# Zirconium Diffusion

Calculates the amount of zirconium that is transported across the mesh

## Description

After all the diffusivities are set, ZirconiumDiffusion is used to transport the zirconium across the mesh.

A pseudo-binary model for U-19Pu-10Zr fuel was implemented into Bison with two essential numerical modifications: addition of an artificial diffusion term in the 2-phase region to stabilize the algorithm, and smoothing of the model coefficients near phase diagram curves. We solve the coupled system of equations, (1) for an evolving temperature distribution and Zr atom fraction distribution . This is a departure from the model considered in Kim et al. (2004) and Kim et al. (2006) which used a fixed temperature profile. We use a variable heat source that will depend on the axial position in the fuel rod and the local Zr fraction (actually the local fraction of actinides, which is the complement of the Zr fraction). Additionally a thermal conductivity model that accounts for constituent migration as well as porosity effects was used (Galloway model described in ThermalUPuZr).

In a single-phase region of the phase diagram (here denotes the arbitrary phase) we have (2) where is the diffusivity and the heat of transport of Zr in the phase . In the 2-phase region bounded by the solubility limit curves and , the Soret term coefficient is (3) where is the phase fraction of according to the lever rule. In a 2-phase region one ought to take , however this would result in Eq. 1 being purely advective with a transport velocity proportional to the temperature gradient. Mathematically this leads to jump discontinuities in the Zr atom fraction profile at domain boundaries and boundaries between single and 2-phase regions. Moreover, it is well-known that standard centered finite difference or Galerkin finite element schemes are unstable for pure advection, leading to spurious oscillations. Thus to stabilize our Galerkin finite element implementation we add some artificial diffusion and take, (4) where and are dimensionless numerical parameters taken as small as possible while maintaining stability. In future work we plan to replace this simple stabilization approach with the SUPG (streamline upwind Petrov-Galerkin) finite element method, which can be also implemented within Bison. The and coefficients in the 2-phase region are defined in an analogous way.

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As defined above, and do not vary smoothly (or even continuously) with and when crossing phase diagram boundaries, and consequently the discretized system will be much more difficult to solve than it would otherwise be. We have ameliorated this effect by smoothing the relative phase contributions at each location in the phase diagram, discussed in detail in the documentation for PhaseUPuZr.

## Example Input Syntax


[./ZrDiffusion]
type = ZirconiumDiffusion
variable = zirconium
temp = T
[../]
(test/tests/zirconium_diffusion/test.i)

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

### Required Parameters

• Zr_diffusivity_dTZr_diffusivity_dTZr_diffusivity_dT

Default:Zr_diffusivity_dT

C++ Type:MaterialPropertyName

Description:Zr_diffusivity_dT

• Zr_diffusivityZr_diffusivityZr_diffusivity

Default:Zr_diffusivity

C++ Type:MaterialPropertyName

Description:Zr_diffusivity

• Zr_Soret_dZrZr_Soret_dZrZr_Soret_dZr

Default:Zr_Soret_dZr

C++ Type:MaterialPropertyName

Description:Zr_Soret_dZr

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

• Zr_SoretZr_SoretZr_Soret

Default:Zr_Soret

C++ Type:MaterialPropertyName

Description:Zr_Soret

• Zr_diffusivity_dZrZr_diffusivity_dZrZr_diffusivity_dZr

Default:Zr_diffusivity_dZr

C++ Type:MaterialPropertyName

Description:Zr_diffusivity_dZr

• Zr_Soret_dTZr_Soret_dTZr_Soret_dT

Default:Zr_Soret_dT

C++ Type:MaterialPropertyName

Description:Zr_Soret_dT

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• 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

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.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• vector_tagsnontimeThe tag for the vectors this Kernel should fill

Default:nontime

C++ Type:MultiMooseEnum

Description:The tag for the vectors this Kernel should fill

• extra_vector_tagsThe extra tags for the vectors this Kernel should fill

C++ Type:std::vector

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

Description:The tag for the matrices this Kernel should fill

• extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

C++ Type:std::vector

Description:The extra tags for the matrices this Kernel should fill