
  [temp]
    initial_condition = 800
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
  [heat_source_fuel]
    type = CoupledForce
    variable = temp
    block = fuel
    v = power_density
  []
  [heat_sink_center_comp]
    type = CoupledForce
    variable = temp
    block = heat_pipes
    v = hp_flux_aux
  []
[]

[AuxVariables]
  [power_density]
    block = fuel
    family = L2_LAGRANGE
    order = FIRST
    initial_condition = 3.4e6
  []
  [Tfuel]
    block = fuel
  []
  [fuel_thermal_conductivity]
    block = fuel
    order = CONSTANT
    family = MONOMIAL
  []
  [fuel_specific_heat]
    block = fuel
    order = CONSTANT
    family = MONOMIAL
  []
  [monolith_thermal_conductivity]
    block = 'monolith '
    order = CONSTANT
    family = MONOMIAL
  []
  [monolith_specific_heat]
    block = 'monolith '
    order = CONSTANT
    family = MONOMIAL
  []
  [temp_uo] #auxvariable to hold heat pipe surface temperature from UserObject
    initial_condition = 800
    block = 'heat_pipes'
  []
  [hp_flux_aux]
    block = 'heat_pipes'
  []
[]

[AuxKernels]
  [assign_tfuel]
    type = NormalizationAux
    variable = Tfuel
    source_variable = temp
    execute_on = 'timestep_end'
  []
  [fuel_thermal_conductivity]
    type = MaterialRealAux
    variable = fuel_thermal_conductivity
    property = thermal_conductivity
    execute_on = timestep_end
  []
  [fuel_specific_heat]
    type = MaterialRealAux
    variable = fuel_specific_heat
    property = specific_heat
    execute_on = timestep_end
  []
  [monolith_thermal_conductivity]
    type = MaterialRealAux
    variable = monolith_thermal_conductivity
    property = thermal_conductivity
    execute_on = timestep_end
  []
  [monolith_specific_heat]
    type = MaterialRealAux
    variable = monolith_specific_heat
    property = specific_heat
    execute_on = timestep_end
  []
  [temp_uo]
    type = SpatialUserObjectAux
    variable = temp_uo
    user_object = temp_uo
  []
[]

[BCs]
  [outside_bc] # An "inefficient" cooling mechanism through outer surfaces
    type = ConvectiveFluxFunction # (Robin BC)
    variable = temp
    boundary = 'reflector_surface_all B4C_surface_all air_surface_all'
    coefficient = 1e3 # W/K/m^2
    T_infinity = 800 # K air temperature at the top of the core
  []
[]

[Materials]
  [fuel_matrix_thermal]
    type = GraphiteMatrixThermal
    block = fuel
    packing_fraction = 0.4
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    graphite_grade = IG_110
    fast_neutron_fluence = 0 # Fresh matrix (near BoC)
    temperature = temp
  []
  [monolith_matrix_thermal]
    type = GraphiteMatrixThermal
    block = 'monolith '
    packing_fraction = 0
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    graphite_grade = IG_110
    fast_neutron_fluence = 0 # Fresh matrix (near BoC)
    temperature = temp
  []
  [moderator_thermal]
    type = HeatConductionMaterial
    block = moderator
    temp = temp
    thermal_conductivity = 20 # W/m/K
    specific_heat = 500 # arbitrary value
  []
  [heat_pipes_thermal]
    type = HeatConductionMaterial
    block = 'heat_pipes' # Vapor with high thermal conductivity
    temp = temp
    thermal_conductivity = 1e4 # W/m/K
    specific_heat = 5 # arbitrary value
  []
  [airgap_thermal]
    type = HeatConductionMaterial
    block = 'air_gap' # Helium gap
    temp = temp
    thermal_conductivity = 0.15 # W/m/K
    specific_heat = 5197 # arbitrary value
  []
  [axial_reflector_thermal]
    type = HeatConductionMaterial
    block = 'reflector'
    temp = temp
    thermal_conductivity = 199 # W/m/K
    specific_heat = 1867 # arbitrary value
  []
  [B4C_thermal]
    type = HeatConductionMaterial
    block = 'B4C'
    temp = temp
    thermal_conductivity = 92 # W/m/K
    specific_heat = 960 # arbitrary value
  []
  [fuel_density]
    type = Density
    block = fuel
    density = 2276.5
  []
  [moderator_density]
    type = Density
    block = moderator
    density = 4.3e3
  []
  [monolith_density]
    type = Density
    block = 'monolith'
    density = 1806
  []
  [heat_pipes_density]
    type = Density
    block = 'heat_pipes'
    density = 180 #random number for vapor
  []
  [airgap_density]
    type = Density
    block = air_gap #helium
    density = 180
  []
  [axial_reflector_density]
    type = Density
    block = reflector
    density = 1848
  []
  [B4C_density]
    type = Density
    block = B4C
    density = 2510
  []
[]

[MultiApps]
  [sockeye]
    type = TransientMultiApp
    positions_file = 'hp_centers.txt'
    input_files = 'MP_FC_ss_sockeye.i'
    execute_on = 'timestep_begin' # execute on timestep begin because hard to have a good initial guess on heat flux
    max_procs_per_app = 1
    output_in_position = true
  []
[]

[Transfers]
  [from_sockeye_flux] # Transfer heat pipe heat flux from Sockeye subapps
    type = MultiAppGeneralFieldNearestLocationTransfer
    from_multi_app = sockeye
    source_variable = flux_uo
    variable = hp_flux_aux
    execute_on = 'timestep_begin'
    # Reduces transfers efficiency for now, can be removed once transferred fields are checked
    bbox_factor = 10
  []
  [to_sockeye_temp] # Transfer heat pipe surface temperature to Sockeye subapps
    type = MultiAppGeneralFieldNearestLocationTransfer
    to_multi_app = sockeye
    source_variable = temp_uo
    variable = T_wall_var
    execute_on = 'timestep_begin'
    # Reduces transfers efficiency for now, can be removed once transferred fields are checked
    bbox_factor = 10
  []
[]

[UserObjects]
  [temp_uo]
    type = NearestPointLayeredSideAverage
    direction = z
    num_layers = 100
    points_file = 'hp_centers.txt'
    variable = temp
    execute_on = linear
    boundary = 'heat_pipe_boundary_up_side heat_pipe_boundary_core_side'
  []
[]

[Executioner]
  type = Transient

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  nl_rel_tol = 1e-7
  nl_abs_tol = 1e-8

  start_time = -2e4 # negative start time so we can start running from t = 0
  end_time = 0
  dtmin = 1
  dt = 50

  # Time integration scheme
  scheme = 'implicit-euler'
[]

[Postprocessors]
  [total_evap_heat_exam]
    type = ElementIntegralVariablePostprocessor
    variable = hp_flux_aux
    block = 'heat_pipes'
    execute_on = 'initial timestep_end'
  []
  [hp_heat_integral]
    type = SideDiffusiveFluxIntegral
    variable = temp
    boundary = 'heat_pipe_boundary_up_side heat_pipe_boundary_core_side'
    diffusivity = thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [fuel_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = fuel
  []
  [fuel_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = fuel
  []
  [fuel_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = fuel
    value_type = min
  []
  [mod_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = moderator
  []
  [mod_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = moderator
  []
  [mod_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = moderator
    value_type = min
  []
  [monolith_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = monolith
  []
  [monolith_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = monolith
  []
  [monolith_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = monolith
    value_type = min
  []
  [heatpipe_surface_temp_avg]
    type = SideAverageValue
    variable = temp
    boundary = heat_pipe_boundary_all
  []
  [power_density]
    type = ElementIntegralVariablePostprocessor
    block = fuel
    variable = power_density
    execute_on = 'initial timestep_end'
  []
  [fuel_cP]
    type = ElementExtremeValue
    value_type = 'max'
    variable = fuel_specific_heat
    block = fuel
    execute_on = 'initial timestep_end'
  []
  [fuel_k]
    type = ElementExtremeValue
    value_type = 'max'
    variable = fuel_thermal_conductivity
    block = fuel
    execute_on = 'initial timestep_end'
  []
  [monolith_cP]
    type = ElementExtremeValue
    value_type = 'max'
    variable = monolith_specific_heat
    block = 'monolith '
    execute_on = 'initial timestep_end'
  []
  [monolith_k]
    type = ElementExtremeValue
    value_type = 'max'
    variable = monolith_thermal_conductivity
    block = 'monolith '
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  perf_graph = true
  exodus = true
  color = true
  csv = false
  #checkpoint = true
[]

Sockeye Model

Sockeye is used for steady-state heat pipe thermal performance using the effective thermal conductivity model, i.e., a 2D axisymmetric conduction model with a very high thermal conductivity of 2×10⁵ W/m-K is applied to the vapor core. A heat flux boundary condition is applied to the exterior of the casing in the evaporator section, which is provided by the bulk conduction model. A convective boundary condition is applied to the exterior of the envelope in the condenser section, with an external temperature of 800 K and h=10⁶ W/m²-K.

################################################################################
## NEAMS Micro-Reactor Application Driver                                     ##
## Sockeye Sub Application input file                                         ##
## Heat Pipe Effective Heat Conduction Model                                  ##
## POC: Nicholas Stauff, nstauff at anl.gov                                   ##
################################################################################
# If using or referring to this model, please cite as explained in
# https://mooseframework.inl.gov/virtual_test_bed/citing.html

# Average heat removed/added to heat pipe (W)
Q_hp = 1800.

# Wick characteristics
R_pore = 15.0e-6 # m
D_h_pore = ${fparse 2.0 * R_pore}
permeability = 2e-9
porosity =  0.70

# Effective "super" conductivity (W/m-K)
k_vapor = 1.0e6

# Lengths (m)
length_evap = 180.0e-2
length_adia =  30.0e-2
length_cond =  90.0e-2

# Mesh density
# The dimensions are nicely divisible by 3 cm mesh.
nelem_base_evap = 50
nelem_base_adia = 10
nelem_base_cond = 30
mesh_density = 3
nelem_evap = ${fparse mesh_density*nelem_base_evap}
nelem_adia = ${fparse mesh_density*nelem_base_adia}
nelem_cond = ${fparse mesh_density*nelem_base_cond}

# Envelope thickness (m)
t_env = 0.08e-2
# Liquid annulus thickness (m)
t_ann = 0.07e-2
# Wick thickness (m)
t_wick = 0.1e-2

# Radial geometry
# Envelope outer (m)
R_hp_o = 1.05e-2
D_hp_o = ${fparse 2.0 * R_hp_o}
# Inner Envelope/outer annulus (m)
R_hp_i = ${fparse R_hp_o - t_env}
D_hp_i = ${fparse 2.0 * R_hp_i}
# Inner annulus/wick outer (m)
R_wick_o = ${fparse R_hp_i - t_ann}
D_wick_o = ${fparse 2.0 * R_wick_o}
# Inner wick/vapor core outer (m)
R_wick_i = ${fparse R_wick_o - t_wick}
D_wick_i = ${fparse 2.0 * R_wick_i}

# BCs for condenser
T_ext_cond = 800. # K
htc_ext_cond = 1.0e6 # (W/m2/K)

# Evaporator parameters
htc_wall_initial = 750 # air gap k=0.15 W/mK, thickness=0.0002 m
S_evap = ${fparse pi * D_hp_o * length_evap}
q_evap = ${fparse Q_hp / S_evap}

# Flux correction
R_clad_o = 0.0105 # heat pipe outer radius (m)
R_hp_hole = 0.0107 # heat pipe + gap (m)
num_sides = 28 # full_core level 9
alpha = ${fparse 2 * pi / num_sides}
area_correction = ${fparse sqrt(alpha / sin(alpha))} # polygonization correction factor for area
corr_factor = ${fparse 2 * R_clad_o / R_hp_hole / R_hp_hole / area_correction / area_correction} #full-core

[FluidProperties]
  [fp_2phase]
    type = PotassiumTwoPhaseFluidProperties
    emit_on_nan = none
  []
[]

[SolidProperties]
  [sp_ss316]
    type = ThermalSS316Properties
  []
[]

[Components]
  [hp]
    type = HeatPipeConduction

    position = '0 0 0'
    orientation = '0 0 1'
    length = '${length_evap} ${length_adia} ${length_cond}'
    n_elems = '${nelem_evap} ${nelem_adia} ${nelem_cond}'
    gravity_vector = '0 0 -9.8'
    D_wick_i = ${D_wick_i}
    D_wick_o = ${D_wick_o}
    R_pore = ${R_pore}
    porosity = ${porosity}
    permeability = ${permeability}

    # Axial dimensions (for heat transfer & analytic limits)
    axial_region_names = 'evap adia cond'
    L_evap = ${length_evap}
    L_adia = ${length_adia}
    L_cond = ${length_cond}
    # Radial dimensions, mesh, and materials for heat transfer problem
    D_clad_o = ${D_hp_o}
    D_clad_i = ${D_hp_i}
    D_h_pore = ${D_h_pore}
    # Mesh
    n_elems_clad = 4
    n_elems_wick = 8
    n_elems_core = 10

    # Solid properties for the cladding and wick structure
    sp_clad = sp_ss316
    sp_wick = sp_ss316
    # Temperature at which to evaluate constant density
    T_ref_density = 900

    # Core thermal conductivity, which is controlled
    k_core = ${k_vapor}

    fp_2phase = fp_2phase
    evaporator_at_start_end = true

    # Initial temperature of block
    initial_T = ${T_ext_cond}

    # Temperature to evaluate heat pipe limit approximations
    T_ref = T_evap_inner

    make_pressure_corrections = true
  []

  [condenser_boundary]
    type = HSBoundaryAmbientConvection
    boundary = 'hp:cond:outer'
    hs = hp
    T_ambient = ${T_ext_cond}
    htc_ambient = ${htc_ext_cond} # large value to approach an effective DirichletBC
    scale = 1
  []
  [evaporator_boundary]
    type = HSBoundaryExternalAppConvection
    boundary = 'hp:evap:outer'
    hs = hp
    T_ext = T_wall_var
    htc_ext = htc_wall_var
    scale = 1
  []
[]

[AuxKernels]
  [flux_uo]
    type = SpatialUserObjectAux
    variable = flux_uo
    user_object = flux_uo
  []
[]

[UserObjects]
  [flux_uo]
    type = LayeredSideDiffusiveFluxAverage
    direction = z
    num_layers = 100
    variable = T_solid
    execute_on = linear
    boundary = 'hp:evap:outer'
    diffusivity = ${fparse 22.6 * corr_factor}
  []
[]

[Functions]
  [hp_ax1_vf]
    type = PiecewiseLinearFromVectorPostprocessor
    argument_column = z
    component  = z
    value_column = T_solid
    vectorpostprocessor_name = hp_ax1
  []
  [flux_vf]
    type = PiecewiseLinearFromVectorPostprocessor
    argument_column = z
    component  = z
    value_column = main_flux
    vectorpostprocessor_name = flux_vpp
  []
  [scale_fcn]
    type = ParsedFunction
    symbol_names = 'catastrophic_pp recoverable_pp operational_pp'
    symbol_values = 'catastrophic_pp recoverable_pp operational_pp'
    expression = 'catastrophic_pp*recoverable_pp*operational_pp'
  []
[]

[AuxVariables]
  [T_wall_var]
    initial_condition = ${T_ext_cond}
  []
  [htc_wall_var]
    initial_condition = ${htc_wall_initial}
  []
  [operational_aux]
    initial_condition = 1
  []
  [main_flux]
    initial_condition = ${q_evap}
  []
  [hp_temp_aux]
    initial_condition = ${T_ext_cond}
  []
  [flux_uo]
    initial_condition = 0.0
  []
[]

[Postprocessors]
  [total_evap_heat]
    type = SideIntegralVariablePostprocessor
    variable = flux_uo
    boundary = 'hp:evap:outer'
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [total_evap_heat_exam]
    type = LinearCombinationPostprocessor
    pp_names = 'total_evap_heat'
    pp_coefs = '${fparse 2 * pi * R_hp_hole}'
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [avg_flux_uo]
    type = ElementAverageValue
    variable = flux_uo
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [Integral_BC_Total]
    type = LinearCombinationPostprocessor
    pp_names = 'condenser_boundary_integral evaporator_boundary_integral'
    pp_coefs = '1 1'
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [ZeroPP]
    type = EmptyPostprocessor
  []
  [Integral_BC_Cond]
    type = DifferencePostprocessor
    value1 = ZeroPP
    value2 = condenser_boundary_integral
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [Integral_BC_RelErr]
    type = RelativeDifferencePostprocessor
    value1 = evaporator_boundary_integral
    value2 = Integral_BC_Cond
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [operational_pp]
    type = ElementAverageValue
    variable = operational_aux
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [catastrophic_pp]
    type = HeatRemovalRateLimitScale
    heat_addition_pps = 'evaporator_boundary_integral'
    limit_condenser_side = false
    catastrophic_heat_removal_limit_pps = 'hp_boiling_limit hp_capillary_limit '
                                          'hp_entrainment_limit'
    recoverable_heat_removal_limit_pps = ''
    fp_2phase = fp_2phase
    T = T_inner_avg
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [recoverable_pp]
    type = HeatRemovalRateLimitScale
    heat_addition_pps = 'evaporator_boundary_integral'
    limit_condenser_side = false
    catastrophic_heat_removal_limit_pps = ''
    recoverable_heat_removal_limit_pps = 'hp_sonic_limit hp_viscous_limit'
    fp_2phase = fp_2phase
    T = T_inner_avg
    execute_on = 'INITIAL linear nonlinear TIMESTEP_END'
  []
  [T_evap_inner]
    type = SideAverageValue
    boundary = hp:evap:inner
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_cond_inner]
    type = SideAverageValue
    boundary = hp:cond:inner
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_evap_outer]
    type = SideAverageValue
    boundary = hp:evap:outer
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_cond_outer]
    type = SideAverageValue
    boundary = hp:cond:outer
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_wall_var_avg]
    type = ElementAverageValue
    variable = T_wall_var
    execute_on = 'Initial timestep_end'
  []
  [T_inner_avg]
    type = SideAverageValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_inner_max]
    type = NodalExtremeValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_inner_min]
    type = NodalExtremeValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
    value_type = min
  []
  [DT_outer]
    type = DifferencePostprocessor
    value1 = T_evap_outer
    value2 = T_cond_outer
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [DT_inner]
    type = DifferencePostprocessor
    value1 = T_evap_inner
    value2 = T_cond_inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[VectorPostprocessors]
  [hp_ax1]
    type = SideValueSampler
    variable = T_solid
    boundary = 'hp:evap:outer'
    sort_by = z
    execute_on = 'timestep_begin TIMESTEP_END'
  []
  [env_vpp]
    type = NodalValueSampler
    variable = T_solid
    block = 'hp:clad'
    sort_by = x
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [flux_vpp]
    type = SideValueSampler
    variable = main_flux
    boundary = 'hp:evap:inner'
    sort_by = z
    execute_on = 'timestep_begin TIMESTEP_END'
  []
[]

[Preconditioning]
  [pc]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  scheme = bdf2
  line_search = none

  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-9
  nl_max_its = 15

  l_tol = 1e-3
  l_max_its = 10

  automatic_scaling = true
  compute_scaling_once = false

  start_time = -3e4 # Usually, this needs to be compatible with the Main App
  end_time = 0
  dtmin = 1
  dt = 1000
[]

[Outputs]
  [console]
    type = Console
    execute_postprocessors_on = 'NONE'
  []
  [csv]
    type = CSV
    show = 'T_evap_inner T_evap_outer T_cond_inner T_cond_outer'
    execute_on = 'INITIAL TIMESTEP_END FINAL FAILED'
    execute_vector_postprocessors_on = 'NONE'
  []
[]

MultiApp

Multiphysics simulations performed leverage the MOOSE MultiApps system to couple thermo-mechanics and heat pipe heat energy transfer systems, which are simulated by BISON, and Sockeye, respectively. Each code has its own mesh and corresponding space and time scales. The MOOSE MultiApp system is leveraged to tightly couple the different codes via Picard iterations as illustrated below. In the absence of a neutronic model, constant power density is transferred to the BISON sub-application into the thermal simulation. Before solving for the temperature, BISON passes the surface temperature at the heat pipes to the Sockeye sub-apps, which compute the temperature distribution in every single heat pipe. Based on the calculated temperature gradient in each heat pipe, the consequent heat flux is passed back to BISON and converted into a heat sink for the thermal calculation.

The list and location of each heat-pipe in the 1/6 code is provided in:

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(microreactors/mrad/legacy/steady/MP_FC_ss_bison.i)

################################################################################
## NEAMS Micro-Reactor Application Driver                                     ##
## BISON Main Application input file                                          ##
## Thermal (Heat Conduction) model only                                       ##
## Constant and uniform power profile in fuel region                          ##
## POC: Nicholas Stauff, nstauff at anl.gov                                   ##
################################################################################
# If using or referring to this model, please cite as explained in
# https://mooseframework.inl.gov/virtual_test_bed/citing.html

[GlobalParams]
  flux_conversion_factor = 1
[]

# Note: the mesh is stored using git large file system (LFS)
[Mesh]
  file = ../mesh/mrad_mesh.e
[]

[Variables]
  [temp]
    initial_condition = 800
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
  [heat_source_fuel]
    type = CoupledForce
    variable = temp
    block = fuel
    v = power_density
  []
  [heat_sink_center_comp]
    type = CoupledForce
    variable = temp
    block = heat_pipes
    v = hp_flux_aux
  []
[]

[AuxVariables]
  [power_density]
    block = fuel
    family = L2_LAGRANGE
    order = FIRST
    initial_condition = 3.4e6
  []
  [Tfuel]
    block = fuel
  []
  [fuel_thermal_conductivity]
    block = fuel
    order = CONSTANT
    family = MONOMIAL
  []
  [fuel_specific_heat]
    block = fuel
    order = CONSTANT
    family = MONOMIAL
  []
  [monolith_thermal_conductivity]
    block = 'monolith '
    order = CONSTANT
    family = MONOMIAL
  []
  [monolith_specific_heat]
    block = 'monolith '
    order = CONSTANT
    family = MONOMIAL
  []
  [temp_uo] #auxvariable to hold heat pipe surface temperature from UserObject
    initial_condition = 800
    block = 'heat_pipes'
  []
  [hp_flux_aux]
    block = 'heat_pipes'
  []
[]

[AuxKernels]
  [assign_tfuel]
    type = NormalizationAux
    variable = Tfuel
    source_variable = temp
    execute_on = 'timestep_end'
  []
  [fuel_thermal_conductivity]
    type = MaterialRealAux
    variable = fuel_thermal_conductivity
    property = thermal_conductivity
    execute_on = timestep_end
  []
  [fuel_specific_heat]
    type = MaterialRealAux
    variable = fuel_specific_heat
    property = specific_heat
    execute_on = timestep_end
  []
  [monolith_thermal_conductivity]
    type = MaterialRealAux
    variable = monolith_thermal_conductivity
    property = thermal_conductivity
    execute_on = timestep_end
  []
  [monolith_specific_heat]
    type = MaterialRealAux
    variable = monolith_specific_heat
    property = specific_heat
    execute_on = timestep_end
  []
  [temp_uo]
    type = SpatialUserObjectAux
    variable = temp_uo
    user_object = temp_uo
  []
[]

[BCs]
  [outside_bc] # An "inefficient" cooling mechanism through outer surfaces
    type = ConvectiveFluxFunction # (Robin BC)
    variable = temp
    boundary = 'reflector_surface_all B4C_surface_all air_surface_all'
    coefficient = 1e3 # W/K/m^2
    T_infinity = 800 # K air temperature at the top of the core
  []
[]

[Materials]
  [fuel_matrix_thermal]
    type = GraphiteMatrixThermal
    block = fuel
    packing_fraction = 0.4
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    graphite_grade = IG_110
    fast_neutron_fluence = 0 # Fresh matrix (near BoC)
    temperature = temp
  []
  [monolith_matrix_thermal]
    type = GraphiteMatrixThermal
    block = 'monolith '
    packing_fraction = 0
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    graphite_grade = IG_110
    fast_neutron_fluence = 0 # Fresh matrix (near BoC)
    temperature = temp
  []
  [moderator_thermal]
    type = HeatConductionMaterial
    block = moderator
    temp = temp
    thermal_conductivity = 20 # W/m/K
    specific_heat = 500 # arbitrary value
  []
  [heat_pipes_thermal]
    type = HeatConductionMaterial
    block = 'heat_pipes' # Vapor with high thermal conductivity
    temp = temp
    thermal_conductivity = 1e4 # W/m/K
    specific_heat = 5 # arbitrary value
  []
  [airgap_thermal]
    type = HeatConductionMaterial
    block = 'air_gap' # Helium gap
    temp = temp
    thermal_conductivity = 0.15 # W/m/K
    specific_heat = 5197 # arbitrary value
  []
  [axial_reflector_thermal]
    type = HeatConductionMaterial
    block = 'reflector'
    temp = temp
    thermal_conductivity = 199 # W/m/K
    specific_heat = 1867 # arbitrary value
  []
  [B4C_thermal]
    type = HeatConductionMaterial
    block = 'B4C'
    temp = temp
    thermal_conductivity = 92 # W/m/K
    specific_heat = 960 # arbitrary value
  []
  [fuel_density]
    type = Density
    block = fuel
    density = 2276.5
  []
  [moderator_density]
    type = Density
    block = moderator
    density = 4.3e3
  []
  [monolith_density]
    type = Density
    block = 'monolith'
    density = 1806
  []
  [heat_pipes_density]
    type = Density
    block = 'heat_pipes'
    density = 180 #random number for vapor
  []
  [airgap_density]
    type = Density
    block = air_gap #helium
    density = 180
  []
  [axial_reflector_density]
    type = Density
    block = reflector
    density = 1848
  []
  [B4C_density]
    type = Density
    block = B4C
    density = 2510
  []
[]

[MultiApps]
  [sockeye]
    type = TransientMultiApp
    positions_file = 'hp_centers.txt'
    input_files = 'MP_FC_ss_sockeye.i'
    execute_on = 'timestep_begin' # execute on timestep begin because hard to have a good initial guess on heat flux
    max_procs_per_app = 1
    output_in_position = true
  []
[]

[Transfers]
  [from_sockeye_flux] # Transfer heat pipe heat flux from Sockeye subapps
    type = MultiAppGeneralFieldNearestLocationTransfer
    from_multi_app = sockeye
    source_variable = flux_uo
    variable = hp_flux_aux
    execute_on = 'timestep_begin'
    # Reduces transfers efficiency for now, can be removed once transferred fields are checked
    bbox_factor = 10
  []
  [to_sockeye_temp] # Transfer heat pipe surface temperature to Sockeye subapps
    type = MultiAppGeneralFieldNearestLocationTransfer
    to_multi_app = sockeye
    source_variable = temp_uo
    variable = T_wall_var
    execute_on = 'timestep_begin'
    # Reduces transfers efficiency for now, can be removed once transferred fields are checked
    bbox_factor = 10
  []
[]

[UserObjects]
  [temp_uo]
    type = NearestPointLayeredSideAverage
    direction = z
    num_layers = 100
    points_file = 'hp_centers.txt'
    variable = temp
    execute_on = linear
    boundary = 'heat_pipe_boundary_up_side heat_pipe_boundary_core_side'
  []
[]

[Executioner]
  type = Transient

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  nl_rel_tol = 1e-7
  nl_abs_tol = 1e-8

  start_time = -2e4 # negative start time so we can start running from t = 0
  end_time = 0
  dtmin = 1
  dt = 50

  # Time integration scheme
  scheme = 'implicit-euler'
[]

[Postprocessors]
  [total_evap_heat_exam]
    type = ElementIntegralVariablePostprocessor
    variable = hp_flux_aux
    block = 'heat_pipes'
    execute_on = 'initial timestep_end'
  []
  [hp_heat_integral]
    type = SideDiffusiveFluxIntegral
    variable = temp
    boundary = 'heat_pipe_boundary_up_side heat_pipe_boundary_core_side'
    diffusivity = thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [fuel_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = fuel
  []
  [fuel_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = fuel
  []
  [fuel_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = fuel
    value_type = min
  []
  [mod_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = moderator
  []
  [mod_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = moderator
  []
  [mod_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = moderator
    value_type = min
  []
  [monolith_temp_avg]
    type = ElementAverageValue
    variable = temp
    block = monolith
  []
  [monolith_temp_max]
    type = ElementExtremeValue
    variable = temp
    block = monolith
  []
  [monolith_temp_min]
    type = ElementExtremeValue
    variable = temp
    block = monolith
    value_type = min
  []
  [heatpipe_surface_temp_avg]
    type = SideAverageValue
    variable = temp
    boundary = heat_pipe_boundary_all
  []
  [power_density]
    type = ElementIntegralVariablePostprocessor
    block = fuel
    variable = power_density
    execute_on = 'initial timestep_end'
  []
  [fuel_cP]
    type = ElementExtremeValue
    value_type = 'max'
    variable = fuel_specific_heat
    block = fuel
    execute_on = 'initial timestep_end'
  []
  [fuel_k]
    type = ElementExtremeValue
    value_type = 'max'
    variable = fuel_thermal_conductivity
    block = fuel
    execute_on = 'initial timestep_end'
  []
  [monolith_cP]
    type = ElementExtremeValue
    value_type = 'max'
    variable = monolith_specific_heat
    block = 'monolith '
    execute_on = 'initial timestep_end'
  []
  [monolith_k]
    type = ElementExtremeValue
    value_type = 'max'
    variable = monolith_thermal_conductivity
    block = 'monolith '
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  perf_graph = true
  exodus = true
  color = true
  csv = false
  #checkpoint = true
[]

(microreactors/mrad/legacy/steady/MP_FC_ss_sockeye.i)

################################################################################
## NEAMS Micro-Reactor Application Driver                                     ##
## Sockeye Sub Application input file                                         ##
## Heat Pipe Effective Heat Conduction Model                                  ##
## POC: Nicholas Stauff, nstauff at anl.gov                                   ##
################################################################################
# If using or referring to this model, please cite as explained in
# https://mooseframework.inl.gov/virtual_test_bed/citing.html

# Average heat removed/added to heat pipe (W)
Q_hp = 1800.

# Wick characteristics
R_pore = 15.0e-6 # m
D_h_pore = ${fparse 2.0 * R_pore}
permeability = 2e-9
porosity =  0.70

# Effective "super" conductivity (W/m-K)
k_vapor = 1.0e6

# Lengths (m)
length_evap = 180.0e-2
length_adia =  30.0e-2
length_cond =  90.0e-2

# Mesh density
# The dimensions are nicely divisible by 3 cm mesh.
nelem_base_evap = 50
nelem_base_adia = 10
nelem_base_cond = 30
mesh_density = 3
nelem_evap = ${fparse mesh_density*nelem_base_evap}
nelem_adia = ${fparse mesh_density*nelem_base_adia}
nelem_cond = ${fparse mesh_density*nelem_base_cond}

# Envelope thickness (m)
t_env = 0.08e-2
# Liquid annulus thickness (m)
t_ann = 0.07e-2
# Wick thickness (m)
t_wick = 0.1e-2

# Radial geometry
# Envelope outer (m)
R_hp_o = 1.05e-2
D_hp_o = ${fparse 2.0 * R_hp_o}
# Inner Envelope/outer annulus (m)
R_hp_i = ${fparse R_hp_o - t_env}
D_hp_i = ${fparse 2.0 * R_hp_i}
# Inner annulus/wick outer (m)
R_wick_o = ${fparse R_hp_i - t_ann}
D_wick_o = ${fparse 2.0 * R_wick_o}
# Inner wick/vapor core outer (m)
R_wick_i = ${fparse R_wick_o - t_wick}
D_wick_i = ${fparse 2.0 * R_wick_i}

# BCs for condenser
T_ext_cond = 800. # K
htc_ext_cond = 1.0e6 # (W/m2/K)

# Evaporator parameters
htc_wall_initial = 750 # air gap k=0.15 W/mK, thickness=0.0002 m
S_evap = ${fparse pi * D_hp_o * length_evap}
q_evap = ${fparse Q_hp / S_evap}

# Flux correction
R_clad_o = 0.0105 # heat pipe outer radius (m)
R_hp_hole = 0.0107 # heat pipe + gap (m)
num_sides = 28 # full_core level 9
alpha = ${fparse 2 * pi / num_sides}
area_correction = ${fparse sqrt(alpha / sin(alpha))} # polygonization correction factor for area
corr_factor = ${fparse 2 * R_clad_o / R_hp_hole / R_hp_hole / area_correction / area_correction} #full-core

[FluidProperties]
  [fp_2phase]
    type = PotassiumTwoPhaseFluidProperties
    emit_on_nan = none
  []
[]

[SolidProperties]
  [sp_ss316]
    type = ThermalSS316Properties
  []
[]

[Components]
  [hp]
    type = HeatPipeConduction

    position = '0 0 0'
    orientation = '0 0 1'
    length = '${length_evap} ${length_adia} ${length_cond}'
    n_elems = '${nelem_evap} ${nelem_adia} ${nelem_cond}'
    gravity_vector = '0 0 -9.8'
    D_wick_i = ${D_wick_i}
    D_wick_o = ${D_wick_o}
    R_pore = ${R_pore}
    porosity = ${porosity}
    permeability = ${permeability}

    # Axial dimensions (for heat transfer & analytic limits)
    axial_region_names = 'evap adia cond'
    L_evap = ${length_evap}
    L_adia = ${length_adia}
    L_cond = ${length_cond}
    # Radial dimensions, mesh, and materials for heat transfer problem
    D_clad_o = ${D_hp_o}
    D_clad_i = ${D_hp_i}
    D_h_pore = ${D_h_pore}
    # Mesh
    n_elems_clad = 4
    n_elems_wick = 8
    n_elems_core = 10

    # Solid properties for the cladding and wick structure
    sp_clad = sp_ss316
    sp_wick = sp_ss316
    # Temperature at which to evaluate constant density
    T_ref_density = 900

    # Core thermal conductivity, which is controlled
    k_core = ${k_vapor}

    fp_2phase = fp_2phase
    evaporator_at_start_end = true

    # Initial temperature of block
    initial_T = ${T_ext_cond}

    # Temperature to evaluate heat pipe limit approximations
    T_ref = T_evap_inner

    make_pressure_corrections = true
  []

  [condenser_boundary]
    type = HSBoundaryAmbientConvection
    boundary = 'hp:cond:outer'
    hs = hp
    T_ambient = ${T_ext_cond}
    htc_ambient = ${htc_ext_cond} # large value to approach an effective DirichletBC
    scale = 1
  []
  [evaporator_boundary]
    type = HSBoundaryExternalAppConvection
    boundary = 'hp:evap:outer'
    hs = hp
    T_ext = T_wall_var
    htc_ext = htc_wall_var
    scale = 1
  []
[]

[AuxKernels]
  [flux_uo]
    type = SpatialUserObjectAux
    variable = flux_uo
    user_object = flux_uo
  []
[]

[UserObjects]
  [flux_uo]
    type = LayeredSideDiffusiveFluxAverage
    direction = z
    num_layers = 100
    variable = T_solid
    execute_on = linear
    boundary = 'hp:evap:outer'
    diffusivity = ${fparse 22.6 * corr_factor}
  []
[]

[Functions]
  [hp_ax1_vf]
    type = PiecewiseLinearFromVectorPostprocessor
    argument_column = z
    component  = z
    value_column = T_solid
    vectorpostprocessor_name = hp_ax1
  []
  [flux_vf]
    type = PiecewiseLinearFromVectorPostprocessor
    argument_column = z
    component  = z
    value_column = main_flux
    vectorpostprocessor_name = flux_vpp
  []
  [scale_fcn]
    type = ParsedFunction
    symbol_names = 'catastrophic_pp recoverable_pp operational_pp'
    symbol_values = 'catastrophic_pp recoverable_pp operational_pp'
    expression = 'catastrophic_pp*recoverable_pp*operational_pp'
  []
[]

[AuxVariables]
  [T_wall_var]
    initial_condition = ${T_ext_cond}
  []
  [htc_wall_var]
    initial_condition = ${htc_wall_initial}
  []
  [operational_aux]
    initial_condition = 1
  []
  [main_flux]
    initial_condition = ${q_evap}
  []
  [hp_temp_aux]
    initial_condition = ${T_ext_cond}
  []
  [flux_uo]
    initial_condition = 0.0
  []
[]

[Postprocessors]
  [total_evap_heat]
    type = SideIntegralVariablePostprocessor
    variable = flux_uo
    boundary = 'hp:evap:outer'
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [total_evap_heat_exam]
    type = LinearCombinationPostprocessor
    pp_names = 'total_evap_heat'
    pp_coefs = '${fparse 2 * pi * R_hp_hole}'
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [avg_flux_uo]
    type = ElementAverageValue
    variable = flux_uo
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [Integral_BC_Total]
    type = LinearCombinationPostprocessor
    pp_names = 'condenser_boundary_integral evaporator_boundary_integral'
    pp_coefs = '1 1'
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [ZeroPP]
    type = EmptyPostprocessor
  []
  [Integral_BC_Cond]
    type = DifferencePostprocessor
    value1 = ZeroPP
    value2 = condenser_boundary_integral
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [Integral_BC_RelErr]
    type = RelativeDifferencePostprocessor
    value1 = evaporator_boundary_integral
    value2 = Integral_BC_Cond
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [operational_pp]
    type = ElementAverageValue
    variable = operational_aux
    execute_on = 'initial timestep_begin TIMESTEP_END'
  []
  [catastrophic_pp]
    type = HeatRemovalRateLimitScale
    heat_addition_pps = 'evaporator_boundary_integral'
    limit_condenser_side = false
    catastrophic_heat_removal_limit_pps = 'hp_boiling_limit hp_capillary_limit '
                                          'hp_entrainment_limit'
    recoverable_heat_removal_limit_pps = ''
    fp_2phase = fp_2phase
    T = T_inner_avg
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [recoverable_pp]
    type = HeatRemovalRateLimitScale
    heat_addition_pps = 'evaporator_boundary_integral'
    limit_condenser_side = false
    catastrophic_heat_removal_limit_pps = ''
    recoverable_heat_removal_limit_pps = 'hp_sonic_limit hp_viscous_limit'
    fp_2phase = fp_2phase
    T = T_inner_avg
    execute_on = 'INITIAL linear nonlinear TIMESTEP_END'
  []
  [T_evap_inner]
    type = SideAverageValue
    boundary = hp:evap:inner
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_cond_inner]
    type = SideAverageValue
    boundary = hp:cond:inner
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_evap_outer]
    type = SideAverageValue
    boundary = hp:evap:outer
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_cond_outer]
    type = SideAverageValue
    boundary = hp:cond:outer
    variable = T_solid
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_wall_var_avg]
    type = ElementAverageValue
    variable = T_wall_var
    execute_on = 'Initial timestep_end'
  []
  [T_inner_avg]
    type = SideAverageValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_inner_max]
    type = NodalExtremeValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [T_inner_min]
    type = NodalExtremeValue
    variable = T_solid
    boundary = hp:inner
    execute_on = 'INITIAL TIMESTEP_END'
    value_type = min
  []
  [DT_outer]
    type = DifferencePostprocessor
    value1 = T_evap_outer
    value2 = T_cond_outer
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [DT_inner]
    type = DifferencePostprocessor
    value1 = T_evap_inner
    value2 = T_cond_inner
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[VectorPostprocessors]
  [hp_ax1]
    type = SideValueSampler
    variable = T_solid
    boundary = 'hp:evap:outer'
    sort_by = z
    execute_on = 'timestep_begin TIMESTEP_END'
  []
  [env_vpp]
    type = NodalValueSampler
    variable = T_solid
    block = 'hp:clad'
    sort_by = x
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [flux_vpp]
    type = SideValueSampler
    variable = main_flux
    boundary = 'hp:evap:inner'
    sort_by = z
    execute_on = 'timestep_begin TIMESTEP_END'
  []
[]

[Preconditioning]
  [pc]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  scheme = bdf2
  line_search = none

  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-9
  nl_max_its = 15

  l_tol = 1e-3
  l_max_its = 10

  automatic_scaling = true
  compute_scaling_once = false

  start_time = -3e4 # Usually, this needs to be compatible with the Main App
  end_time = 0
  dtmin = 1
  dt = 1000
[]

[Outputs]
  [console]
    type = Console
    execute_postprocessors_on = 'NONE'
  []
  [csv]
    type = CSV
    show = 'T_evap_inner T_evap_outer T_cond_inner T_cond_outer'
    execute_on = 'INITIAL TIMESTEP_END FINAL FAILED'
    execute_vector_postprocessors_on = 'NONE'
  []
[]

(microreactors/mrad/legacy/steady/hp_centers.txt)

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Mesh File
BISON Model
Sockeye Model
MultiApp