# Uvar element = document.getElementById("moose-equation-6197a9a3-91d8-4bfa-902b-e8c85187e54d");katex.render("_3", element, {displayMode:false,throwOnError:false});Sivar element = document.getElementById("moose-equation-299c6790-da15-46e2-bf6d-aaa360752620");katex.render("_2", element, {displayMode:false,throwOnError:false}); Fission Gas Behavior

Calculates fission gas release and swelling in U3Si2 through a physically-based model.

## Description

The processes induced by the generation of the fission gases xenon and krypton in nuclear fuel have a strong impact on the thermo-mechanical performance of the fuel rods. On the one hand, the fission gases tend to precipitate into bubbles resulting in fuel swelling, which promotes pellet-cladding gap closure and the ensuing pellet-cladding mechanical interaction (PCMI). On the other hand, fission gas release (FGR) to the fuel rod free volume causes pressure build-up and thermal conductivity degradation of the rod filling gas.

The fundamental physical processes, which control the kinetics of fission gas swelling and release in irradiated fuel, may be summarized as follows. Fission gas atoms generated in the fuel grains diffuse towards the grain boundaries through repeated trapping in and irradiation-induced resolution from nanometre-size intra-granular gas bubbles. Although a part of the gas atoms that reach the grain boundaries is dissolved back to the grain interior by irradiation, the majority of the gas diffuses into grain-face gas bubbles, giving rise to grain-face swelling. Bubble growth brings about bubble coalescence and inter-connection, eventually leading to the formation of a tunnel network through which a fraction of the gas is released to the fuel rod free volume.

In Bison, fission gas behavior is computed for each integration point in the fuel finite element mesh. The gas produced at each integration point is computed by a numerical time integration of the gas production rate, given as the product of the fission rate and fractional yield of gas atoms per fission.

## Physics-Based Model

The U3Si2FissionGas model handles fission gas swelling and release in USi under power reactor conditions. The model calculates the coupled fission gas swelling and release concurrently and is physically based. This model relies on the current understanding of microstructure and fission gas behavior in USi, including the recent findings from lower-length scale modeling and the available experimental data. Based on the experimental evidence from Shimizu (1965), we assume USi remains crystalline at power reactor temperatures. We also assume both intra-granular and grain-boundary gas bubbles develop, as in UO. Accordingly, the same basic structure of the Bison UO ( Sifgrs ) model is applied in this model, with a two-stage description of intra-granular and inter-granular processes.

In order to mitigate the scarcity of experimental data, new physically based descriptions of specific processes can be informed with USi material parameters which have been extracted from lower-length scale modeling. Furthermore, the physical interpretation of some relevant processes differs from the interpretation in the UO model to better conform to the current understanding of fission gas behavior in USi. These processes include

1) modeling of intra-granular bubble nucleation and re-solution based on the so-called homogeneous mechanisms and

2) an intra-granular bubble growth model considering absorption of vacancies by the bubbles and is based on an adaptation of the Speight-Beere model. Both capabilities were developed specifically for the U3Si2FissionGas model.

Material parameters are taken from lower-length scale calculations for USi, where available. Parameters for which specific USi values are not yet available are given acceptable values based on data for metals, theoretical considerations or the best fitting of model results to experimental data. The U3Si2FissionGas model is an initial engineering-scale fission gas model for USi that incorporates state-of-the-art understanding and lower-length scale modeling data and will be progressively updated as new data become available.

### Intra-granular gas behavior

The model accounts for nucleation of bubbles, re-solution of gas from bubbles to the matrix, and trapping of gas from the matrix into the bubbles. Fission gas transport from within the fuel grains to the grain faces is computed by the numerical solution of the diffusion equation in one-dimensional spherical geometry.

Nucleation and re-solution may occur by different mechanisms, i.e., heterogeneous and homogeneous (Olander and Wongsawaeng, 2006). Heterogeneous nucleation and re-solution refer to the creation of new bubbles nuclei as a direct consequence of the interaction of fission fragments with the lattice and the bubbles destruction occurring en-bloc by passing fission fragments, respectively. The homogeneous mechanisms accounts for the nucleation of bubbles by diffusion-driven interactions of dissolved gas atoms and re-solution occurring gradually by ejection of individual atoms. The dominant mechanisms depend upon the nature of the interactions between fission fragments and lattice (electronic or phononic). Based on Matthews et al. (2016), we assume the homogeneous mechanisms to dominate in USi . The equations for the evolution of the intra-granular gas bubble number density and gas atom concentrations are: (1)

(2)

(3) where (m) is the number density of intra-granular bubbles, is the number of gas atoms per bubble, and (m) are the intra-granular gas concentration in the matrix and in the bubbles, respectively, (s) the time, (ms) the single-atom gas diffusion coefficient, (m) the radial coordinate in the spherical grain, (ms) the gas generation rate, (s) the trapping rate, (s) the re-solution rate. The coefficient of 2 for the nucleation rate (atm ) in Eq. 2 and Eq. 3 represents the fact that bubbles are nucleated as dimers.

The nucleation rate is calculated as: (4) where (m) is the radius of a single fission gas atom and is the nucleation factor (dimensionless), equal to (e.g., Veshchunov (2000)). This system of equations is solved with an advanced version of the PolyPole algorithm (Pizzocri et al., 2016).

Intra-granular bubble growth is treated using a modified Speight and Beere (1975) model. The mechanical equilibrium of an intra-granular bubble, assumed to be spherical, is governed by the Young-Laplace equation (5) where (Pa) is the equilibrium pressure, (J ) is the USi/gas surface energy and (Pa) is the hydrostatic stress. In general, the bubbles are in a non-equilibrium state and tend to the equilibrium condition absorbing or emitting vacancies. The vacancy absorption/emission rate can be calculated starting from the approach in Speight and Beere (1975) as (6) being (dimensionless) the number of vacancies per intra-granular bubble, ( ) the intra-granular vacancy diffusion coefficient, (m) is the radius of the equivalent Wigner-Seitz cell surrounding a bubble and influenced by the vacancy absorption/emission, (J K) is the Boltzmann constant, (K) is the local temperature, and (dimensionless) is an dimensionless factor, which is calculated as (Pizzocri et al., 2016) (7) where is the ratio between the bubble and the cell radii. The approach here presented differs from the original proposed in Speight and Beere (1975) because involves a 3D representation of the absorption/emission phenomena, rather than a 2D description, which better suits the absorption/emission of vacancy at grain boundary. For the diffusion coefficients, and , and the re-solution rate, , we use values for USi from the atomistic work of Andersson (2017) and Matthews et al. (2016), respectively.

The pressure of the bubble is expressed as, considering a van der Waals gas, (8) where () is the vacancy volume, (dimensionless) is the ratio between and , and () is the van der Waals atomic volume for xenon.

The volume of intra-granular bubbles is calculated as (9) such that the radius is evaluated considering spherical bubbles (10) The intra-granular swelling is calculated as the average volume of the bubble with respect to the bubble number density. The formation of the high burnup structure is not currently represented in the model, due to the lack of data on this phenomenon in USi.

## Grain-face gas behavior

The numerical solution of Eq. 2 allows estimating the arrival rate of gas at the grain faces, providing the source term for the grain-face gas behavior module. This computes both the fission gas swelling and release through a direct description of the grain-face bubble development, including bubble growth and coalescence (which are reflected in fuel swelling), and the eventual inter-connection (leading to thermal fission gas release).

These conceptual steps and the related equations are identical to those applied in UO ( Sifgrs ) model. However, the material parameters are specific to the U3Si2FissionGas model. Nevertheless, in this model an initial concentration of grain-face bubbles equal to is employed. The value is one order of magnitude lower than the one employed in UO, reflecting the lower bubble density noticeable in uranium silicide from the available experimental data Shimizu (1965).

The fractional volume grain-face fission gas swelling is given by (11) where is the number density of grain-face bubbles per unit surface, the grain radius, the bubble semi-dihedral angle, the geometric factor relating the volume of a lenticular-shape bubble to that of a sphere, which is , and the bubble radius of curvature. The factor 1/2 is introduced in Eq. 11 because a grain-face bubble is shared by two neighboring grains.

Bubble growth is calculated with the model from Speight and Beere (1975) to describe the growth (or shrinkage) of grain-face bubbles as proceeding by absorption (or emission) of vacancies in grain boundaries, induced by the difference between the pressure of the gas in the bubble, (Pa), and the mechanical equilibrium pressure, (Pa).

The approach is conceptually analogous to that applied in for the growth and shrinkage of intra-granular bubbles and is described in Sifgrs. The diffusion coefficient of vacancies at grain boundaries is estimated multiplying the intra-granular one by a factor of .

This approach computes the bubble growth rate from the rate of inflow of gas atoms and the rate of absorption (emission) of vacancies at the bubble. The combined effects of gas atom inflow and vacancy absorption (emission) are interactive, since the addition of fission gas atoms gives rise to a change in the bubble pressure, which affects the propensity of the bubble to absorb (or emit) vacancies. Given the volume, , of a lenticular bubble of circular projection, the bubble radius of curvature is calculated as (12) The process of grain-face bubble coalescence, which leads to a progressive decrease of the bubble number density throughout irradiation, is described with a model based on Pastore et al. (2013) and White (2004). According to this model, the rate of loss of bubbles by coalescence is given by (13) where and represent the number density and projected area of grain-face bubbles, respectively.

The release of fission gas to the fuel rod free volume after the inter-connection of grain-face bubbles and the consequent formation of pathways for gas venting to the fuel exterior (thermal release) is based on the principle of grain face saturation. More specifically, a saturation coverage concept is adopted that assumes once the fractional coverage, , attains a saturation value, , the bubble number density and projected area obey the saturation coverage condition (14) where is the bubble number density and is the bubble projected area on the grain face. In absence of experimental data on the maximum grain-face bubble coverage in USi, the theoretical value is used.

Eq. 14 implies that, after attainment of the saturation coverage, a fraction of the gas on the grain faces is released to the fuel exterior and thereby compensates for continuing bubble growth.

## Example Input Syntax


[./Fission_Gas_Release]
type = U3Si2FissionGas
block = 0
temp = T
fission_rate = fission_rate
[../]
(test/tests/tensor_mechanics/u3si2_eigenstrains/u3si2_vswelling/VSwellingU3Si2_mech_test_tm.i)

## Input Parameters

• trap_param_option0Select trapping parameter

Default:0

C++ Type:short

Description:Select trapping parameter

• temperature_scalef1scaling factor for temperature

Default:1

C++ Type:double

Description:scaling factor for temperature

• gbdiffcoeff_scalef1scaling factor for grain boundary diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for grain boundary diffusion coefficient

• eff_diff_coeff_option0Select effective diffusion coefficient

Default:0

C++ Type:short

Description:Select effective diffusion coefficient

• diff_coeff_option0Select diffusion coefficient

Default:0

C++ Type:short

Description:Select diffusion coefficient

• burnup_functionBurnup function

C++ Type:FunctionName

Description:Burnup function

• ig_diff_algorithm0Select intra-granular diffusion algorithm

Default:0

C++ Type:short

Description:Select intra-granular diffusion algorithm

Default:1

C++ Type:double

• resolutionp_scalef1scaling factor for resolution parameter

Default:1

C++ Type:double

Description:scaling factor for resolution parameter

• hydrostatic_stress_const0constant hydrostatic stress (Pa)

Default:0

C++ Type:double

Description:constant hydrostatic stress (Pa)

Default:2.5e-05

C++ Type:double

• ig_bubble_model2Select bubble evolution model

Default:2

C++ Type:short

Description:Select bubble evolution model

• effdiffcoeff_scalef1scaling factor for intragranular effective diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for intragranular effective diffusion coefficient

• 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

• nuclerate_scalef1scaling factor for nucleation rate

Default:1

C++ Type:double

Description:scaling factor for nucleation rate

• fract_yield0.3fractional yield of fission gas atoms (Xe + Kr) (/)

Default:0.3

C++ Type:double

Description:fractional yield of fission gas atoms (Xe + Kr) (/)

• axial_power_profileaxial power peaking function.

C++ Type:FunctionName

Description:axial power peaking function.

• hydrostatic_stressCoupled Hydrostatic Stress

C++ Type:std::vector

Description:Coupled Hydrostatic Stress

• rod_ave_lin_powlinear power function.

C++ Type:FunctionName

Description:linear power function.

• res_param_option0Select resolution parameter

Default:0

C++ Type:short

Description:Select resolution parameter

• ig_fully_coupled1Solving diffusion coupled to bubble evolution

Default:1

C++ Type:short

Description:Solving diffusion coupled to bubble evolution

• igdiffcoeff_scalef1scaling factor for intragranular diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for intragranular diffusion coefficient

• 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

• fission_rateCoupled Fission Rate

C++ Type:std::vector

Description:Coupled Fission Rate

• initial_porosity0initial fuel porosity (/)

Default:0

C++ Type:double

Description:initial fuel porosity (/)

• saturation_coverage0.785398saturation coverage (/)

Default:0.785398

C++ Type:double

Description:saturation coverage (/)

• fission_gas_concCoupled Fission Gas Concentration

C++ Type:std::vector

Description:Coupled Fission Gas Concentration

• burnupCoupled Burnup

C++ Type:std::vector

Description:Coupled Burnup

• nucleation_option0Select intragranular bubble nucleation model

Default:0

C++ Type:short

Description:Select intragranular bubble nucleation model

• trappingp_scalef1scaling factor for trapping parameter

Default:1

C++ Type:double

Description:scaling factor for trapping parameter

• 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

### 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

• testing_outputFalseProvides an analytic reference for the value of the intra-granular fission gas release

Default:False

C++ Type:bool

Description:Provides an analytic reference for the value of the intra-granular fission gas release

• skip_bdr_modelFalseSkips the grain-boundary model

Default:False

C++ Type:bool

Description:Skips the grain-boundary model

• 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. D.A. Andersson. Density functional theory calculations of the defect and fission gas properties in U-Si fuels. Technical Report, Los Alamos National Laboratory, 2017.[BibTeX]
2. C. Matthews, D.A. Andersson, and C. Unal. Radiation re-solution calculation in uranium-silicide fuels. Technical Report, Los Alamos National Laboratory, 2016.[BibTeX]
3. D. R. Olander and D. Wongsawaeng. Re-solution of fission gas â€“ A review: Part I. Intragranular bubbles. Journal of Nuclear Materials, 354:94â€“109, 2006.[BibTeX]
4. G. Pastore, L. Luzzi, V. Di Marcello, and P. Van Uffelen. Physics-based modelling of fission gas swelling and release in UO$_2$ applied to integral fuel rod analysis. Nuclear Engineering and Design, 256:75â€“86, 2013.[BibTeX]
5. D. Pizzocri, F. Cappia, V. V. Rondinella, and P. Van Uffelen. Preliminary model for the pore growth in the HBS. Technical Report, JRC103064, European Commission, Directorate for Nuclear Safety and Security, JRC-Karlsruhe, 2016.[BibTeX]
6. D. Pizzocri, C. Rabiti, L. Luzzi, T. Barani, P. Van Uffelen, and G. Pastore. PolyPole-1: An accurate numeical algorithm for intra-granular fission gas release. Journal of Nuclear Materials, 478:333â€“342, 2016.[BibTeX]
7. H. Shimizu. The properties and irradiation behavior of U$_3$Si$_2$. Technical Report NAA-SR-10621, Atomics International, 1965.[BibTeX]
8. M.V. Speight and W. Beere. Vacancy potential and void growth on grain boundaries. Metal Science, 9:190â€“191, 1975.[BibTeX]
9. M. S. Veshchunov. On the theory of fission gas bubble evolution in irradiated UO$_2$ fuel. Journal of Nuclear Materials, 277:67â€“81, 2000.[BibTeX]
10. R.J. White. The development of grain-face porosity in irradiated oxide fuel. Journal of Nuclear Materials, 325:61â€“77, 2004.[BibTeX]