MOXPoreVelocityVaporPressure

Computes pore speed from author Kato. Used with vapor pressure calculations from MOXVaporPressure.

Description

Pore speed is calculated in this class, which is used in the kernel MOXPoreContinuity shown in the second term of the following equation: $var element = document.getElementById("moose-equation-35e0cf9e-1564-4c98-8129-2a7ac09a0ba4");katex.render(" \\frac{\\partial p}{\\partial t} + \\nabla \\cdot \\left(p v_p \\nabla T - \\nu \\nabla p \\right) = 0", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-de83673c-6668-4cd2-a808-631dd78c8bb4");katex.render("p", element, {displayMode:false,throwOnError:false});$ is the porosity, $var element = document.getElementById("moose-equation-e095dfc1-d2be-4ae6-9cbb-8718d3860431");katex.render("v_p", element, {displayMode:false,throwOnError:false});$ is the pore velocity, $var element = document.getElementById("moose-equation-f66ac8f9-c291-442e-8016-6f9bcd41b34e");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature, and $var element = document.getElementById("moose-equation-3e6187b8-c95d-4457-bdaa-089569e5943f");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ is the diffusion coefficient MOXPoreDiffusion. Usually, the temperature gradient is included in the pore velocity term, but here, the temperature gradient is written separately to emphasize the dependence of pore migration on temperature gradient. The difference between the equation for pore speed found here and in MOXPoreVelocity is vapor pressure terms for UO $var element = document.getElementById("moose-equation-bcc40a45-453c-4deb-868e-9a945a4beaf4");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , PuO $var element = document.getElementById("moose-equation-710a445d-4bdb-4c3e-8945-ae77bb083f9d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , UO, U, UO $var element = document.getElementById("moose-equation-825651ae-46cb-485d-880a-7c3fb1eb242a");katex.render("_3", element, {displayMode:false,throwOnError:false});$ , PuO, Pu, AmO $var element = document.getElementById("moose-equation-15ff106b-6def-403e-bb41-8559ff5f7f3d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , Am, AmO, and O $var element = document.getElementById("moose-equation-05f5d2c3-a523-4f8e-9c20-1cb1db46f155");katex.render("_2", element, {displayMode:false,throwOnError:false});$ .

From Ikusawa et al. (2014) $var element = document.getElementById("moose-equation-bf7756eb-6ce5-432f-9df9-661398964bae");katex.render("\\begin{aligned} v_p & = \\Omega \\text{ } D_{12} \\frac{dn}{dT} \\frac{dT}{dr}\\\\ \\Omega & = \\frac{1}{4}(X_{PuO_{2}}C_{PuO_{2}}+X_{UO_{2}}C_{UO_{2}})^3\\\\ D_{12} & = \\frac{3}{8}\\frac{1}{\\pi nd^2}\\sqrt{\\frac{\\pi kT}{2m^*}}\\\\ \\frac{1}{m^*} & = \\left(\\frac{1}{18}+\\frac{1}{270}\\right)6.023E23\\\\ \\frac{dn}{dT} & = \\frac{d}{dT}\\frac{P}{kT} = \\frac{1}{kT^2}\\left(\\frac{dP}{dT}-P\\right)\\\\ \\left(\\frac{dT}{dr}\\right)_{void} & = \\alpha\\left(\\frac{dT}{dr}\\right)_{matrix} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-613ddc81-e84f-42da-8959-78b6577c2028");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ is fuel molecular volume in cm $var element = document.getElementById("moose-equation-f47c1341-2e32-4cdd-8102-ce85e70acef8");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-06880738-2ee3-446d-be39-592383975899");katex.render("X_{PuO_{2}}", element, {displayMode:false,throwOnError:false});$ is the mol fraction of PuO $var element = document.getElementById("moose-equation-bc26b328-f16f-4f21-b479-1a9655a40e73");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-e9d10e2a-bc71-4c32-9233-c10d5783fda9");katex.render("X_{UO_{2}}", element, {displayMode:false,throwOnError:false});$ is the mol fraction of UO $var element = document.getElementById("moose-equation-293e66d5-9647-40a2-83a1-b223ab1dd4ba");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-c4a0e2b7-ede1-4644-a95b-50951021bf61");katex.render("C_{PuO_{2}}", element, {displayMode:false,throwOnError:false});$ is the lattice parameter of PuO $var element = document.getElementById("moose-equation-65e92495-6994-4b6a-a37c-dc6c55162902");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ (5.396E-8 cm), $var element = document.getElementById("moose-equation-b21362b9-7e06-4d8a-ad48-cccbe24e0fe2");katex.render("C_{UO_{2}}", element, {displayMode:false,throwOnError:false});$ is the lattice parameter of UO $var element = document.getElementById("moose-equation-3904840c-74a1-4165-a99b-62185fc53893");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ (5.4704E-8 cm), D $var element = document.getElementById("moose-equation-f347fb5e-e8f5-415e-a137-8418f300c453");katex.render("_{12}", element, {displayMode:false,throwOnError:false});$ is the diffusion coefficient of PuO $var element = document.getElementById("moose-equation-6082feb0-bc86-4406-948d-71b77e522462");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ and UO $var element = document.getElementById("moose-equation-15688e46-ea60-4462-879a-d6b2ae2a105a");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ gas in the void, $var element = document.getElementById("moose-equation-4d4b9208-bfa0-41cf-a956-602ecec7341b");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the Boltzmann constant (1.3708E-16 cm $var element = document.getElementById("moose-equation-76ebd501-795d-4340-a801-c63d2ad074b1");katex.render("^2", element, {displayMode:false,throwOnError:false});$ g/sK), m $var element = document.getElementById("moose-equation-ba559479-edd9-4453-8aed-f3f1c867bcef");katex.render("^*", element, {displayMode:false,throwOnError:false});$ is a mass conversion factor, $var element = document.getElementById("moose-equation-ee5d3c6b-942d-493d-9437-857e993b8675");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the number of gas molecules captured within voids during sintering, $var element = document.getElementById("moose-equation-5cb010ef-7726-4d5e-8369-e70727fbe348");katex.render("d", element, {displayMode:false,throwOnError:false});$ is the gas molecule diameter (4.33E-8 cm), $var element = document.getElementById("moose-equation-49aa3452-3e7c-4c4b-8ffc-7985d7b61155");katex.render("\\frac{dn}{dT}", element, {displayMode:false,throwOnError:false});$ is the concentration gradient of gas molecules in a void, $var element = document.getElementById("moose-equation-12f149c5-2a3f-4a4a-b515-7a16697ab8f4");katex.render("T", element, {displayMode:false,throwOnError:false});$ is temperature in K, $var element = document.getElementById("moose-equation-683d8b80-1ce1-48f2-af9c-e759821e8dc2");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the correction factor of void migration velocity, and $var element = document.getElementById("moose-equation-5546a136-795d-459e-b126-31413ad3e244");katex.render("P", element, {displayMode:false,throwOnError:false});$ is calculated from correlations between vapor pressure of vapor species and Gibbs free energy. Specifically, $var element = document.getElementById("moose-equation-a7817865-d48e-4a9f-9d7e-76dfa7e715b5");katex.render("P", element, {displayMode:false,throwOnError:false});$ is the summation of the vapor pressure terms from the following fuel constituents: UO $var element = document.getElementById("moose-equation-a9e335fd-dab9-448c-b569-361ec9e04fb1");katex.render("_{2}", element, {displayMode:false,throwOnError:false});$ , PuO2, UO, U, UO $var element = document.getElementById("moose-equation-be532b5b-3167-4fa8-99fd-9fbfd2ccd2d3");katex.render("_3", element, {displayMode:false,throwOnError:false});$ , PuO, Pu, AmO $var element = document.getElementById("moose-equation-85af6ba1-f231-43b5-86aa-8057036d1e64");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , Am, AmO, and O $var element = document.getElementById("moose-equation-2c9097de-e716-4e0e-b628-cb719ab066c2");katex.render("_2", element, {displayMode:false,throwOnError:false});$ . The term $var element = document.getElementById("moose-equation-284f4c4d-017b-4063-921a-cfae5e779071");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ corresponds to a factor to assume the temperature gradient in a pore from that in fuel.

For $var element = document.getElementById("moose-equation-c516491f-3ec0-4b7f-b4bc-af55a21b0683");katex.render("q", element, {displayMode:false,throwOnError:false});$ = Pu concentration, $var element = document.getElementById("moose-equation-43344e1e-3c60-487e-9d4f-72e896dc6890");katex.render("r", element, {displayMode:false,throwOnError:false});$ = Am concentration, $var element = document.getElementById("moose-equation-49744633-da05-49ba-8131-11e72b00ad49");katex.render("x", element, {displayMode:false,throwOnError:false});$ = deviation from stoichiometry of the metal oxide (MO $var element = document.getElementById("moose-equation-2a305a4b-f672-48d3-b30e-a895bc7ffe20");katex.render("_{2.00 - x}", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-a35c8b77-f4c7-429f-92cf-ab42550b0b21");katex.render("p\\text{\\_}O_{2}", element, {displayMode:false,throwOnError:false});$ = partial pressure of oxygen, and Gibbs free energy via the Rand-Markin model ( $var element = document.getElementById("moose-equation-9131440b-425b-40bd-903c-eab2cd1b5d7e");katex.render("\\Delta G", element, {displayMode:false,throwOnError:false});$ s defined subsequently), the following equations are the vapor pressure terms as a function of temperature ( $var element = document.getElementById("moose-equation-0c6682a4-74fb-45d2-aaec-a52e17a74aeb");katex.render("T", element, {displayMode:false,throwOnError:false});$ ):

$var element = document.getElementById("moose-equation-fef88d55-8dec-4e7b-93bb-ecadbed52bc7");katex.render(" \\begin{aligned} p\\text{\\_}UO_{2} & = (1-q-r) \\text{ } \\text{exp}\\left(-\\frac{\\Delta G_{UO_{2},\\text{vap}}}{RT}\\right)\\\\ \\text{ }\\\\ p\\text{\\_}PuO_{2} & = q \\text{ } (p\\text{\\_}O_{2})^{m/2}\\text{exp}\\left(-\\frac{\\Delta G_{A}}{RT}\\right)\\\\ \\text{ }\\\\ p\\text{\\_}UO & = \\frac{p\\text{\\_}UO_{2}}{\\text{exp}\\left(-\\frac{\\Delta G_{UO/UO_{2}}}{RT}\\right) (p\\text{\\_}O_{2})^{1/2}}\\\\ \\text{ }\\\\ p\\text{\\_}U & = \\frac{p\\text{\\_}UO}{\\text{exp}\\left(-\\frac{\\Delta G_{U/UO}}{RT}\\right) (p\\text{\\_}O_{2})^{1/2}}\\\\ \\text{ }\\\\ p\\text{\\_}UO_{3} & = p\\text{\\_}UO_{2} \\text{ } (p\\text{\\_}O_{2})^{1/2} \\text{ } \\text{exp}\\left(-\\frac{\\Delta G_{UO_{2}/UO_{3}}}{RT}\\right) \\\\ \\text{ }\\\\ p\\text{\\_}PuO & = \\frac{p\\text{\\_}PuO_{2}}{\\text{exp}\\left(-\\frac{\\Delta G_{PuO/PuO_{2}}}{RT}\\right) (p\\text{\\_}O_{2})^{1/2}}\\\\ \\text{ }\\\\ p\\text{\\_}Pu & = \\frac{p\\text{\\_}PuO}{\\text{exp}\\left(-\\frac{\\Delta G_{Pu/PuO_{2}}}{RT}\\right) (p\\text{\\_}O_{2})^{1/2}}\\\\ \\text{ }\\\\ p\\text{\\_}AmO_{2} & = r \\text{ } (p\\text{\\_}O_{2})^{n/2} \\text{ } \\text{exp}\\left(-\\frac{\\Delta G_{B}}{RT}\\right) \\\\ \\text{ }\\\\ p\\text{\\_}Am & = \\frac{p\\text{\\_}AmO_{2}}{\\text{exp}\\left(-\\frac{\\Delta G_{AmO_{2}/Am}}{RT}\\right) p\\text{\\_}O_{2}}\\\\ \\text{ }\\\\ p\\text{\\_}AmO & = p\\text{\\_}Am \\text{ } (p\\text{\\_}O_{2})^{1/2} \\text{ } \\text{exp}\\left(-\\frac{\\Delta G_{AmO/Am}}{RT}\\right) \\\\ \\text{ }\\\\ \\Delta G_{A} & = -\\frac{1}{2}\\int_0^m \\! RT \\text{ }\\text{ln}(p\\text{\\_}O_{2}) \\, \\mathrm{d}x + \\Delta G_{PuO_{2},\\text{vap}}\\\\ \\text{ }\\\\ \\Delta G_{B} & = -\\frac{1}{2}\\int_0^n \\! RT \\text{ }\\text{ln}(p\\text{\\_}O_{2}) \\, \\mathrm{d}x + \\Delta G_{AmO_{2},\\text{vap}}\\\\ \\text{ }\\\\ m & = \\frac{x}{q}\\\\ \\text{ }\\\\ n & = \\frac{x}{r}\\\\ \\text{ }\\\\ \\Delta G_{UO_{2},\\text{vap}} & = 567000 - 150T \\\\ \\text{ }\\\\ \\Delta G_{PuO_{2},\\text{vap}} & = 571000 - 150T \\\\ \\text{ }\\\\ \\Delta G_{UO/UO_{2}} & = -471000 + 71T \\\\ \\text{ }\\\\ \\Delta G_{U/UO} & = -528000 + 62T \\\\ \\text{ }\\\\ \\Delta G_{UO_{2}/UO_3} & = -404000 + 90T \\\\ \\text{ }\\\\ \\Delta G_{PuO/PuO_{2}} & = -352000 + 69T \\\\ \\text{ }\\\\ \\Delta G_{Pu/PuO} & = -498000 + 46T \\\\ \\text{ }\\\\ \\Delta G_{AmO_2,\\text{vap}} & = -R \\text{ } T \\text{ } ln(10^{(7.28-28260T)}) \\\\ \\text{ }\\\\ \\Delta G_{AmO/Am} & = (-137700 - 28.8T) - (263650 - 112.32T) \\\\ \\text{ }\\\\ \\Delta G_{AmO_2/Am} & = (-388300 + 30.7T) - (263650 - 112.32T) \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Oxygen partial pressure, p $var element = document.getElementById("moose-equation-35a7b563-74f3-4596-b83d-aff680273047");katex.render("_{O_2}", element, {displayMode:false,throwOnError:false});$ , can be calculated from the following equation: $var element = document.getElementById("moose-equation-513890a3-97a4-41c2-8d20-ea9b77659472");katex.render("O/M = 2 + \\left[ \\text{exp}\\left(-\\frac{22.8+84.5C_{Pu}}{R}\\right)\\text{exp}\\left(\\frac{105000}{RT}\\right)(p\\text{\\_}O_2)^{1/2} \\right] - \\left[ \\frac{1}{T_1 + T_2 + T_3 + T_4+T_5} \\right]", element, {displayMode:true,throwOnError:false});$ where the coefficients in the denominator of the third term are defined as $var element = document.getElementById("moose-equation-6ab9bd93-04fe-441e-a436-c70015b536e0");katex.render(" \\begin{aligned} T_1 & = \\left\\{\\text{exp}\\left(\\frac{44+55.8C_{Pu}}{R}\\right)\\text{exp}\\left(-\\frac{376000}{RT}\\right)(p\\text{\\_}O_2)^{-1/2}\\right\\}^{-5} \\\\ T_2 & = \\left\\{\\left(\\text{exp}\\left(\\frac{68.8+131.3C_{Pu}}{R}\\right)\\text{exp}\\left(\\frac{-515000}{RT}\\right)\\right)^{1/2}p\\text{\\_}O_2^{-1/4}\\right\\}^{-5}\\\\ T_3 & = \\left\\{ \\left(2\\text{ exp} \\left(\\frac{153.5+96.5C_{Pu}+331{C_{Pu}}^2}{R}\\right)\\text{exp}\\left( \\frac{-891000}{RT}\\right)\\right)^{1/3} p\\text{\\_}O_{2}^{-1/3} \\right\\}^{-5}\\\\ T_4 & = \\left\\{ \\left(\\text{exp} \\left(\\frac{111.2+20C_{Pu}}{R}\\right)\\text{exp}\\left( \\frac{-445000}{RT}\\right)\\right)^{1/2} p\\text{\\_}O_{2}^{-1/4} \\right\\}^{-5} \\\\ T_5 & = \\left(\\frac{C_{Pu}}{2}\\right)^{-5} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ and where $var element = document.getElementById("moose-equation-28187205-10ae-4f68-bb53-690ba67a37d7");katex.render(" \\begin{aligned} \\text{for MOX} \\text{, } C_{Pu} & = \\frac{Pu}{U+Pu}\\\\ \\text{for Am-MOX} \\text{, } C_{Pu} & = \\frac{Pu}{U+Pu+Am}+2.5\\frac{Am}{U+Pu+Am} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Again, the pressure in the velocity equation is the sum of all these individual vapor pressures: $var element = document.getElementById("moose-equation-a882298f-b345-4927-99f8-91aff541598a");katex.render("P = p\\text{\\_}UO_{2} + p\\text{\\_}PuO_{2} + p\\text{\\_}UO + p\\text{\\_}U + p\\text{\\_}UO_{3} + p\\text{\\_}PuO + p\\text{\\_}Pu + p\\text{\\_}AmO_{2} + p\\text{\\_}Am + p\\text{\\_}AmO", element, {displayMode:true,throwOnError:false});$

The vapor pressure $var element = document.getElementById("moose-equation-012dff31-00cf-4624-a3f7-45d99c1b885a");katex.render("P", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-5b162268-45b5-4cec-89bd-4364860dcefe");katex.render("\\frac{dP}{dT}", element, {displayMode:false,throwOnError:false});$ come from the class MOXVaporPressure.

construction

This class is still under development!

See bison/test/mox_pore_velocity/ for examples of how this class should be used.

Input Parameters

temperatureCoupled Temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled Temperature

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.
scale_factor1scale the velocity to account for uncertainty
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:scale the velocity to account for uncertainty

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

Input Files

(test/tests/mox_pore_velocity/MOXPoreVelocityVaporPressure.i)

References

Yoshihisa Ikusawa, Takayuki Ozawa, Shun Hirooka, Koji Maeda, Masato Kato, and Seiichiro Maeda. Development and verification of the thermal behavior analysis code for ma containing mox fuels. In International Conference on Nuclear Engineering. Prague, Czech Republic, July 7–11 2014.

@inproceedings{ikusawa_icone_2014,
    Author = "Ikusawa, Yoshihisa and Ozawa, Takayuki and Hirooka, Shun and Maeda, Koji and Kato, Masato and Maeda, Seiichiro",
    Title = "Development and Verification of the Thermal Behavior Analysis Code for MA Containing MOX Fuels",
    Year = "2014",
    Booktitle = "International Conference on Nuclear Engineering",
    Month = "July 7--11",
    Address = "Prague, Czech Republic"
}

(test/tests/mox_pore_velocity/MOXPoreVelocityVaporPressure.i)

# This input files uses the pore difusion kernels

[UserObjects]
  [pin_geometry]
    type = Layered1DFuelPinGeometry
    include_clad = false
    mesh_generator = layered1D_mesh
  []
[]

[Mesh]
  coord_type = RZ
  [layered1D_mesh]
    type = Layered1DMeshGenerator
    fuel_height = 0.1
    pellet_outer_radius = 0.0041
    include_clad = false
    pellet_bottom_coor = 0.0
    pellet_mesh_density = customize
    nx_p = 200
    elem_type = EDGE2
    slices_per_block = 1
    include_plenum = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 1400.0
  []
  [pore]
    initial_condition = 0.12
    scaling = 1e14
  []
[]

[AuxVariables]
  [pore_speed_aux]
    order = constant
    family = monomial
  []
  [fission_rate_aux_variable_mox]
    order = first
    family = lagrange
  []
[]

[Functions]
  [power_history1]
    type = PiecewiseLinear
    x = '0 10000'
    y = '0 50000'
  []
[]

[Kernels]

  [heat] # gradient term in heat conduction equation
    type = HeatConduction
    variable = temperature
  []
  [heat_ie] # time term in heat conduction equation
    type = HeatConductionTimeDerivative
    variable = temperature
  []
  [heat_source] # source term in heat conduction equation
    type = NeutronHeatSource
    variable = temperature
    block = fuel # fission rate applied to the fuel (block 2) only
    fission_rate = fission_rate_aux_variable_mox
  []

  [pore_diffusion]
    type = MOXPoreDiffusion
    variable = pore
    debug = 0
    # nu = 3.25e-8 #seems to be THE value to use... result is super sensitive to this number
    # nu = 10e-10
    nu = 1e-12
    heating_function = power_history1
    v_upper = 1e-12
    v_lower = 1e-20
    # v_upper = 1
    # v_lower = 1
  []

  [pore_continuity]
    type = MOXPoreContinuity
    variable = pore
    temperature = temperature
    debug = 0
    alpha = 0.25
    beta = 1
    heating_function = power_history1
  []
  [poretimederivative]
    type = CoefTimeDerivative
    variable = pore
    Coefficient = 1
  []
[]

[AuxKernels]
  [pore_speed_aux]
    type = MaterialRealAux
    variable = pore_speed_aux
    property = pore_velocity
    block = fuel
    execute_on = 'initial timestep_end'
  []
  [fission_rate_aux_kernel_mox]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate_aux_variable_mox
    block = fuel
    porosity = pore
    initial_porosity = 0.12
    rod_ave_lin_pow = power_history1
    pellet_diameter = 0.0082
    pellet_inner_diameter = 0
    energy_per_fission = 3.2e-11
    execute_on = 'initial timestep_end'
  []
[]

[BCs]
  [temp_outside] # pin pellets and clad along axis of symmetry (y)
    type = DirichletBC
    variable = temperature
    boundary = 10
    value = 1400
  []
[]

[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    block = fuel
    temperature = temperature
    porosity = pore
    porosity_limit = 0.9
  []
  [density_block]
    type = GenericConstantMaterial
    block = fuel
    prop_names = density
    prop_values = 10431.0
  []
  [pore_velocity]
    type = MOXPoreVelocityVaporPressure
    block = fuel
    temperature = temperature
    scale_factor = 1e0
    # limit = 1e-3
    # scale_factor = 0.05 # go back to this if necessary
    # scale_factor = 0.1
    # oxygen_partial_pressure = PO2
  []
  [oxygen_partial_pressure_integral]
    type = MOXOxygenPartialPressure
    block = fuel
    temperature = temperature
    o2m_deviation = 0.02
    po2_initial = 0.01
    outputs = exodus
    # type = GenericConstantMaterial
    # block = fuel
    # prop_names = PO2
    # prop_values = 1.0
  []
  [Sum]
    type = MOXVaporPressure
    temperature = temperature
    block = fuel
    evalerror_behavior = error
    outputs = exodus
  []
[]

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temperature
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  snesmf_reuse_base = false

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package' # -mat_superlu_dist_fact'
  petsc_options_value = 'lu superlu_dist' # SamePattern_SameRowPerm'

  line_search = 'none'

  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 25
  nl_rel_tol = 1e-5
  nl_abs_tol = 1e-8 #1e-10
  n_startup_steps = 1
  end_time = 1.5e5
  num_steps = 2
  dtmax = 1000
  dtmin = 1

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2e2
    optimal_iterations = 8
    iteration_window = 2
    linear_iteration_ratio = 100
    growth_factor = 2
    cutback_factor = .5
    force_step_every_function_point = true
    timestep_limiting_function = power_history1
  []
[]

[Postprocessors]
  [_dt] # time step
    type = TimestepSize
  []
  [z_nonlinear_its] # number of nonlinear iterations at each timestep
    type = NumNonlinearIterations
  []
  [power_input]
    type = FunctionValuePostprocessor
    function = power_history1
    scale_factor = 0.1 # rod height
  []

  [rod_total_power_mox]
    type = LayeredElementIntegralPowerPostprocessor
    variable = temperature
    block = fuel
    fission_rate = fission_rate_aux_variable_mox
    fuel_pin_geometry = pin_geometry
  []

  [ave_fuel_temp]
    type = ElementAverageValue
    block = fuel
    variable = temperature
  []
  [max_fuel_temp]
    type = NodalExtremeValue
    block = fuel
    value_type = max
    variable = temperature
  []

  [ave_pore]
    type = ElementAverageValue
    block = fuel
    variable = pore
  []
  [max_pore]
    type = NodalExtremeValue
    block = fuel
    value_type = max
    variable = pore
  []
  [min_pore]
    type = NodalExtremeValue
    block = fuel
    value_type = min
    variable = pore
  []
  [max_pore_speed]
    type = ElementExtremeValue
    block = fuel
    value_type = max
    variable = pore_speed_aux
  []
  [center_PO2]
    type = ElementalVariableValue
    elementid = 0
    variable = PO2
  []
[]

# The MOX capabilities are under active development and the blocks below are useful for
# development and debugging by providing the profiles of the desired quantities.
# They are commented out for the tests, as it would unnecessarily increase computational costs
# and memory requirements.
# [VectorPostprocessors]
#   [line_value_vector_postprocessor_pore]
#     type = LineValueSampler
#     variable = pore
#     start_point = '0.0 0.05 0'
#     end_point = '0.0041 0.05 0'
#     num_points = 100
#     sort_by = x
#     execute_on = linear
#     outputs = stuff_v_rad
#     control_tags = a
#   []
#   [line_value_vector_postprocessor_pore_speed]
#     type = LineValueSampler
#     variable = pore_speed_aux
#     start_point = '0.0 0.05 0'
#     end_point = '0.0041 0.05 0'
#     num_points = 100
#     sort_by = x
#     execute_on = linear
#     outputs = stuff_v_rad
#   []
#   [line_value_vector_postprocessor_temperature]
#     type = LineValueSampler
#     variable = temperature
#     start_point = '0.0 0.05 0'
#     end_point = '0.0041 0.05 0'
#     num_points = 100
#     sort_by = x
#     execute_on = linear
#     outputs = stuff_v_rad
#   []
# []

[Outputs]
  exodus = true
  csv = false
  color = false
  [console]
    type = Console
    max_rows = 25
    all_variable_norms = true
  []
  # [stuff_v_rad]
  #   type = CSV
  #   execute_on = 'FINAL'
  # []
[]

[Debug]
  show_var_residual_norms = true
[]

Description
Input Parameters
Input Files
References