Pd Source Material

Calculates Pd production rate density using simplified one-group approximations to account for fuel burnout and breeding.

Description

The PdSourceMaterial class is used to model Pd production in tri-structural isotropic (TRISO) particle fuel kernels. It applies one-group approximations and simplified transmutation chains to account for the effects of fuel burnout and breeding on the production rate density of Pd.

The transmutation chain relevant to Pd production is shown below: $(1)var element = document.getElementById("moose-equation-60a40ce5-1b30-4536-8ecd-9d48eb1387d7");katex.render(" ^{238}\\text{U} \\xrightarrow{\\text{n},\\gamma}\\: ^{239}\\text{U} \\xrightarrow{\\beta-}\\: ^{239}\\text{Np} \\xrightarrow{\\beta-}\\: ^{239}\\text{Pu} \\xrightarrow{\\text{n},\\gamma}\\: ^{240}\\text{Pu} \\xrightarrow{\\text{n},\\gamma}\\: ^{241}\\text{Pu} \\xrightarrow{\\beta-}\\: ^{241}\\text{Am} ", element, {displayMode:true,throwOnError:false});$ TRISO particle irradiations are typically on the order of months to years. $var element = document.getElementById("moose-equation-8abf1867-4438-49cf-aaea-a2b06df2a3ca");katex.render("^{239}\\text{U}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-19131917-1e07-41f7-9617-b55f7d605155");katex.render("^{239}\\text{Np}", element, {displayMode:false,throwOnError:false});$ have short half-lives on the order of minutes and days, respectively. Conversely, the Pu isotopes have long half-lives on the order of tens of years to tens of thousands of years. Comparing the TRISO irradiation time scale to the half-lives of the radioactive nuclides allows the transmutation chain to be simplified considerably: $(2)var element = document.getElementById("moose-equation-f1af3552-2615-4920-8b8c-98c16ba5c7b3");katex.render(" ^{238}\\text{U} \\xrightarrow{\\sim\\ \\text{n},\\gamma}\\: ^{239}\\text{Pu} \\xrightarrow{\\text{n}\\,\\gamma}\\: ^{240}\\text{Pu} \\xrightarrow{\\text{n},\\gamma}\\: ^{241}\\text{Pu} ", element, {displayMode:true,throwOnError:false});$

Assuming that there is initially no Pu present, and given the initial fuel density $var element = document.getElementById("moose-equation-2255b315-79d0-4efb-b941-2236529b072e");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ , mass fraction of U in the fuel $var element = document.getElementById("moose-equation-a9f7144b-404c-4618-b9ed-0e636df7e3be");katex.render("f_\\text{U}", element, {displayMode:false,throwOnError:false});$ , and initial $var element = document.getElementById("moose-equation-5fb6702a-d5f5-4020-ad77-2cbf12ccf364");katex.render("^{235}\\text{U}", element, {displayMode:false,throwOnError:false});$ enrichment $var element = document.getElementById("moose-equation-bdf37913-0826-4d05-ab04-a9ed413c70b2");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ , the initial number densities $var element = document.getElementById("moose-equation-1a6d965a-a96d-4178-a755-ea873cb9f95b");katex.render("N_i", element, {displayMode:false,throwOnError:false});$ of $var element = document.getElementById("moose-equation-cf4e316f-5617-41dc-9519-52a6f918717e");katex.render("^{235}\\text{U}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-1f7f7b90-17f1-44ac-a1aa-5b47b1a2847c");katex.render("^{238}\\text{U}", element, {displayMode:false,throwOnError:false});$ are calculated using: $(3)var element = document.getElementById("moose-equation-d377b451-654b-417e-a7cc-0ada70090474");katex.render(" N_i = \\begin{dcases} \\frac{\\epsilon \\rho f_\\text{U} N_A}{M_i}; & ^{235}\\text{U}, \\\\ \\\\ \\frac{(1 - \\epsilon) \\rho f_\\text{U} N_A}{M_i}; & ^{238}\\text{U}, \\\\ \\end{dcases} ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-d00bf0b9-66f2-4751-aec6-ee7d821ebece");katex.render("N_A", element, {displayMode:false,throwOnError:false});$ is Avogadro's number, and $var element = document.getElementById("moose-equation-5bc5924d-9ee1-419c-973e-7f9bbad1b50f");katex.render("M_i", element, {displayMode:false,throwOnError:false});$ are the molar masses of the nuclides.

PdSourceMaterial then uses a provided fission rate density $var element = document.getElementById("moose-equation-8a22f763-c35e-4d14-b7e5-2f0790a1b338");katex.render("\\dot{F}", element, {displayMode:false,throwOnError:false});$ and the microscopic cross sections for fission $var element = document.getElementById("moose-equation-8e07629c-2d93-4020-bf89-e474b685d3ec");katex.render("\\sigma_f^i", element, {displayMode:false,throwOnError:false});$ and neutron capture $var element = document.getElementById("moose-equation-a9d81d64-c634-4839-ad81-c6b61cb5b268");katex.render("\\sigma_c^i", element, {displayMode:false,throwOnError:false});$ to explicitly model evolution of the $var element = document.getElementById("moose-equation-855e3fca-d519-454c-a417-dccf665139cd");katex.render("^{235}\\text{U}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-cf0145b9-6fcd-4ad1-9520-7506e9c09cf4");katex.render("^{238}\\text{U}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a3aad1c6-399b-4c20-b6f0-8ddb4e44f7ec");katex.render("^{239}\\text{Pu}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-03adc084-cccf-415a-8f51-e7687065fe3f");katex.render("^{240}\\text{Pu}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-ad9e1737-0fd7-4185-b3d3-c9b0a5c83b69");katex.render("^{241}\\text{Pu}", element, {displayMode:false,throwOnError:false});$ number densities over time. At each time step, a representative flux $var element = document.getElementById("moose-equation-e050020e-057c-4a63-8671-89c2c1746f92");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is calculated using: $(4)var element = document.getElementById("moose-equation-76187105-2fd7-46ca-adef-5bba92b38d51");katex.render(" \\phi = \\frac{\\dot{F}}{\\sum\\limits_{i=1}^M N_i^\\text{old} \\sigma_f^i}, ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-8086c78a-e83e-4cf9-9e10-f8875697186a");katex.render("M", element, {displayMode:false,throwOnError:false});$ is the number of fissile/fertile nuclides being considered (5), and $var element = document.getElementById("moose-equation-adb177fc-99d0-418f-9c06-87f92b0fa4a9");katex.render("N_i^\\text{old}", element, {displayMode:false,throwOnError:false});$ are number densities evaluated at the previous time step.

The represenative flux is then used to calculate fission densities $var element = document.getElementById("moose-equation-f37b8303-9559-4ced-9746-c192480abd9d");katex.render("F_i", element, {displayMode:false,throwOnError:false});$ , capture densities $var element = document.getElementById("moose-equation-7b49f612-6121-41f9-a9b4-a4e9faeee130");katex.render("C_i", element, {displayMode:false,throwOnError:false});$ , and production densities $var element = document.getElementById("moose-equation-01c72c8a-e723-4999-ba22-d2777bb6b0be");katex.render("P_i", element, {displayMode:false,throwOnError:false});$ : $(5)var element = document.getElementById("moose-equation-89911778-3c4f-4e17-abcb-b2f236e00369");katex.render(" F_i = N_i^\\text{old} \\phi \\sigma_f^i \\Delta t, ", element, {displayMode:true,throwOnError:false});$ $(6)var element = document.getElementById("moose-equation-1976612e-2e19-407e-a148-f2df2e30026a");katex.render(" C_i = N_i^\\text{old} \\phi \\sigma_c^i \\Delta t, ", element, {displayMode:true,throwOnError:false});$ $(7)var element = document.getElementById("moose-equation-d42420f4-0e49-4642-bf38-13cf01cd48fb");katex.render(" P_i = \\begin{dcases} 0; & i \\leq 2, \\\\ C_{i - 1}; & i > 2, \\\\ \\end{dcases} ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-17644913-0653-4052-b576-0e78227c8148");katex.render("\\Delta t", element, {displayMode:false,throwOnError:false});$ is the time step size. Finally, the number densities at the current time step are calculated using: $(8)var element = document.getElementById("moose-equation-59a2ea4f-b08d-4bb4-963d-6406f90e6f49");katex.render(" N_i = N_i^\\text{old} - F_i - C_i + P_i. ", element, {displayMode:true,throwOnError:false});$ Cross sections are taken from NEA (1985) and calibrated to cumulative specific fission density calculations performed for Advanced Gas Reactor (AGR) experiments 3/4 Sterbentz (2015).

The number densities are then used to weight nuclide-specific Pd fission yields $var element = document.getElementById("moose-equation-aaf04966-f09d-45d7-839b-adafee6b2066");katex.render("\\gamma_i", element, {displayMode:false,throwOnError:false});$ taken from IAEA (2023). For each fissile/fertile nuclide, its total Pd yield is taken to be the sum of its yields for all stable and long-lived Pd isotopes ( $var element = document.getElementById("moose-equation-5e0f9574-7701-47ed-b877-db27eb8aabba");katex.render("^{102}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-e60f6654-2e6c-42ea-8ecc-2699a42e1e71");katex.render("^{104}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-39bbae21-78cb-4cb5-a84e-14f8997a14b0");katex.render("^{105}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-fb8fdd57-3596-4223-ad04-462d75a6b528");katex.render("^{106}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-5100481f-1461-4fea-98f1-7a1ce3e1daf4");katex.render("^{107}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-8fbbbb70-3e9a-404e-9f1d-0ec7d1ca3681");katex.render("^{108}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-496db512-3877-409a-8abf-0f8ded6437bf");katex.render("^{110}\\text{Pd}", element, {displayMode:false,throwOnError:false});$ ). The total Pd yield $var element = document.getElementById("moose-equation-fb791f72-ee0f-4e0c-997d-aff5ac94a79d");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ is calculated using: $(9)var element = document.getElementById("moose-equation-5b206ff7-ab27-458f-806c-ecd76fe24726");katex.render(" \\gamma = s\\; \\frac{\\sum\\limits_{i=1}^M F_i \\gamma_i}{\\sum\\limits_{i=1}^M F_i}, ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-2348c8e9-df83-4999-b040-b9ebe46198f5");katex.render("s", element, {displayMode:false,throwOnError:false});$ is an optional constant that can be applied to scale the total Pd yield. Finally, the Pd production rate density $var element = document.getElementById("moose-equation-01556a61-10b4-4777-8c1a-21f40827a93c");katex.render("\\dot{P_\\text{Pd}}", element, {displayMode:false,throwOnError:false});$ (in mol/m $var element = document.getElementById("moose-equation-91a16364-e186-47e7-beae-c8ca781f9b61");katex.render("^3\\cdot", element, {displayMode:false,throwOnError:false});$ s) is calculated using: $(10)var element = document.getElementById("moose-equation-10257723-d384-4fe4-873f-79c937f6eada");katex.render(" \\dot{P_\\text{Pd}} = \\frac{\\gamma \\sum\\limits_{i=1}^M F_i}{N_A \\Delta t}, ", element, {displayMode:true,throwOnError:false});$

Note that while this model accounts for the effects of fuel burnout and breeding to deliver reasonable Pd production rate densities for prototypic TRISO geometries and operating conditions, it employs numerous assumptions and simplifications and relies on parameters calibrated to a limited dataset. The represenative flux, cross sections, and number densities associated with this model should be used with caution if applied to analyses outside the scope of Pd production in prototypic TRISO fuels. This model is not intended to take the place of proper neutronic and depletion calculations performed with realistic transmutation chains, energy-resolved flux spectra, and appropriate multi-group cross sections.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [pd_source]
    type = ADPdSourceMaterial<<<{"description": "Calculates Pd production rate density using simplified one-group approximations to account for fuel burnout and breeding.", "href": "PdSourceMaterial.html"}>>>
    energy_per_fission<<<{"description": "The energy per fission in J"}>>> = 3.20435e-11
    fission_rate<<<{"description": "The fission rate density in 1/m^3-s"}>>> = fission_rate
    initial_enrichment<<<{"description": "The initial U-235 enrichment in kg/kg"}>>> = 0.19717
    initial_fuel_density<<<{"description": "The initial density of the fuel in kg/m^3"}>>> = 10400
    mass_fraction_of_U_in_fuel<<<{"description": "The initial mass fraction of U in the fuel in kg/kg"}>>> = 0.885
    scaling_factor<<<{"description": "The Pd production scaling factor"}>>> = 1
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

Input Parameters

initial_enrichmentThe initial U-235 enrichment in kg/kg
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The initial U-235 enrichment in kg/kg
initial_fuel_densityThe initial density of the fuel in kg/m^3
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The initial density of the fuel in kg/m^3
mass_fraction_of_U_in_fuelThe initial mass fraction of U in the fuel in kg/kg
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The initial mass fraction of U in the fuel in kg/kg

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
energy_per_fission3.20435e-11The energy per fission in J
Default:3.20435e-11
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The energy per fission in J
fission_ratefission_rateThe fission rate density in 1/m^3-s
Default:fission_rate
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The fission rate density in 1/m^3-s
scaling_factor1The Pd production scaling factor
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The Pd production scaling factor

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
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
Controllable:Yes
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
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
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
Controllable:No
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

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

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
Unit:(no unit assumed)
Controllable:No
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.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

References

International Atomic Energy Agency IAEA. Live chart of nuclides - nuclear structure and decay data. https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html, 2023.

@MISC{chartOfNuclides2023,
    author = "IAEA, International Atomic Energy Agency",
    title = "{Live chart of nuclides - nuclear structure and decay data}",
    howpublished = "https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html",
    year = "2023"
}

Nuclear Energy Agency Data Bank NEA. JEF report 3 - table of simple integral neutron cross-section data from JEF-1, ENDF/B-V, ENDL-82, JENDL-2, KEDAK-4, RCN-3. https://www.oecd-nea.org/dbdata/nds_jefreports/jefreport-3.pdf, 1985.

@MISC{JEFReport1985,
    author = "NEA, Nuclear Energy Agency Data Bank",
    title = "{JEF} report 3 - table of simple integral neutron cross-section data from {JEF}-1, {ENDF/B-V}, {ENDL}-82, {JENDL}-2, {KEDAK}-4, {RCN}-3",
    howpublished = "https://www.oecd-nea.org/dbdata/nds\_jefreports/jefreport-3.pdf",
    year = "1985"
}

J. Sterbentz. JMOCUP as-run daily physics depletion calculation for AGR-3/4 TRISO particle experiment in ATR northeast flux trap. Technical Report TEM-10200-1 Revision 6, Idaho National Laboratory, June 2015.

@TECHREPORT{ECAR-2753,
    author = "Sterbentz, J.",
    title = "{JMOCUP} as-run daily physics depletion calculation for {AGR}-3/4 {TRISO} particle experiment in {ATR} northeast flux trap",
    year = "2015",
    number = "TEM-10200-1 Revision 6",
    month = "June",
    publisher = "INL Technical Report",
    institution = "Idaho National Laboratory"
}

(test/tests/pd_source_material/ad.i)

# This test is to verify the implementation of PdSourceMaterial material.

# AD-generated Jacobian entries are compared to finite difference calculations.

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Variables]
  [var_production]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1
  []
[]

[Kernels]
  [var_production_time]
    type = ADTimeDerivative
    variable = var_production
  []
  [var_production_force]
    type = ADMatHeatSource
    variable = var_production
    material_property = pd_production_rate_density
  []
[]

[Materials]
  [fission_rate]
    type = ADDerivativeParsedMaterial
    property_name = fission_rate
    coupled_variables = var_production
    expression = var_production
    derivative_order = 1
    outputs = all
  []
  [pd_source]
    type = ADPdSourceMaterial
    energy_per_fission = 3.20435e-11
    fission_rate = fission_rate
    initial_enrichment = 0.19717
    initial_fuel_density = 10400
    mass_fraction_of_U_in_fuel = 0.885
    scaling_factor = 1
    outputs = all
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  line_search = none
  nl_abs_tol = 1e-15
  start_time = 0
  num_steps = 10
  dt = 1e6
[]

[Postprocessors]
  # inputs to PdSourceMaterial
  [fission_rate]
    type = ElementAverageValue
    variable = fission_rate
  []

  # coupled variable values
  [var_production]
    type = ElementAverageValue
    variable = var_production
  []
[]

(test/tests/pd_source_material/exact.i)

# This test is to verify the implementation of PdSourceMaterial material.

# PdSourceMaterial is used to model fuel burnout and breeding for a prototypic TRISO kernel.

# The following quanitities are compared to hand-calculated results:
#   - nuclide number densities and
#   - weighted Pd yield.

initial_fuel_density = 10400
mass_fraction_of_U_in_fuel = 0.885
initial_enrichment = 0.19717
energy_per_fission = 3.20435e-11

[Problem]
  solve = false
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Materials]
  [fission_rate]
    type = ParsedMaterial
    property_name = fission_rate
    expression = '3.57e9 / ${energy_per_fission}'
  []
  [burnup]
    type = Burnup
    fission_rate = fission_rate
    atoms_heavy_metal_per_volume = '${fparse initial_fuel_density * mass_fraction_of_U_in_fuel * 6.0221e23 * (initial_enrichment / 0.235043923 + (1.0 - initial_enrichment) / 0.238050783)}'
  []
  [pd_source]
    type = PdSourceMaterial
    energy_per_fission = ${energy_per_fission}
    fission_rate = fission_rate
    initial_enrichment = ${initial_enrichment}
    initial_fuel_density = ${initial_fuel_density}
    mass_fraction_of_U_in_fuel = ${mass_fraction_of_U_in_fuel}
    scaling_factor = 1
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  start_time = 0
  end_time = 32014766
  dt = 1114456
[]

[Postprocessors]
  [fission_rate]
    type = ElementAverageMaterialProperty
    mat_prop = fission_rate
  []
  [burnup]
    type = ElementAverageMaterialProperty
    mat_prop = burnup
  []
  [weighted_pd_yield]
    type = ElementAverageMaterialProperty
    mat_prop = weighted_pd_yield
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [pd_production_rate_density]
    type = ElementAverageMaterialProperty
    mat_prop = pd_production_rate_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [u235_number_density]
    type = ElementAverageMaterialProperty
    mat_prop = u235_number_density
  []
  [u238_number_density]
    type = ElementAverageMaterialProperty
    mat_prop = u238_number_density
  []
  [pu239_number_density]
    type = ElementAverageMaterialProperty
    mat_prop = pu239_number_density
  []
  [pu240_number_density]
    type = ElementAverageMaterialProperty
    mat_prop = pu240_number_density
  []
  [pu241_number_density]
    type = ElementAverageMaterialProperty
    mat_prop = pu241_number_density
  []
[]

[Outputs]
  csv = true
[]

(test/tests/pd_source_material/ad.i)

# This test is to verify the implementation of PdSourceMaterial material.

# AD-generated Jacobian entries are compared to finite difference calculations.

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Variables]
  [var_production]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1
  []
[]

[Kernels]
  [var_production_time]
    type = ADTimeDerivative
    variable = var_production
  []
  [var_production_force]
    type = ADMatHeatSource
    variable = var_production
    material_property = pd_production_rate_density
  []
[]

[Materials]
  [fission_rate]
    type = ADDerivativeParsedMaterial
    property_name = fission_rate
    coupled_variables = var_production
    expression = var_production
    derivative_order = 1
    outputs = all
  []
  [pd_source]
    type = ADPdSourceMaterial
    energy_per_fission = 3.20435e-11
    fission_rate = fission_rate
    initial_enrichment = 0.19717
    initial_fuel_density = 10400
    mass_fraction_of_U_in_fuel = 0.885
    scaling_factor = 1
    outputs = all
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  line_search = none
  nl_abs_tol = 1e-15
  start_time = 0
  num_steps = 10
  dt = 1e6
[]

[Postprocessors]
  # inputs to PdSourceMaterial
  [fission_rate]
    type = ElementAverageValue
    variable = fission_rate
  []

  # coupled variable values
  [var_production]
    type = ElementAverageValue
    variable = var_production
  []
[]

Description
Example Input Syntax
Input Parameters
Input Files
References