# Thermal Properties for Silicide Fuel

Computes the specific heat and thermal conductivity for different phases of uranium silicide fuel

## Description

The ThermalSilicideFuel model computes the specific heat and thermal conductivity for different phases of uranium silicide fuel including pure silicon, pure uranium metal, USi, USi, and USi.

The ThermalSilicideFuel material model contains four different options to model thermal conductivity of USi: WHITE, SHIMIZU, ZHANG, and ARGONNE and three different options to model specific heat of USi: WHITE, IAEA, HANDBOOK.

## Thermal Conductivity Models

### Default WHITE Model

The default model (WHITE) for thermal conductivity is given by equation 4 in White et al. (2015): (1) where is temperature in K. This expression is valid for temperatures up to 1773 K.

### SHIMIZU Model

An alternative model (SHIMIZU) is available for use in ThermalSilicide by using experimental data from figure 4 of Shimizu (1965). The conservative expression for thermal conductivity k (W/m-K) of arc cast USi pellets is: (2) where is temperature in K. This expression is valid for temperatures from room temperature to 1473.15 K. This expression may underestimate the true thermal conductivity of USi.

### ZHANG Model

The third thermal conductivity option known as the ZHANG model is more sophisticated because it is able to determine the thermal conductivity of pure uranium metal, pure silicon, USi, USi, and USi. By utilizing details from Ho et al. (1978), Tsiovkin et al. (2010), and Glassbrenner and Slack (1964), Zhang arrived at an equation of the form: (3) where is the silicon concentration (given as mole fraction in the fuel). For example, for USi =0.4. and are the conductivities of U and Si, respectively. and are fitting parameters. The first step is to find the values of and . Zhang found that the exponential decay function can be used to reproduce these values well: (4) where T is the temperature in K and, m, m, m, T, T, and T are parameters unique to U or Si. The values of these parameters are summarized in Table 1.

Table 1: Parameters used to fit the intrinsic thermal resistivity of U and Si

Parameters
U0.004480.00890.032670.0500.769171555.4716
Si0.0830329.1523.8884164587.48315252.19318

Next, Zhang used the data from White et al.'s references for USi (White et al., 2015) and USi (White et al., 2015) to fit the parameters and . The equation for these parameters are 5th order polynomials of temperature given by: (5) (6)

### ARGONNE Model

The final thermal conductivity available is known as the ARGONNE model (Miao et al., 2017). This model uses the U3Si2TricubicInterpolationUserObject to calculate temperature, temperature gradient, and fission density (burnup) dependent degradation factors applied to the intrinsic thermal conductivity calculated from the unirradiated thermal conductivity computed by the WHITE model. The range of applicability of the model is temperatures from 390 K to 1190 K, temperature gradients from 0 to 160 K/mm, and fission densities of 0 to 2.5755 10 fissions/cm.

The intrinsic thermal conductivity is calculated as follows:

(7)

where is thermal conductivity calculated by the WHITE model, is the Kapitza resistance (2.5e-8 m-K/W), and is the grain size (taken as 35 m).

The modified Kapitza resistance is determined based upon the amount of grain boundary coverage: (8) where is the grain boundary coverage, and is computed by U3Si2TricubicInterpolationUserObject. The modified Kapitza resistance is then determined by: (9)

The first degradation factor known as the intergranular factor is then computed by: (10) where is the grain size specified in the GRASS-SST rate theory calculation. This value is taken as 5.0 m. The second degradation factor known as the intragranular factor is calculated by: (11) where is the intragranular gaseous swelling strain due to intragranular bubbles calculated by U3Si2TricubicInterpolationUserObject. Finally the thermal conductivity is then given as: (12)

## Specific Heat Capacity

### Default WHITE model

The default correlation for the specific heat of C (J/kg-K) of USi is equation 2 from White et al. (2015): (13) where is temperature in K.

### IAEA model

An alternative correlation that can be used is taken from Matos and Snelgrove (1992): (14) where is temperature in K and C is the specific heat capacity is in J/kg-K. The reference does not state the validity range of this expression.

### HANDBOOK model

A third option is available from the USi handbook White (2017). (15) where is temperature in K and C is the specific heat capacity is in J/kg-K.

## Example Input Syntax

[./fuel_thermalU3Si2]
type = ThermalSilicideFuel
block = 1
temp = T
silicon_mole_fraction = 1.5
specific_heat_model = WHITE # This is the default
thermal_conductivity_model = ZHANG
[../]
(test/tests/thermalSilicideFuel/thermalU3Si2_zhang_error.i)

## Input Parameters

• originOrigin of cylinder axis of rotation for 2D and 3D Cartesian models when using the Argonne thermal conductivity model.

C++ Type:libMesh::VectorValue

Description:Origin of cylinder axis of rotation for 2D and 3D Cartesian models when using the Argonne thermal conductivity model.

• silicon_mole_fraction0.4The mole fraction of silicon in the fuel. For example for U3Si2 this parameter would be 0.4

Default:0.4

C++ Type:double

Description:The mole fraction of silicon in the fuel. For example for U3Si2 this parameter would be 0.4

• thermal_conductivity_scale_factor1The scaling factor on the thermal conductivity.

Default:1

C++ Type:double

Description:The scaling factor on the thermal conductivity.

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

• specific_heat_conductivity_scale_factor1The scaling factor on the specific heat.

Default:1

C++ Type:double

Description:The scaling factor on the specific heat.

• specific_heat_modelWHITEThe chosen model to use for specific heat: WHITE or IAEA

Default:WHITE

C++ Type:MooseEnum

Description:The chosen model to use for specific heat: WHITE or IAEA

• thermal_conductivity_modelWHITEThe chosen model to use for thermal conductivity: WHITE, SHIMIZU, ZHANG, or ARGONNE

Default:WHITE

C++ Type:MooseEnum

Description:The chosen model to use for thermal conductivity: WHITE, SHIMIZU, ZHANG, or ARGONNE

• axis_vectorVector defining direction of cylindrical axis (3D Cartesian models) when using the Argonne thermal conductivity model.

C++ Type:libMesh::VectorValue

Description:Vector defining direction of cylindrical axis (3D Cartesian models) when using the Argonne thermal conductivity model.

• 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

• thermal_conductivity_degradationName of the UserObject that is used to calculate the intergranular and intragranular degradation of thermal conductivity. Must be supplied when using the Argonne thermal conductivity model.

C++ Type:UserObjectName

Description:Name of the UserObject that is used to calculate the intergranular and intragranular degradation of thermal conductivity. Must be supplied when using the Argonne thermal conductivity model.

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Description:The list of boundary IDs from the mesh where this boundary condition applies

• burnup_functionBurnup function

C++ Type:FunctionName

Description:Burnup function

• burnupCoupled Burnup

C++ Type:std::vector

Description:Coupled Burnup

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• 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

• 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

• 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 computeSubdomainProperties() 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

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 computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

## References

1. C.J. Glassbrenner and G.A. Slack. Thermal conductivity of silicon and germanium from 3k to the melting point. Journal of Physical Review, 134:A1059, 1964.[BibTeX]
2. C.Y. Ho, M.W. Ackerman, K.Y. Wu, S.G. Oh, and T.N. Havill. Thermal conductivity of ten selected binary alloy systems. Journal of Physical and Chemical Reference Data, 7:959, 1978.[BibTeX]
3. J. E. Matos and J. L. Snelgrove. Research reactor core conversion guidebook-Vol 4: Fuels (Appendices I-K). Technical Report IAEA-TECDOC-643, IAEA, 1992.[BibTeX]
4. Y. Miao, K. A. Gamble, D. Andersson, B. Ye, Z. Mei, G. Hofman, and A. M. Yacout. Gaseous swelling of U$_3$Si$_2$ during steady-state lwr operation: a rate theory investigation. Nuclear Engineering and Design, 322:336â€“344, 2017.[BibTeX]
5. H. Shimizu. The properties and irradiation behavior of U$_3$Si$_2$. Technical Report NAA-SR-10621, Atomics International, 1965.[BibTeX]
6. Yu. Tsiovkin, V.V. Dremov, E.S. Koneva, A.A. Povzner, A.N. Filanovich, and A.N. Petrova. Theory of the residual electrical resistivity of binary actinide alloys. Journal of Physics of the Solid State, 52:1–5, 2010.[BibTeX]
7. J. T. White. Issue draft U$_3$Si$_2$ fuel property handbook. Technical Report LA-UR-17-20609, Los Alamos National Laboratory, 2017.[BibTeX]
8. J. T. White, A. T. Nelson, D. D. Byler, D. J. Safarik, J.T. Dunwoody, and K. J. McClellan. Thermophysical properties of U$_3$Si$_5$ to 1773K. Journal of Nuclear Materials, 456:442â€“448, 2015.[BibTeX]
9. J. T. White, A. T. Nelson, J. T. Dunwoody, D. D. Byler, D. J. Safarik, and K. J. McClellan. Thermophysical properties of U$_3$Si$_2$ to 1773K. Journal of Nuclear Materials, 464:275â€“280, 2015.[BibTeX]