Plenum Pressure

Overview

An important feature in a fuel performance code is the ability to compute an internal gas pressure. This is necessary for computing the plenum pressure, for example.

Starting with the ideal gas law, the procedure is as follows. $var element = document.getElementById("moose-equation-7ff2ed2b-dce1-4fc3-83ba-b952ec6764b0");katex.render("PV = nRT", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-d92e21fd-324a-48a4-89ee-c052abb1cdec");katex.render("P", element, {displayMode:false,throwOnError:false});$ is the pressure (Pa), $var element = document.getElementById("moose-equation-4ae1eaab-f09f-4d85-8d82-f99112cc8c88");katex.render("V", element, {displayMode:false,throwOnError:false});$ is volume (m $var element = document.getElementById("moose-equation-989a168d-f31a-43a5-90ec-8ff93d6e0b8c");katex.render("^3", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-3b80c834-ea9d-41b1-8bc6-76852a082780");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the amount of material in the volume (moles), $var element = document.getElementById("moose-equation-f5172198-caaf-4f20-a667-487b4fc511fd");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the universal gas constant (8.314472 J/K/mol), and $var element = document.getElementById("moose-equation-55054c44-60dc-45e3-bc44-5b0e59619fe2");katex.render("T", element, {displayMode:false,throwOnError:false});$ is temperature (K).

Demonstration

As a demonstration, we take an axisymmetric mesh composed of two blocks. One block has an interior cavity of volume 8 m $var element = document.getElementById("moose-equation-728f672a-8621-488e-992d-11e035c9ef68");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . A second block is inside with a volume of 1 m $var element = document.getElementById("moose-equation-a2bdecfd-d99c-434e-b862-59d09a4b1b23");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , giving a net volume of 7 m $var element = document.getElementById("moose-equation-fdd3650b-5c2e-4bf0-8abc-bbf0a33c9cfa");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . Temperature is defined as $var element = document.getElementById("moose-equation-c78c24e0-61da-46c5-a842-cc6e0cb24f79");katex.render("T=T_0 + \\beta t", element, {displayMode:true,throwOnError:false});$ Also,

$var element = document.getElementById("moose-equation-4aafed43-01b0-41a6-bdd6-29dc55fe83b6");katex.render("\\beta = T_0", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-3422d7a3-2798-4d9d-b445-6a8a21b49375");katex.render("T_0 = 240.54443866068704", element, {displayMode:false,throwOnError:false});$ K
$var element = document.getElementById("moose-equation-b83b143e-a446-4887-bf9b-6b15fb46ec47");katex.render("p(0) = 100", element, {displayMode:false,throwOnError:false});$ Pa

This gives $var element = document.getElementById("moose-equation-e3ef891f-aa5a-4819-8200-42a6e8abe35d");katex.render("n(0) = \\frac{p(0)V}{RT(0)} = 0.35", element, {displayMode:true,throwOnError:false});$

At $var element = document.getElementById("moose-equation-3febb285-d243-4e98-9cee-319dac0b8419");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 1 s, $var element = document.getElementById("moose-equation-c4cdceaf-4bb6-4a08-8101-76a6cae1d5ba");katex.render("P", element, {displayMode:false,throwOnError:false});$ = 200 Pa. Bison computes this value exactly.

Input Syntax

#
# Cavity Pressure Test
#
# This test is designed to compute an internal pressure based on
#   p = n * R * T / V
# where
#   p is the pressure
#   n is the amount of material in the volume (moles)
#   R is the universal gas constant
#   T is the temperature
#   V is the volume
#
# The mesh is composed of one block (2) with an interior cavity of volume 8.
#   Block 1 sits in the cavity and has a volume of 1.  Thus, the total
#   initial volume is 7.
# The test adjusts T in the following way:
#   T => T0 + beta * t
# with
#   beta = T0
#   T0 = 240.54443866068704
#   V0 = 7
#   n0 = f(p0)
#   p0 = 100
#   R = 8.314472 J * K^(-1) * mol^(-1)
#
# So, n0 = p0 * V0 / R / T0 = 100 * 7 / 8.314472 / 240.544439
#        = 0.35
#
# At t = 1, p = 200.
#

[GlobalParams<<<{"href": "../../../syntax/GlobalParams/index.html"}>>>]
  displacements = 'disp_r disp_z'
[]

[Mesh<<<{"href": "../../../syntax/Mesh/index.html"}>>>]
  coord_type = RZ
  [mesh]
    type = FuelPinMeshGenerator<<<{"description": "Creates a 2D-RZ axisymmetric mesh of a fuel rod.", "href": "../../../source/meshgenerators/FuelPinMeshGenerator.html"}>>>
    pellet_quantity<<<{"description": "Number of pellets in smeared mesh"}>>> = 1
    pellet_outer_radius<<<{"description": "Pellet outer radius"}>>> = 0.5
    pellet_height<<<{"description": "Pellet height"}>>> = 1.2732395447351628
    clad_gap_width<<<{"description": "Gap between outer radius of pellet and inside surface of cladding"}>>> = 0.5
    clad_thickness<<<{"description": "Thickness of cladding side wall"}>>> = 0.5
    bottom_clad_height<<<{"description": "Height of bottom cladding wall."}>>> = 0.5
    top_clad_height<<<{"description": "Height of top cladding wall."}>>> = 0.5
    clad_top_gap_height<<<{"description": "Gap between top of pellet and inside top surface of cladding"}>>> = 0.6366197723675814
    clad_bot_gap_height<<<{"description": "Gap between bottom of pellet and inside bottom surface of cladding"}>>> = 0.6366197723675814
    nx_p<<<{"description": "number of fuel elements in radial direction"}>>> = 1
    ny_p<<<{"description": "Number of fuel elements in axial direction"}>>> = 1
    nx_c<<<{"description": "Number of cladding elements in radial direction."}>>> = 1
    ny_c<<<{"description": "Number of cladding elements in axial direction."}>>> = 1
    ny_cu<<<{"description": "Number of cladding elements in upper plug in axial direction."}>>> = 1
    ny_cl<<<{"description": "Number of cladding elements in lower plug in  axial direction."}>>> = 1
    pellet_mesh_density<<<{"description": "Sets the mesh density of the pellet (really_coarse coarse medium fine customize)."}>>> = customize
    clad_mesh_density<<<{"description": "Sets the mesh density of the cladding (coarse, medium, fine)"}>>> = customize
  []
[]

[Functions<<<{"href": "../../../syntax/Functions/index.html"}>>>]
  [temperature]
    type = PiecewiseLinear<<<{"description": "Linearly interpolates between pairs of x-y data", "href": "../../../source/functions/PiecewiseLinear.html"}>>>
    x<<<{"description": "The abscissa values"}>>> = '0 1'
    y<<<{"description": "The ordinate values"}>>> = '1 2'
    scale_factor<<<{"description": "Scale factor to be applied to the ordinate values"}>>> = 240.54443866068704
  []
[]

[Variables<<<{"href": "../../../syntax/Variables/index.html"}>>>]
  [disp_r]
  []
  [disp_z]
  []
  [temp]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 240.54443866068704
  []
[]

[Physics<<<{"href": "../../../syntax/Physics/index.html"}>>>/SolidMechanics<<<{"href": "../../../syntax/Physics/SolidMechanics/index.html"}>>>/QuasiStatic<<<{"href": "../../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
  [all]
    strain<<<{"description": "Strain formulation"}>>> = finite
  []
[]

[Kernels<<<{"href": "../../../syntax/Kernels/index.html"}>>>]
  [heat]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = temp
    use_displaced_mesh<<<{"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."}>>> = true
  []
[]

[BCs<<<{"href": "../../../syntax/NuclearMaterials/BCs/index.html"}>>>]
  [no_x]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_r
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = '9 12'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []
  [no_y]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_z
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = '9 12'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []
  [temperatureInterior]
    type = FunctionDirichletBC<<<{"description": "Imposes the essential boundary condition $u=g(t,\\vec{x})$, where $g$ is a (possibly) time and space-dependent MOOSE Function.", "href": "../../../source/bcs/FunctionDirichletBC.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 9
    function<<<{"description": "The forcing function."}>>> = temperature
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = temp
  []
  [CavityPressure<<<{"href": "../../../syntax/BCs/CavityPressure/index.html"}>>>]
    [1]
      boundary<<<{"description": "The list of boundary IDs from the mesh where the pressure will be applied"}>>> = 9
      initial_pressure<<<{"description": "The initial pressure in the cavity.  If not given, a zero initial pressure will be used."}>>> = 100
      R<<<{"description": "The universal gas constant for the units used."}>>> = 8.314472
      temperature<<<{"description": "The name of the average temperature postprocessor value."}>>> = aveTempInterior
      volume<<<{"description": "The name of the postprocessor(s) that holds the value of the internal volume in the cavity"}>>> = internalVolume
      startup_time<<<{"description": "The amount of time during which the pressure will ramp from zero to its true value."}>>> = 0.5
      output<<<{"description": "The name to use for the cavity pressure value"}>>> = bison_ppress
    []
  []
[]

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [elastic_tensor1]
    type = ComputeIsotropicElasticityTensor<<<{"description": "Compute a constant isotropic elasticity tensor.", "href": "../../../source/materials/ComputeIsotropicElasticityTensor.html"}>>>
    youngs_modulus<<<{"description": "Young's modulus of the material."}>>> = 1e6
    poissons_ratio<<<{"description": "Poisson's ratio for the material."}>>> = 0.3
  []
  [stress]
    type = ComputeFiniteStrainElasticStress<<<{"description": "Compute stress using elasticity for finite strains", "href": "../../../source/materials/ComputeFiniteStrainElasticStress.html"}>>>
  []
[]

[Executioner<<<{"href": "../../../syntax/Executioner/index.html"}>>>]
  type = Transient

  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm       lu'

  nl_abs_tol = 1e-10

  l_max_its = 20

  dt = 1.0
  end_time = 1.0
[]

[Postprocessors<<<{"href": "../../../syntax/Postprocessors/index.html"}>>>]
  [internalVolume]
    type = InternalVolume<<<{"description": "Computes the volume of an enclosed area by performing an integral over a user-supplied boundary.", "href": "../../../source/postprocessors/InternalVolume.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 9
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial linear'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = none
  []
  [aveTempInterior]
    type = SideAverageValue<<<{"description": "Computes the average value of a variable on a sideset. Note that this cannot be used on the centerline of an axisymmetric model.", "href": "../../../source/postprocessors/SideAverageValue.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 9
    variable<<<{"description": "The name of the variable which this postprocessor integrates"}>>> = temp
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial linear'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = none
  []
  [analytical_ppress]
    type = ParsedPostprocessor<<<{"description": "Computes a parsed expression with post-processors", "href": "../../../source/postprocessors/ParsedPostprocessor.html"}>>>
    pp_names<<<{"description": "Post-processors arguments"}>>> = ''
    constant_names<<<{"description": "Vector of constants used in the parsed function (use this for kB etc.)"}>>> = 'n0 T0 R V'
    constant_expressions<<<{"description": "Vector of values for the constants in constant_names (can be an FParser expression)"}>>> = '0.35 240.54443866068704 8.314472 7'
    expression<<<{"description": "function expression"}>>> = 'T := 2*T0;
                n0*R*T/V'
  []
  [error]
    type = ParsedPostprocessor<<<{"description": "Computes a parsed expression with post-processors", "href": "../../../source/postprocessors/ParsedPostprocessor.html"}>>>
    expression<<<{"description": "function expression"}>>> = '(bison_ppress-analytical_ppress)/analytical_ppress*100'
    pp_names<<<{"description": "Post-processors arguments"}>>> = 'bison_ppress analytical_ppress'
  []
[]

[Outputs<<<{"href": "../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
[]

(assessment/verification/Other/plenum_pressure/rz.i)

#
# Cavity Pressure Test
#
# This test is designed to compute an internal pressure based on
#   p = n * R * T / V
# where
#   p is the pressure
#   n is the amount of material in the volume (moles)
#   R is the universal gas constant
#   T is the temperature
#   V is the volume
#
# The mesh is composed of one block (2) with an interior cavity of volume 8.
#   Block 1 sits in the cavity and has a volume of 1.  Thus, the total
#   initial volume is 7.
# The test adjusts T in the following way:
#   T => T0 + beta * t
# with
#   beta = T0
#   T0 = 240.54443866068704
#   V0 = 7
#   n0 = f(p0)
#   p0 = 100
#   R = 8.314472 J * K^(-1) * mol^(-1)
#
# So, n0 = p0 * V0 / R / T0 = 100 * 7 / 8.314472 / 240.544439
#        = 0.35
#
# At t = 1, p = 200.
#

[GlobalParams]
  displacements = 'disp_r disp_z'
[]

[Mesh]
  coord_type = RZ
  [mesh]
    type = FuelPinMeshGenerator
    pellet_quantity = 1
    pellet_outer_radius = 0.5
    pellet_height = 1.2732395447351628
    clad_gap_width = 0.5
    clad_thickness = 0.5
    bottom_clad_height = 0.5
    top_clad_height = 0.5
    clad_top_gap_height = 0.6366197723675814
    clad_bot_gap_height = 0.6366197723675814
    nx_p = 1
    ny_p = 1
    nx_c = 1
    ny_c = 1
    ny_cu = 1
    ny_cl = 1
    pellet_mesh_density = customize
    clad_mesh_density = customize
  []
[]

[Functions]
  [temperature]
    type = PiecewiseLinear
    x = '0 1'
    y = '1 2'
    scale_factor = 240.54443866068704
  []
[]

[Variables]
  [disp_r]
  []
  [disp_z]
  []
  [temp]
    initial_condition = 240.54443866068704
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = finite
  []
[]

[Kernels]
  [heat]
    type = Diffusion
    variable = temp
    use_displaced_mesh = true
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_r
    boundary = '9 12'
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_z
    boundary = '9 12'
    value = 0.0
  []
  [temperatureInterior]
    type = FunctionDirichletBC
    boundary = 9
    function = temperature
    variable = temp
  []
  [CavityPressure]
    [1]
      boundary = 9
      initial_pressure = 100
      R = 8.314472
      temperature = aveTempInterior
      volume = internalVolume
      startup_time = 0.5
      output = bison_ppress
    []
  []
[]

[Materials]
  [elastic_tensor1]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1e6
    poissons_ratio = 0.3
  []
  [stress]
    type = ComputeFiniteStrainElasticStress
  []
[]

[Executioner]
  type = Transient

  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm       lu'

  nl_abs_tol = 1e-10

  l_max_its = 20

  dt = 1.0
  end_time = 1.0
[]

[Postprocessors]
  [internalVolume]
    type = InternalVolume
    boundary = 9
    execute_on = 'initial linear'
    outputs = none
  []
  [aveTempInterior]
    type = SideAverageValue
    boundary = 9
    variable = temp
    execute_on = 'initial linear'
    outputs = none
  []
  [analytical_ppress]
    type = ParsedPostprocessor
    pp_names = ''
    constant_names = 'n0 T0 R V'
    constant_expressions = '0.35 240.54443866068704 8.314472 7'
    expression = 'T := 2*T0;
                n0*R*T/V'
  []
  [error]
    type = ParsedPostprocessor
    expression = '(bison_ppress-analytical_ppress)/analytical_ppress*100'
    pp_names = 'bison_ppress analytical_ppress'
  []
[]

[Outputs]
  csv = true
[]

Overview
Demonstration
Input Syntax