FVThermalResistanceBC

Thermal resistance Heat flux boundary condition for the fluid and solid energy equations

Description

Heat flux boundary condition for the fluid or solid energy conservation equations. When used with the fluid energy equation, this boundary condition specifies

$var element = document.getElementById("moose-equation-5768b929-87dd-41e9-b7dc-f671ee9428a8");katex.render("-\\int_\\Gamma\\kappa_f \\nabla T_f\\cdot\\hat{n}d\\Gamma\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where

$var element = document.getElementById("moose-equation-2bdb6b00-41a8-4465-96b6-9d29a8e0c7bf");katex.render("-\\kappa_f\\nabla T_f\\cdot\\hat{n}=\\tilde{q}\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-0714dff4-0efa-49ba-9886-4a8d1a3fc072");katex.render("\\tilde{q}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the imposed heat flux, to be discussed shortly. When used with the solid energy equation, this boundary condition specifies

$var element = document.getElementById("moose-equation-792ed08a-8780-46fc-ae2e-c462e380d608");katex.render("-\\int_\\Gamma\\kappa_s\\cdot\\nabla T_s\\cdot\\hat{n}d\\Gamma\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where

$var element = document.getElementById("moose-equation-3071ac09-e7f9-4faa-a379-8844ff21f8d1");katex.render("-\\kappa_s\\cdot\\nabla T_s\\cdot\\hat{n}=\\tilde{q}\\ .", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

The heat flux $var element = document.getElementById("moose-equation-342cca79-7bb2-42d7-b485-44d084a0e78a");katex.render("\\tilde{q}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is computed based on the thermal resistance concept, where heat is assumed to conduct through multiple parallel slabs of constant-property solid in series, followed by parallel convection and radiation to an ambient temperature. For $var element = document.getElementById("moose-equation-e3ca4a41-5cd3-4296-bc73-48bdd86c9894");katex.render("N", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ conducting layers, the conduction thermal resistance $var element = document.getElementById("moose-equation-7c91693e-1532-44e7-8c30-69079c9c7891");katex.render("R_c", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is computed for Cartesian slabs as

$var element = document.getElementById("moose-equation-dfc6c596-7c1f-47bd-b304-e1840d5830f9");katex.render("R_c=\\sum_{i=1}^N\\frac{\\Delta x_i}{k_i}\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-1d30f217-5985-4540-a4fe-aa23c1eff70c");katex.render("\\Delta x_i", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the thickness of each layer and $var element = document.getElementById("moose-equation-1a803dc1-6a01-411e-9f7f-bb965ffb598c");katex.render("k_i", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the thermal conductivity of each layer. For cylindrical annuli, the conduction thermal resistance is computed as

$var element = document.getElementById("moose-equation-0dcd6275-e805-412f-a963-0cb5b714cdc1");katex.render("R_c=\\sum_{i=1}^N\\frac{\\ln{\\left(\\frac{\\Delta x_i+r_i}{r_i}\\right)}}{2 \\pi k_i}\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-205c67d6-2a95-426d-8a02-c0540a408612");katex.render("r_i", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the inner radius corresponding to each layer. The inner radius of the very first cylindrical annulus is specified by the inner_radius parameter.

warning

The inner radius of a cylindrical bed is also provided by the inner_radius parameter for setting porosity functions and other near-wall behavior. To prevent errors associated with mixing the two different interpretations of the inner_radius parameter, it's best to avoid setting this parameter in the GlobalParams input file block.

The parallel convection and radiation resistance from the surface of the conduction layers, or $var element = document.getElementById("moose-equation-2bb4b453-151e-4db2-9efb-eb232d2ab340");katex.render("R_p", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , is then computed as

$var element = document.getElementById("moose-equation-859871c8-2044-4818-99e0-5f0c5f4d1cb4");katex.render("R_p=\\frac{1}{R\\left(h_r+h_c\\right)}\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-87e13c7d-f671-44c7-80cb-703991ad2ed1");katex.render("R", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is equal to the cylindrical annuli outer radius for cylindrical geometries and unity for Cartesian geometries; $var element = document.getElementById("moose-equation-cce5c27b-e5be-4bd2-9dbd-b6f187c0a4bd");katex.render("h_c", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the surface convection coefficient; and $var element = document.getElementById("moose-equation-b23a6fc6-bc11-4051-9119-4b30eb783cf5");katex.render("h_r", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the radiation heat transfer coefficient,

$var element = document.getElementById("moose-equation-38975cc5-82ef-4c97-8fe2-f55d6ca6f984");katex.render("h_r=\\varepsilon\\sigma\\left(T_s^2+T_\\infty^2\\right)\\left(T_s+T_\\infty\\right)\\ ,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-cc8bed08-2288-4eaa-ac68-1f059c60a6a7");katex.render("\\varepsilon", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the emissivity of the last conduction layer, $var element = document.getElementById("moose-equation-29afdcda-7f19-408e-8b82-d7a22df7e04b");katex.render("\\sigma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the Stefan-Boltzmann constant, $var element = document.getElementById("moose-equation-82872baa-bd96-458c-b352-d59d50e63de9");katex.render("T_s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the surface temperature of the last conduction layer, and $var element = document.getElementById("moose-equation-e5a95ca7-6c33-472e-85b9-1cbe12759852");katex.render("T_\\infty", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the far-field temperature.

The surface temperature is implicitly dependent on the heat flux, so an underrelaxed fixed point iteration is used to solve for $var element = document.getElementById("moose-equation-e0fe9aff-bd8c-4942-b9ed-852710c70977");katex.render("T_s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ at each quadrature point.

warning

The thermal resistance concept does not apply to conducting slabs with heat sources or systems undergoing transients. This boundary condition should only be applied to steady-state or pseudo-steady transients.

note

The emissivity parameter represents the emissivity of the _surface_, or last layer, of the conducting slabs.

Input Parameters

T_ambientconstant ambient temperature
C++ Type:double
Options:
Description:constant ambient temperature
boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector<BoundaryName>
Options:
Description:The list of boundary IDs from the mesh where this boundary condition applies
conduction_thicknessesvector of conduction layer thicknesses
C++ Type:std::vector<double>
Options:
Description:vector of conduction layer thicknesses
emissivityemissivity of the surface
C++ Type:double
Options:
Description:emissivity of the surface
htcheat transfer coefficient
C++ Type:MaterialPropertyName
Options:
Description:heat transfer coefficient
thermal_conductivitiesvector of thermal conductivity values used for the conduction layers
C++ Type:std::vector<double>
Options:
Description:vector of thermal conductivity values used for the conduction layers
variableThe name of the variable that this boundary condition applies to
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this boundary condition applies to

Required Parameters

displacementsThe displacements
C++ Type:std::vector<VariableName>
Options:
Description:The displacements
geometrycartesiantype of geometry
Default:cartesian
C++ Type:MooseEnum
Options:cartesian, cylindrical
Description:type of geometry
inner_radiuscoordinate corresponding to the first resistance layer
C++ Type:double
Options:
Description:coordinate corresponding to the first resistance layer
max_iterations100maximum iterations
Default:100
C++ Type:unsigned int
Options:
Description:maximum iterations
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Options:
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
step_size0.1underrelaxation step size
Default:0.1
C++ Type:double
Options:
Description:underrelaxation step size
temperaturetemperature variable
C++ Type:std::vector<VariableName>
Options:
Description:temperature variable
tolerance0.001tolerance to converge iterations
Default:0.001
C++ Type:double
Options:
Description:tolerance to converge iterations

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Options:
Description:Determines whether this object is calculated using an implicit or explicit form
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Options:
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Options:
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Options:
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

(modules/heat_conduction/test/tests/fvbcs/fv_thermal_resistance/test.i)

(modules/heat_conduction/test/tests/fvbcs/fv_thermal_resistance/test.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 5
    ny = 5
    xmax = 2
  []
[]

[Variables]
  [u]
    type = MooseVariableFVReal
    initial_condition = 0.5
  []
[]

[FVKernels]
  [diff_left]
    type = FVDiffusion
    variable = u
    coeff = 4
  []
  [gradient_creating]
    type = FVBodyForce
    variable = u
  []
[]

[FVBCs]
  [left]
    type = FVThermalResistanceBC
    geometry = 'cartesian'
    variable = u
    T_ambient = 10
    htc = 'htc'
    emissivity = 0.2
    thermal_conductivities = '0.1 0.2 0.3'
    conduction_thicknesses = '1 0.7 0.2'
    boundary = 'left'

    # Test setting iteration parameters
    step_size = 0.02
    max_iterations = 120
    tolerance = 1e-4
  []
  [top]
    type = FVThermalResistanceBC
    geometry = 'cartesian'
    variable = u
    # Test setting it separately
    temperature = 'u'
    T_ambient = 14
    htc = 'htc'
    emissivity = 0
    thermal_conductivities = '0.1 0.2 0.3'
    conduction_thicknesses = '1 0.7 0.4'
    boundary = 'top'
  []
  [other]
    type = FVDirichletBC
    variable = u
    boundary = 'right bottom'
    value = 0
  []
[]

[Materials]
  [cht]
    type = ADGenericConstantMaterial
    prop_names = 'htc'
    prop_values = '1'
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
[]

[Outputs]
  exodus = true
[]

Input Parameters
Input Files

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