UO2Sifgrs

Recommended fission gas model to account for generation of fission gasses in nuclear fuel

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 Simple Integrated Fission Gas Release and Swelling (Sifgrs) model is intended for consistently evaluating the kinetics of both fission gas swelling and release in UO. The model incorporates the fundamental features of fission gas behavior, among which are gas diffusion and precipitation in grains, growth and coalescence of gas bubbles at grain faces, thermal, athermal, steady-state, and transient gas release. Through a direct description of the grain-face gas bubble development, the fission gas swelling and release are calculated as inherently coupled processes, on a physical basis. The level of complexity of the model is consistent with reasonable computational cost and the uncertainties inherent in engineering-scale fuel analysis. The Sifgrs model draws on and extends the approach described in Pastore et al. (2013).

Intra-granular Gas Behavior

Fission gas transport from within the fuel grains to the grain faces is computed through numerical solution of the relevant diffusion equation in one-dimensional spherical geometry (1) where (m) is the intra-granular gas concentration, (s) the time, (ms) the effective gas diffusion coefficient, (m) the radial co-ordinate in the spherical grain, and (ms) the gas generation rate. The effective diffusion coefficient, which accounts for the effects of repeated trapping in and irradiation-induced resolution from intra-granular bubbles, is calculated based on White and Tucker (1983) and Speight (1969). Eq. (1) is solved using dedicated numerical algorithms. Both the algorithm from Hermansonn and Massih (2002) and the more recent one from Pizzocri et al. (2016) are available in BISON.

In alternative to the single diffusion equation with the effective diffusion coefficient , it is possible to solve the coupled equations describing gas diffusion and exchange of gas between the grain matrix and bubbles (2)

(3) where and (m) are the single gas atom concentration and the gas in bubble concentration, respectively, (ms) the single atom diffusion coefficient, (s) the trapping rate, and (s) the re-solution rate. This system of equations is solved in time-varying conditions by a dedicated algorithm, extending the one from Pizzocri et al. (2016). The solution of this system of equations is expected to be needed during fast transients (e.g., RIA). Nevertheless, Eq. (1) provides a good description of the intra-granular gas behavior for the wide majority of temperature and power conditions.

An empirical model (Lassmann et al., 1995) is included to consider intra-granular gas depletion in the high burnup structure (HBS). No specific model for gas release from the HBS is considered.

Evolution of Intra-granular Bubbles

The contribution of intra-granular bubbles to fission gas swelling (intra-granular swelling) is generally less important than the swelling due to grain-face bubbles in normal operation conditions, while at burnup above about 45 GWd/t (Kashibe et al., 1969) and/or in fast temperature transients (White et al., 2006) can constitute a significant portion of the fuel gaseous swelling. In the framework of Sifgrs, a model featuring a mechanistic description of the evolution of the nanometric-sized intra-granular bubbles and the so-called dislocation bubbles, the latter being those heavily contributing to fuel swelling in certain conditions, is available. The model for small intra-granular bubbles is drawn on Pizzocri et al. (2018), while his extension to consider dislocation bubbles is taken from Barani et al. (2018).

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 experimental and theoretical findings, the heterogeneous mechanisms is assumed as dominating in UO. The classical model traces back to the work of Turnbull (1971) as reported by White and Tucker (1983). This model assumes complete destruction of intragranular bubbles after the interaction with a fission fragment, and the re-solution rate is calculated accordingly as

(4)

Veshchunov and Tarasov (2013) proposed a mechanistic model accounting for the non-linear reduced effectiveness of bubble re-solution as the bubble radius increases. The resolution rate is defined as

(5)

with critical distance from the pore surface within which atom re-solution occurs and thickness of the re-solution layer around the pore. Lösönen (2017) then added additional modifications, reading

(6)

Lastly, Setyawan et al. (2018) reviewed previous on irradiation-induced re-solution and, based on molecular dynamics analysis, proposed the following re-solution parameter, also accounting for a size-dependent re-solution

(7)

A portion of the intra-granular bubbles is pinned to dislocation lines, which are postulated to be responsible for the coarsening (i.e., the abrupt growth) of the pinned intra-granular bubbles in temperature transients. In addition to the above mentioned mechanisms governing the small intra-granular bubbles, the evolution of the "dislocation" intra-granular bubbles is governed by an additional trapping of gas along dislocation lines, and eventually precipitating into such bubbles. The additional absorption of fission gas atoms may trigger the mentioned bubble coarsening, thanks to the availability of vacancies in the nearby of the dislocation core (differently from the bulk of the grain, where the so-called "vacancy starvation" is preventing the small intra-granular bubbles to grow). A sketch of the mechanisms acting on dislocation bubbles is represented in Figure 1.

Figure 1: Sketch of the trapping and re-solution mechanisms acting on a dislocation bubble (in orange). In green are represented the fission gas single gas atoms.

The equations for the evolution of the two populations of the intra-granular gas bubbles, in terms of bubble number density and gas atom concentrations are: (8) where and (m) are the number density of small and dislocation intra-granular bubbles, and is the number of gas atoms per small and dislocation bubbles, , , and (m) are the intra-granular gas concentration in the matrix, in the small and dislocation bubbles, respectively, (s) the time, (ms) the single-atom gas diffusion coefficient, (ms) the gas generation rate, and (s) the (reduced) trapping rate into small and dislocation bubbles, (s) is the (reduced) trapping rate along dislocation lines and then into dislocation bubbles, and (s) the (reduced) re-solution rate from small and dislocation bubbles, and () the nucleation rate, which is calculated as: (9) being (bubble/fission-fragment) the average number of bubbles nucleated per fission fragments (e.g., Turnbull (1971)) and the fission rate. Eventually, (bubbles/m) is the nucleation rate of bubbles on dislocations, and () is the dislocation density.

Small intra-granular bubbles radius is calculated, assuming spherical shape of the bubbles, multiplying the number of atoms per intra-granular bubble, i.e. , by the equivalent radius associated to atoms in intra-granular bubbles (derived from experimental measurements, (e.g. Olander and Wongsawaeng (2006)).

As for dislocation bubbles, when they meet the critical condition for coarsening, i.e., when the internal energy exceeds the surface energy of the bubbles, vacancies absorption is triggered, causing the coarsening of this bubble population. The absorption of vacancies is modeled as the result of a two-steps process entailing the precipitation from the bulk and the subsequent absorption, reading

where (Jm) is the UO gas specific surface energy, (m) is the dislocation bubble radius, (Pa) is the hydrostatic stress, (J/K) is the Boltzmann constant, () is the vacancy volume, (dimensionless) is the number of vacancies per dislocation bubble, and are the vacancy diffusion coefficients in the bulk and along dislocations, respectively, and (m) is the radius of the equivalent Wigner-Seitz cell associated to one dislocation bubble.

Moreover, interconnection between dislocation bubbles is accounted for, considering the nearest-neighbor distribution function for mono-disperse hard spheres as proposed by Torquato (2013), reading:

where is the nearest-neighbor distribution function, and are the dislocation bubbles volume (m) and porosity (dimensionless), i.e., the swelling, defined as

Finally, the radius of the dislocation bubbles is evaluated considering spherical bubbles as

Grain-face Gas Behavior

Numerical solution of Eq. (1) allows estimating the arrival rate of gas at the grain faces, thus providing the source term for the grain-face gas behavior module. The latter 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 eventual inter-connection (leading to thermal FGR). In outline:

  • Peculiarities related to the presence of grain edges (where three grains meet) are neglected (e.g., Kogai (1997) and Massih and Forsberg (2008)).

  • The flux of gas atoms dissolved from the grain faces back to the grain interior by irradiation is neglected (Rest, 2003).

  • An initial number density of grain-face bubbles (nucleation centers) is considered, and further nucleation during the irradiation is neglected (one-off nucleation, e.g., White (2004)).

  • The absorption rate of gas at the grain-face bubbles is assumed to equal the arrival rate of gas at the grain faces (White, 2004; Olander and Uffelen, 2001).

  • All grain-face bubbles are considered to have, at any instant, equal size and equal lenticular shape of circular projection (e.g., Veshchunov (2008)); hence, the fractional volume grain-face fission gas swelling is given by

(10) 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. (10) because a grain-face bubble is shared by two neighboring grains.

warningwarning:2D Grain Size to 3D Grain Radius Conversion

When using grain size from experimental data, special care must be taken in ensuring the true grain radius is calculated. See GrainRadiusStandardfor more information.

Bubble Growth

Bubble growth is treated using the model from Speight and Beere (1975), which describes 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 vacancy absorption/emission rate at a bubble is given by

(11) where (dimensionless) is the number of vacancies in the bubble, (m s) the vacancy diffusion coefficient in grain boundaries, (m) the thickness of the diffusion layer in grain boundaries, and the parameter (dimensionless) may be calculated as (White, 2004) with being the fraction of grain faces covered by bubbles (fractional coverage). The mechanical equilibrium pressure, , of the gas in a lenticular bubble of circular projection is given by (12) where (Jm) is the gas specific surface energy, (m) the bubble radius of curvature, and (Pa) the hydrostatic stress (considered to be negative if the solid medium is under compression). For describing the bubble thermodynamic state, the Van der Waals' equation of state is adopted in the following form: (13) where (dimensionless) is the number of fission gas atoms per bubble, (JK) the Boltzmann constant, (K) the temperature, (m) the bubble volume, and (m) the Van der Waals' volume of a fission gas atom. Given that each bubble consists of vacancies and gas atoms, the volume of a bubble comprising fission gas atoms and vacancies is given by (14) where (m) is the atomic (vacancy) volume in the bubble. The combination of Eq. (13) and Eq. (14) gives for the pressure of the gas in the bubble (15) The above approach allows computing the bubble growth rate from the rate of inflow of gas atoms along with 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 via Eq. (15). The bubble pressure affects the propensity of the bubble to absorb (or emit) vacancies through Eq. (11). Given the volume, , of a lenticular bubble of circular projection, the bubble radius of curvature is calculated as

Lower-Length-Scale-Infomed Equation of State (eos_option==YANG)

An equation of state for fission gas has been derived from molecular dynamics simulations by Yang and Wirth (2022). This provides an improved description of bubble pressure as a function of temperature, the number of gas atoms in the bubble, and the number of vacancies that make up the free space (volume) of the bubble. In particular, the model improves the decription of pressure in the high pressure regime, and has been validated up to a gas density of atoms/m. This model can be selected using eos_option = YANG; otherwise, the definition of pressure described above is used by default. In the model of Yang and Wirth, the number density of gas, , is given by where the bubble volume is . The bubble pressure is given by (16) where and m, while , and are temperature-dependent coefficients defined as such where and K. The parameters , , and are listed in Table 1. Additional parameters relating to the containment of nm-sized bubble in UO have been not been included in this implementation, as the impact is negligible for the large bubles that form on grain boundaries.

Table 1: Model parameters from Yang and Wirth (2022).

ParameterValue (unitless)
0.261323
−1.132763
0.028564
0.318315
-0.038613
−0.248067
−0.003885
0.069736
0.018356

The derivative of pressure with respect to volume is given by

Additional Bubble Growth though Vacancies Assisting Gas Diffusion (vacancies_assisting_gas_diffusion_option==COOPER2024)

The default version of the model assumes that gas atoms arrive at bubbles as interstitial defects. However, it has been determined from lower-length-scale simulations that fission gas diffusion can be assisted by vacancies. Following the results of Matthews et al. (2020), for fission rates typically experienced in-reactor, there is a transition from high temperatures, where diffusion is assisted by two vacancies per gas atom (), to intermediate temperatures, where diffusion is assisted by four vacancies (). At low temperatures (below around 1000 K), gas interstitials dominate diffusion (), meaning that no vacancies assist diffusion. Therefore, one can account for the number of vacancies that arrive at intergranular bubbles by accompanying and assisting gas atom transport, as such: where is the number of vacancies that assist the diffusion of a single gas atom at a given temperature. Accounting for the three possible dominant diffusion mechanisms in UO, The denomenator accounts for the arrival rate of gas atoms due to each mechanism, while the numerator accounts for the corresponding number of vacancies. values are fitted, as a function of temperature, to the diffusivity data from the Centipede model from Matthews et al. (2020), which describes gas diffusivity due separate defects. For gas interstitials, their further incorporation into bubbles becomes unfavorable as the bubbles become highly pressurized. This is accounted for by the Y term based on the energy, , to incorporate a gas interstitial defect into a bubble, as such, The above equation is dependent on the number of gas atoms in the bubble per the number of vacancies in the bubble, following the parameterization of Cooper et al. (2024). This model can be activated using vacancies_assisting_gas_diffusion_option==COOPER2024.

Grain-Face Bubble Coalescence

As intergranular bubbles grow and cover a larger area of the grain boundary surface, they start to interact and coalesce and form fewer, larger bubbles. Sifgrs offers two different models to describe this phenomena. The first is from White (2004). Assuming that bubbles are distributed regularly on a square lattice and have a circular projection on the grain surfaces, White determines the number of bubble interactions as a function of bubble growth. The decrease in intergranular bubble number surface density as a result of coalescence is defined as (17) where is the projected area of the lenticular grain face bubbles and denotes the change in as a result of bubble growth. Once integrated, it leads to (18) where and are the initial and values. This model corresponds to the option intergranular_bubble_coalescence_option=WHITE2004 and is used by default.

Pastore et al. (2013) later proposed an update version of this model. The updated model accounts for the fact that the average bubble volume increases while decreases during coalescence to conserve the total bubble volume per unit surface . The decrease in intergranular bubble number surface density as a result of coalescence is defined as (19) This model corresponds to the option intergranular_bubble_coalescence_option=PASTORE2013.

In both cases, a lower limit of m is set for .

Bubble Number Density Saturation Condition

The release of fission gas to the fuel rod free volume following inter-connection of grain-face bubbles and consequent formation of pathways for gas venting to the fuel exterior (thermal release) is modeled based on a principle of grain face saturation. More specifically, a saturation coverage concept is adopted, namely, it is considered that once the fractional coverage, , attains a saturation value, , the bubble number density and projected area obey the saturation coverage condition (20) where is the bubble number density and is the bubble projected area on the grain face. The commonly accepted value for is 0.5. Eq. (20) implies that, after attainment of the saturation coverage, a fraction of the gas reaching the grain faces is released to the fuel exterior to compensate for continuing bubble growth.

Transient Gas Behavior

There are two approaches available for the calculation of transient gas behavior. The transient_option parameter allows for the selection of which model is used. Selecting transient_option==NO_TRANSIENT skips all calculations of transient fission gas release. The microcracking model developed by Pastore et al. is used for transient_option==MICROCRACKING_BURNUP and transient_option==MICROCRACKING (Pastore et al., 2014). Note that the option transient_option==MICROCRACKING_BURNUP differs from transient_option==MICROCRACKING in that it accounts for the burnup effect on the temperature for burst release. The empirical model developed by Capps et al. is used for transient_option==EMPIRICAL_CAPPS (Capps et al., 2023).

Microcracking model (transient_option==MICROCRACKING_BURNUP or transient_option==MICROCRACKING)

Experimental observations relative to both in-reactor irradiation and post-irradiation annealing of oxide nuclear fuel indicate that substantial fission gas release can occur on a small time scale during temperature transients (burst release). The rapid kinetics of the process cannot be interpreted as purely diffusion-controlled. From the available experimental evidence (Rothwell, 1962; Une and Kashibe, 1990; Sartori et al., ; Ducros et al., 2013), the following main aspects of transient fission gas behavior emerge:

  • Burst release occurs through grain-face separation (micro-cracking) which entails gas depletion of a fraction of the grain faces.

  • Release bursts are triggered by temperature variations, both heating and cooling.

  • The rate of gas release during bursts is a peaked function of temperature with the maximum at a 'central' temperature, which is dependent on the burnup.

An extension (transient model) of the treatment of grain-face gas behavior described in the section "Grain-face gas behavior" is available in BISON, which introduces the effect of micro-cracking on fission gas behavior (Pastore et al., 2014). According to the BISON transient model, gas depletion of a fraction of the grain faces is modeled as a reduction of the fractional coverage, . In particular, is scaled by a factor, , corresponding to the fraction of non-cracked (intact) grain faces. The reduction of the fractional coverage effectively leads to a decrease of the amount of gas retained in the fuel – consequently, of fission gas swelling – and to a corresponding increase of FGR. This contribution to thermal FGR supplements the diffusion-interconnection mechanism considered in the basic model (See section "Grain-face gas behavior"). Also, the lost gas storing capacity of cracked grain faces is represented by scaling the saturation coverage, , by the factor f. Moreover, the healing process of cracked grain faces is considered as a progressive restoration of the grain-face gas storing capacity. Therefore, the fractional coverage and saturation coverage obey

where stands for diffusion-controlled processes (basic model in the section "Grain face gas behavior"), stands for micro-cracking, and for micro-crack healing. The value for the maximum (initial) saturation coverage (corresponding to all intact grain faces) is . The calculation of the term representing the effects of micro-cracking is detailed hereinafter.

We simplify the micro-cracking process into a purely temperature-dependent behavior, characterized by a micro-cracking parameter, . We also observe that the process can only affect intact grain faces, and write (21) where is the reduction rate due to micro-cracking of the fraction of intact grain faces, . The micro-cracking parameter is taken as a function of the sole temperature, hence (22) Then, Eq. (21) can be written as implying which conforms to the experimentally observed characteristic of burst release as triggered by temperature variations. Under the condition expressed by Eq. (22), the analytic solution of Eq. (21) with initial conditions and is Based on the available experimental evidence, the functional form of is chosen as a temperature-dependent sigmoid function (23) where (K) is the central temperature, (K) is a measure of the temperature-domain width of the phenomenon, (dimensionless) is a parameter, and is defined as so that increases during both heating and cooling transients. The following values are used for the parameters: = 5 K, = 33. The micro-cracking parameter, , and the parameter derivative, , are plotted in Figure 2 for a value of K. According to Eq. (23) the absolute value of the temperature derivative for is maximum at (see also Figure 2), thus the combined Eq. (21) and Eq. (23) reproduce the maximum rate of burst release at as temperature varies in time.

For the calculation of the parameter in the model, the following correlation from Barani et al. (2017) is used (24) where K, K, GWdt, and (GWdt) is the burnup at current time step. Eq. (24) is derived from the best-estimate fit of quantitative experimental data, Figure 3.

Figure 2: Micro-cracking parameter, , and derivative, , as a function of temperature, considering a central temperature equal to 1773 K.

Figure 3: Experimental data for the temperature of maximum burst release rate from Une and Kashibe (1990) and Baker and Killeen (1987) as a function of burnup and best-estimate fitting curve.

A simple burnup-dependent model is used for micro-crack healing, which is not described here for brevity. Details can be found in Pastore et al. (2014). The above treatment of transient fission gas behavior preserves the continuity in both time and space as well as the consistent coupling of the calculated fission gas release and swelling. Extensive validation has indicated that the model is capable of consistently representing the kinetics of FGR during transient fuel irradiations (Pastore et al., 2014; Pizzocri et al., 2015; Barani et al., 2017).

Empirical transient model (transient_option==EMPIRICAL_CAPPS)

The empirical transient model is a function of local burnup and temperature. Details of the development of the model and the fitting process can be found in Capps et al. (2023). Note, however, that the expression and parameter values do not exactly match the model described in Capps et al. (2023) to represent recent updates to the model.

The equation for transient fission gas release can be written as a function of burnup and temperature : (25) where FGR is the percent of fission gas release. The constants associated with the model are included in Table 2.

Table 2: Model parameters values for tFGR model from Capps et al. (2023).

ParameterValueUnits
65.320
0.03552(dimensionless)
1827.0
0.4077
0.2010(dimensionless)
928.90
20.650
0.2230(dimensionless)

Athermal Gas Release

At low temperature, the fission gas in the matrix of the solid is relatively immobile. Only the gas formed at the external surface of the solid is capable of escape, with an emission rate that is independent of temperature. This athermal contribution to FGR arises from the surface-fission release mechanisms of recoil (direct release of a fission fragment due to its high kinetic energy) and knockout (ejection of a gas atom following elastic interaction with either a primary fragment or energetic particle created in a collision cascade) (Lewis, 1987). These release mechanisms affect only the outer layer of the fuel (within about from the surface). The rate of gas atom release per unit fuel volume due to recoil and knock-out, (ms), may be calculated as (Lewis, 1987) where (dimensionless) is the fractional yield of fission gas atoms, the fission rate density (ms), (m) the volume of fuel, (m) the geometrical surface area of fuel, (m) the total surface area of fuel (including cracked surface), (m) the fission fragment range in the fuel, and (m) the range of the higher order uranium knock-on in UO.

In line with Koo et al. (2000), the number and length of cracks in each fuel pellet is estimated in a simple way. First, radial cracks are considered to cross the outer, brittle region of the fuel pellet with a temperature lower than 1200 C (Olander, 1976). Second, the number of pellet cracks is considered to increase linearly with fuel linear power (Oguma, 1983). Then, once the linear power and pellet dimensions are given, the total pellet surface area available for athermal gas release can be calculated.

Grain Growth and Grain Boundary Sweeping

Being the fission gas behavior physically dependent on the granular structure of the fuel, the Sifgrs model is coupled with the grain growth model (See section on Grain Growth in the Theory Manual Overview of Ceramic Fuels). The grain growth phenomenon affects the fission gas release in three ways. First of all, due to the low solubility of the fission gas, the moving grain boundary does not redeposit any gas in the newly-formed crystal behind it, thus acting as a filter and contributing to the collection of gas at the grain faces (_grain boundary sweeping_). This effect is taken into account in Sifgrs by adding a supplementary fractional release term () from within the grains to the grain faces that is equal to the volume fraction of the fuel swept by the moving boundaries: where the indices and refer to the previous and current time, respectively. Secondly, the diffusion distance for the fission gas atoms created in the grains increases as the grains grow. Unlike the first consequence this tends to reduce the release rate. Thirdly, grain growth reduces the capacity of the grain boundaries to store fission gas, as it results in a decrease of the total grain surface-to-volume ratio.

Extension to high burnup behavior

Specific models have been implemented to describe fission gas behavior at high burnup during important microstructural changes. These developments are described in Simon et al. (2023) and Simon et al. (2024).

Modeling of high burnup structure (HBS) formation

Although several models exist in the literature, BISON currently offers two HBS formation models, which can be used in parallel with Sifgrs using the HighBurnupStructureFormation object. The first one is an empirical description from Lassmann et al. (1995). Barani et al. (2020) later proposed a more descriptive model for HBS formation. See HighBurnupStructureFormation for additional information.

A two-phase model for the non-restructured region and HBS

As in Barani et al. (2020), the fuel is modeled as a two-phase system, with the fuel being part non-restructured (NR) and part HBS. Fission gas concentration is therefore shared between the two phases, with where (mol/m) is the local fission gas concentration, (mol/m) is the local fission gas concentration in the NR phase, and (mol/m) is the local fission gas concentration in the HBS phase. is the local fraction of HBS provided by HighBurnupStructureFormation. Moreover, the grain size, bubble populations, dislocation density, and other microstructure parameters of each phase are tracked for each phase and for the complete fuel. This approach is similar to what is implemented in SCIANTIX (Barani et al., 2020) and MARGARET (Noirot, 2011).

Note that the same description is consistently used in BISON with Lassmann's model. However, as the local HBS volume fraction is either 0 or 1, the two-phase description is greatly simplified.

Modeling of fission gas generation

The amount of fission gas generated is shared between the NR and HBS regions as and

Modeling of fission gas transition from the NR region to HBS

It has been observed experimentally that the gas within the matrix (intragranular gas) is gradually depleted as the HBS forms (Lassmann et al., 1995). This depletion has been fit empirically with an exponential function that is valid when the current burnup is greater than a threshold burnup . Lassmann et al. describe it as (26) where is the Xe concentration in the matrix, is the Xe production rate, and (MWd/kgU) is a constant fit to data. Once and depletion of the intragranular gas begins, it is assumed that the intragranular gas is transferred to the large intergranular bubbles normally observed in the HBS region at a rate proportional to concentration, following (27) (28) where is the concentration throughout the fuel of Xe that is contained in HBS bubbles (to be distinguished from the concentration of Xe in each bubble, which is significantly higher). This expression for rate of transfer, combined with the source term for production of new Xe atoms, can be integrated to obtain Eq. (26).

Another model available corresponds to an adaptation of the model from Barani et al. (2020). As the fuel transitions to HBS, significant microstructural changes happen. These transformations result in significant changes in the location of fission gases. As HBS forms during irradiation, fission gases are swept from the NR region. As the HBS volume fraction increases by , the amount of fission gas swept from the NR region corresponds to (29) The denominator corresponds to the volume fraction of NR fuel. In this model, we assume that the generation of fission products has already been accounted for, so mass conservation imposes that the amount of fission gas swept from the NR fuel equals the amount of fission gas added to the HBS phase. As such, (30)

As HBS formation happens, the amount of fission gases in the NR matrix decreases. As all the fuel becomes covered by HBS, all the fission gases are located in the HBS phase.

Intragranular fission gas modeling in HBS

As a first approximation, the intragranular fission gas behavior in HBS fuel is greatly simplified. HBS grains are assumed to have a constant radius of 150 nm (Barani et al., 2020), and intragranular fission gases are assumed to diffuse toward grain boundaries. The presence of intragranular bubbles and dislocations, which can affect fission gas transport in NR fuel, are neglected. This assumption, however, is reasonable as a first approximation. HBS formation has been observed to reduce the density of dislocations, and the HBS microstructure is dominated by large intergranular bubbles rather than intragranular ones. Moreover, the purely diffusional description of intragranular fission gas behavior is widely used in fuel performance codes (Barani et al., 2020). In the current report, we use the same diffusion coefficient as proposed in Barani et al. (2020), which originates from an empirical study (Brémier and Walker, 2002). The diffusion of gas atoms in intragranular HBS is therefore defined as (31) where the fission rate in ms, leading to being defined in m/s. Fission gases are now shared between HBS matrix and bubbles. The amount of fission gases in the HBS matrix now reaches a non-zero value, which depends on HBS grain size and HBS intragranular diffusivity. This model will be validated in future work. In particular, it will be compared to the experimental data published in Lassmann et al. (1995).

Note that when Lassmann's model of HBS formation is used, the sweeping of fission gases from the intragranular matrix to HBS bubbles already accounts for intragranular diffusion in the HBS region (Lassmann et al., 1995). As such, no further intragranular model is needed.

Intergranular fission gas modeling in HBS

In addition to the intragranular gas described above, the transfer of intergranular gas to the larger, spherical bubbles found in the HBS region must be accounted for. Moreover, it is assumed that gas is transferred from the intergranular bubbles to the HBS bubbles at the same rate as rate of formation of the HBS itself. Taking the derivative of the equation of high burnup structure volume fraction from HighBurnupStructureFormation with respect to burnup gives: (32) Thus, the rate of change of grain boundary bubble density, , is given by (33)

Having determined how much gas is transferred to the HBS bubbles, we now must determine their pressure and size evolution. It is assumed that initial number density of HBS bubbles is m (Barani et al., 2019), and their initial radius is set to 0 m. Note that gas bubbles in the HBS region are assumed to be spherical, in contrast to the lenticular intergranular (grain boundary) bubbles that are normally observed in UO fuel prior to HBS formation. The density of gas in each bubble, , is calculated by where is the bubble volume. Now having determined the concentration of gas in each bubble, we can determine the bubble pressure using the equation of state of our choice.

To evolve the bubble radius as a function of time, an approach similar to that used for the evolution of intragranular bubbles in USi (Barani et al., 2019; Barani et al., 2022) was employed. The growth of the bubbles is assumed to be controlled by vacancy flux, driven by overpressurization of the HBS bubbles. Given this, the rate of change of bubble volume is given by (34) where m is the volume of a U lattice site in UO Kogai (1997), and is the number of vacancies per intragranular bubble. For overpressured bubbles, the vacancy flux is given by Barani et al. (2019) and Barani et al. (2022): (35) where is the intragranular vacancy diffusion coefficient, is the radius of the equivalent Wigner-Seitz cell surrounding a bubble, and is a dimensionless factor defined as: (36) where . is the equilibrium pressure for a bubble of radius , as calculated using the Laplace-Young equation (37) where J/m (Hall et al., 1987; Kogai, 1997) is the surface tension of the bubble-matrix interface, and is the hydrostatic stress, considered to be negative for a solid in compression. Since the bubbles are expected to be overpressurized shortly after formation, Sifgrs enforces to prevent transient bubble shrinking during the brief initial stage after bubble formation.

Another HBS bubble evolution model is available in UO2Sifgrs. This model builds on the HBS formation model of the previous approach, as well as its fission gas transition model, and its intragranular model (Simon et al., 2023; Simon et al., 2024). However, the HBS intergranular was revisited to more accurately capture grain boundaries (GBs) diffusion and HBS pore evolution. This approach is a mix between (1) a mechanistic description of HBS pore density to alleviate the previous assumption that the pore density remains constant and (2) an empirical model for HBS pore radius (Simon et al., 2024).

The pore density model is from a simplification of the model described in Barani et al. (2022). It states that

with the HBS pore number density in m, the gas concentration in HBS pores in mol/m, the gas single atom concentration HBS GBs in mol/m, the HBS pore nucleation rate in ms, the resolution rate in s, the trapping rate in s, and the rate at which gas atoms reach the HBS GBs from the grain bulk in mol/m/s and is derived from intragranular behavior. The HBS pore nucleation rate is defined as proportional to the local restructuring rate (Barani et al., 2022), meaning with MWdkgUms, the HBS volume fraction, and the effective burnup for HBS formation (Barani et al., 2020; Simon et al., 2023). For the derivative of the HBS volume fraction as a function of burnup, Sifgrs uses the version of the model from Simon et al. (2023), which has different model parameters from Barani et al. (2020) and Barani et al. (2022) to better match experimental data. See HighBurnupStructureFormation for additional information. Note that the value was updated from in the original paper to better match experimental data (Barani et al., 2020). The resolution rate is from a model proposed by Veshchunov and Tarasov (2013). It defines as (38) where (dimensionless), m is the critical distance from the pore surface within which atom resolution occurs, m is the thickness of the resolution layer around the pore, and the average HBS pore radius in m. The trapping rate model is originally from Gösele (1978). It states that (39) with the single atom diffusion coefficient in GBs, and (40) corresponding to the local porosity (dimensionless) assuming HBS pores are spherical. The HBS grain boundary (GB) diffusivity is not well characterized. However, Olander et al. have determined the GB diffusivity for NR fuel (Olander and Uffelen, 2001), and Barani et al. have proposed a corrective factor for high angle GBs in the HBS Barani et al. (2022). The diffusivity by Olander et al. is equal to (41) where is the temperature in Kelvin and is the ideal gas constant in J/K/mol. The HBS GBs diffusivity can then be obtained using (42)

As stated above, this approach uses an empirical model to capture the HBS local porosity and drive bubble growth. This model is from the stand alone fission gas behavior SCIANTIX. This model was provided by Davide Pizzocri from Politecnico di Milano in private communications and is available on SCIANTIX (see Pizzocri et al. (2020) and Zullo et al. (2023)). The same model was utilized as is, but the model parameter was updated. Based on data from Spino et al. (2006) and Cappia et al. (2016), the HBS local porosity is defined as with kgU/MWd. The pore radius can then be obtained using Eq. (40) assuming spherical pores.

After an increase in HBS pore density and radius, the number density can be reduced by pore interconnection (Barani et al., 2020). The pore number density is adjusted as where is a correction factor limiting the interconnection rate when high local porosity is achieved and accounting for the non-superposition of hard spheres (Barani et al., 2022). Bubble interconnection also affects the pore radius to ensure that interconnection does not affect the overall pore volume.

While the original model described in Simon et al. (2023) assumed a constant pore density and used a mechanistic model for pore radius evolution, the new model described above utilizes a mechanistic description of the pore density evolution and an empirical model for porosity and radius calculations Simon et al. (2024).

Preliminary application to FBR MOX

For fast MOX fuels, higher temperatures are reached compared to LWR fuels. The associated pronounced bubble growth and coalescence can lead to the attainment of the lower limit for the number density of grain boundary bubbles systematically. Preliminary simulations of FBR MOX irradiations have indicated that a value of leads to a more accurate result in terms of both FGR and swelling compared to the default LWR UO value. A value of is compatible with experimental observations, e.g., White (2004).

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [fission_gas_release]
    type = UO2Sifgrs<<<{"description": "Recommended fission gas model to account for generation of fission gasses in nuclear fuel", "href": "UO2Sifgrs.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temp
    fission_rate<<<{"description": "Coupled fission rate variable (fiss/m^3/s)"}>>> = fission_rate
  []
[]
(test/tests/element_integral_power/fission_gas_sifgrs_1D.i)

Input Parameters

  • temperatureCoupled temperature

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled temperature

Required Parameters

  • atomic_covolume8.47705e-29Van der Waals covolume for Xe (m^3/atm)

    Default:8.47705e-29

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Van der Waals covolume for Xe (m^3/atm)

  • axial_power_profileAxial power peaking function.

    C++ Type:FunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Axial power peaking function.

  • 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

  • bubble_gb_limit1e+10grain-boundary bubble number density limit (bbl/m**2)

    Default:1e+10

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:grain-boundary bubble number density limit (bbl/m**2)

  • burnupCoupled burnup

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled burnup

  • burnup_functionBurnup function

    C++ Type:BurnupFunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Burnup function

  • 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

  • cr_doped_optionBEST_ESTIMATE_1773Select Cr-doped option when doping_type = CR2O3_DOPED

    Default:BEST_ESTIMATE_1773

    C++ Type:MooseEnum

    Options:CORRECTION, TRANSITION_TEMPERATURE_1525, TRANSITION_TEMPERATURE_1800, TRANSITION_TEMPERATURE_1673, REFINED_1673, BEST_ESTIMATE_1773, UPPER_LIMIT_1773

    Controllable:No

    Description:Select Cr-doped option when doping_type = CR2O3_DOPED

  • 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.

  • diff_coeff_optionTURNBULL_D1_D2Select diffusion coefficient

    Default:TURNBULL_D1_D2

    C++ Type:MooseEnum

    Options:TURNBULL_D1_4D2, ANDERSSON, TURNBULL_D1_D2_D3, TURNBULL_D1_D2, TURNBULL_D1_4D2_4D3, TURNBULL_D1_4D2_D3, HBS_BREMIER, TEST_CASE

    Controllable:No

    Description:Select diffusion coefficient

  • dislocation_bubble_nucleation_factor1e+06dislocation bubble nucleation factor, i.e., number of bubbles per dislocation line density (bubbles/(m/m2))

    Default:1e+06

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:dislocation bubble nucleation factor, i.e., number of bubbles per dislocation line density (bubbles/(m/m2))

  • dislocation_core_radius3.85e-10dislocation core radius (m), i.e., burgers vector length

    Default:3.85e-10

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:dislocation core radius (m), i.e., burgers vector length

  • dislocation_density40000000000000.0Coupled dislocation density (m/m^3)

    Default:40000000000000.0

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled dislocation density (m/m^3)

  • dislocation_density_materialdislocation density material property (m/m^3)

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:dislocation density material property (m/m^3)

  • dislocation_punchingFalseFlag to allow dislocation punching, effectively limiting bubble pressures and tracking the rate of dislocation production.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Flag to allow dislocation punching, effectively limiting bubble pressures and tracking the rate of dislocation production.

  • dislocation_trap_Zfactor25Z * dislocation core radius = dislocation_trap_radius (m)

    Default:25

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Z * dislocation core radius = dislocation_trap_radius (m)

  • doping_typeUNDOPEDThe selected dopant

    Default:UNDOPED

    C++ Type:MooseEnum

    Options:UNDOPED, CR2O3_DOPED

    Controllable:No

    Description:The selected dopant

  • eff_diff_coeff_optionINCLUDING_BUBBLESelect effective diffusion coefficient

    Default:INCLUDING_BUBBLE

    C++ Type:MooseEnum

    Options:INCLUDING_BUBBLE, BULK, LASSMANN, TEST_CASE

    Controllable:No

    Description:Select effective diffusion coefficient

  • effdiffcoeff_scalef1Scaling factor for intragranular effective diffusion coefficient

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for intragranular effective diffusion coefficient

  • eos_optionORIGINAL_MODELSelect Equation of State

    Default:ORIGINAL_MODEL

    C++ Type:MooseEnum

    Options:ORIGINAL_MODEL, YANG

    Controllable:No

    Description:Select Equation of State

  • evolve_bubble_pressure_hbsTrueAllow bubble pressure to evolve in hbs region from initial conditions based on evolution equations

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Allow bubble pressure to evolve in hbs region from initial conditions based on evolution equations

  • fission_gas_concCoupled fission gas concentration

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled fission gas concentration

  • fission_rateCoupled fission rate variable (fiss/m^3/s)

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled fission rate variable (fiss/m^3/s)

  • fission_rate_materialFission rate material property (fiss/m^3/s)

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:Fission rate material property (fiss/m^3/s)

  • fract_yield0.3017fractional yield of fission gas atoms (Xe + Kr) (/)

    Default:0.3017

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

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

  • gas_atom_dislocation_diffusion_multiplier1Factor by which gas atom diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Factor by which gas atom diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion

  • gas_diffusivity_functionOptional function for gas atomic diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.

    C++ Type:FunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional function for gas atomic diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.

  • gbdiffcoeff_scalef1Scaling factor for grain-boundary vacancy diffusion coefficient

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for grain-boundary vacancy diffusion coefficient

  • grain_radiusCoupled grain Radius

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled grain Radius

  • grain_radius_HBS_constant1.5e-07Constant grain radius for HBS structure (m)

    Default:1.5e-07

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Constant grain radius for HBS structure (m)

  • grain_radius_const5e-06constant grain radius (m)

    Default:5e-06

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:constant grain radius (m)

  • grainradius_scalef1Scaling factor for grain radius

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for grain radius

  • hbs_materialThe name of the HighBurnupStructureFormation material, if any. This is needed when hbs_model=true.

    C++ Type:MaterialName

    Controllable:No

    Description:The name of the HighBurnupStructureFormation material, if any. This is needed when hbs_model=true.

  • hbs_pore_evolution_model_optionCONSTANT_DENSITY_MECHANISTIC_RADIUSSelect HBS pore evolution model

    Default:CONSTANT_DENSITY_MECHANISTIC_RADIUS

    C++ Type:MooseEnum

    Options:CONSTANT_DENSITY_MECHANISTIC_RADIUS, MECHANISTIC_DENSITY_EMPIRICAL_RADIUS

    Controllable:No

    Description:Select HBS pore evolution model

  • hbs_volume_fraction_threshold0.5Minimum local volume fraction of HBS for pulverization to occur

    Default:0.5

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Minimum local volume fraction of HBS for pulverization to occur

  • hydrostatic_stressCoupled hydrostatic Stress

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled hydrostatic Stress

  • hydrostatic_stress_const0constant hydrostatic stress (Pa)

    Default:0

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:constant hydrostatic stress (Pa)

  • ig_bubble_coarseningNO_COARSENINGSelect intra-granular diffusion algorithm

    Default:NO_COARSENING

    C++ Type:MooseEnum

    Options:NO_COARSENING, WITH_COARSENING

    Controllable:No

    Description:Select intra-granular diffusion algorithm

  • ig_bubble_modelEMP_BAKER_WHITESelect bubble evolution model

    Default:EMP_BAKER_WHITE

    C++ Type:MooseEnum

    Options:EMP_BAKER_WHITE, FIXED, NUCLEATION_RESOLUTION, MECHANISTIC_AAGESEN

    Controllable:No

    Description:Select bubble evolution model

  • ig_diff_algorithmFORMASSelect intra-granular diffusion algorithm

    Default:FORMAS

    C++ Type:MooseEnum

    Options:FORMAS, POLYPOLE1, POLYPOLE2

    Controllable:No

    Description:Select intra-granular diffusion algorithm

  • ig_fully_coupledLOOSELY_COUPLEDSolving diffusion coupled to bubble evolution

    Default:LOOSELY_COUPLED

    C++ Type:MooseEnum

    Options:LOOSELY_COUPLED, FULLY_COUPLED

    Controllable:No

    Description:Solving diffusion coupled to bubble evolution

  • igdiffcoeff_scalef1Scaling factor for intragranular diffusion coefficients

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for intragranular diffusion coefficients

  • igdiffcoeff_scalef_HBS1Scaling factor for HBS intragranular diffusion coefficients

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for HBS intragranular diffusion coefficients

  • initial_burnup0Burnup reached before annealing/separated effect experiment (GWd/tUO2)

    Default:0

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Burnup reached before annealing/separated effect experiment (GWd/tUO2)

  • initial_porosity0.05initial fuel porosity (/)

    Default:0.05

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:initial fuel porosity (/)

  • intergranular_bubble_coalescence_optionWHITE2004Select intergranular bubble coalescence model

    Default:WHITE2004

    C++ Type:MooseEnum

    Options:WHITE2004, PASTORE2013

    Controllable:No

    Description:Select intergranular bubble coalescence model

  • minimum_pressure_ratio1Factor to limit the equilibrium pressure of bubbles by enforcing a minimum ratio of the equilibrium pressure to the pressure due to surface energies alone. If greater than one, tensile stresses do not impact the equilibrium bubble size. This factor is not allowed to be below 0 since it can result in negative equilibrium pressures. A value of of 0.1 - 0.5 is reasonable for capturing tensile stress impacts on the equilibrium bubble pressure.

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Factor to limit the equilibrium pressure of bubbles by enforcing a minimum ratio of the equilibrium pressure to the pressure due to surface energies alone. If greater than one, tensile stresses do not impact the equilibrium bubble size. This factor is not allowed to be below 0 since it can result in negative equilibrium pressures. A value of of 0.1 - 0.5 is reasonable for capturing tensile stress impacts on the equilibrium bubble pressure.

  • nucleation_optionHETEROGENEOUSSelect intragranular bubble nucleation model

    Default:HETEROGENEOUS

    C++ Type:MooseEnum

    Options:HETEROGENEOUS, HOMOGENEOUS, TEST_CASE

    Controllable:No

    Description:Select intragranular bubble nucleation model

  • nuclerate_scalef1Scaling factor for nucleation rate

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for nucleation rate

  • pellet_brittle_zoneThe name of the UserObject that computes the width of the brittle zone in the fuel pellet

    C++ Type:UserObjectName

    Controllable:No

    Description:The name of the UserObject that computes the width of the brittle zone in the fuel pellet

  • pellet_idCoupled pellet ID

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled pellet ID

  • percolation_to_surface1.0Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in the gas release calculation.

    Default:1.0

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in the gas release calculation.

  • pressure_bubbles_hbs_initial1e+08Initial pressure of bubbles in HBS region (Pa)

    Default:1e+08

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Initial pressure of bubbles in HBS region (Pa)

  • pulverization_transient_fission_gas_release_materialThe name of the UO2PulverizationTransientFissionGasRelease material, if any. This is needed when pulverization_model=true.

    C++ Type:MaterialName

    Controllable:No

    Description:The name of the UO2PulverizationTransientFissionGasRelease material, if any. This is needed when pulverization_model=true.

  • res_param_optionHETEROGENEOUS_WHITESelect resolution parameter

    Default:HETEROGENEOUS_WHITE

    C++ Type:MooseEnum

    Options:HETEROGENEOUS_WHITE, HOMOGENEOUS_LOSONEN, HOMOGENEOUS_PASTORE, HETEROGENEOUS_VESHCHUNOV_TARASOV, HETEROGENEOUS_VESHCHUNOV_LOSONEN, HETEROGENEOUS_SETYAWAN, TEST_CASE

    Controllable:No

    Description:Select resolution parameter

  • resolutionp_scalef1Scaling factor for resolution parameter

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for resolution parameter

  • resolutionp_scalef_dislocation1Scaling factor for resolution parameter in dislocations

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for resolution parameter in dislocations

  • rod_ave_lin_powLinear power function.

    C++ Type:FunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Linear power function.

  • saturation_coverage0.5initial grain boundary saturation coverage (/)

    Default:0.5

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:initial grain boundary saturation coverage (/)

  • semidihedral_angle0.872665Semi-dihedral angle of grain-boundary bubbles (rad)

    Default:0.872665

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Semi-dihedral angle of grain-boundary bubbles (rad)

  • shear_modulusShear modulus material property (Pa). Required if dislocation_punching=true

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:Shear modulus material property (Pa). Required if dislocation_punching=true

  • surface_energy0.85UO2 fuel-gas surface energy (J/m^2)

    Default:0.85

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:UO2 fuel-gas surface energy (J/m^2)

  • temperature_reference_bubble_hbs_initial673.15Reference temperature for initial pressure of bubbles in HBS region (K)

    Default:673.15

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Reference temperature for initial pressure of bubbles in HBS region (K)

  • temperature_scalef1Scaling factor for temperature

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for temperature

  • transient_optionNO_TRANSIENTSelect transient release model]

    Default:NO_TRANSIENT

    C++ Type:MooseEnum

    Options:NO_TRANSIENT, MICROCRACKING_BURNUP, MICROCRACKING, EMPIRICAL_CAPPS

    Controllable:No

    Description:Select transient release model]

  • trap_param_optionDEFAULTSelect trapping parameter

    Default:DEFAULT

    C++ Type:MooseEnum

    Options:DEFAULT, TEST_CASE, GOSELE

    Controllable:No

    Description:Select trapping parameter

  • trappingp_scalef1Scaling factor for trapping parameter

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for trapping parameter

  • trappingp_scalef_dislocation1Scaling factor for trapping parameter in dislocation bubbles

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for trapping parameter in dislocation bubbles

  • trappingp_scalef_dislocation_line1Scaling factor for trapping parameter in dislocation lines

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scaling factor for trapping parameter in dislocation lines

  • vacancies_assisting_gas_diffusion_optionNO_VACANCY_ASSISTSelect intergranular model for vacancy-assisted gas atom diffusion

    Default:NO_VACANCY_ASSIST

    C++ Type:MooseEnum

    Options:NO_VACANCY_ASSIST, COOPER2024

    Controllable:No

    Description:Select intergranular model for vacancy-assisted gas atom diffusion

  • vacancies_per_atom_diffusion_functionOptional function for the number of vacancies assisting gas atom diffusion.

    C++ Type:FunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional function for the number of vacancies assisting gas atom diffusion.

  • vacancy_diffusivity_functionOptional function for vacancy diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.

    C++ Type:FunctionName

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional function for vacancy diffusivity (m^2/s). For implementation of the function, the x-axis corresponds to temperature and y-axis corresponds to fission rate. The z-axis is unused, and time is the standard current time.

  • vacancy_dislocation_diffusion_multiplier1Factor by which vacancy diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Factor by which vacancy diffusion is multiplied along dislocations compared to the bulk to capture pipe diffusion

  • vacancy_volume4.09e-29Atomic (vacancy) volume in bubbles (m^3)

    Default:4.09e-29

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Atomic (vacancy) volume in bubbles (m^3)

Optional Parameters

  • ath_modelFalseActivates the athermal release model

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Activates the athermal release model

  • 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.

  • gbs_modelFalseActivates the grain-boundary sweeping model

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Activates the grain-boundary sweeping model

  • hbs_modelFalseActivates the high burnup structure model

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Activates the high burnup structure model

  • 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

  • load_ig_gas_concFalseNeed to initialize ig diffusion algorithm modes?

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Need to initialize ig diffusion algorithm modes?

  • pulverization_modelFalseActivates the pulverization release model (currently also requires the hbs_model=true)

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Activates the pulverization release model (currently also requires the hbs_model=true)

  • 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

  • skip_bdr_modelFalseSkips the grain-boundary model

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Skips the grain-boundary model

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

    Default:False

    C++ Type:bool

    Controllable:No

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

  • 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

References

  1. C. Baker and J. C. Killeen. Fission gas release during post irradiation annealing of UO$_2$. In International Conference on Materials for Nuclear Reactor Core Applications, Bristol, United Kingdom, October 27-29. 1987.[BibTeX]
  2. T. Barani, E. Bruschi, D. Pizzocri, G. Pastore, P. Van Uffelen, R.L. Williamson, and L. Luzzi. Analysis of transient fission gas behaviour in oxide fuel using \mbox BISON and \mbox TRANSURANUS. Journal of Nuclear Materials, 486:96–110, 2017.[BibTeX]
  3. T. Barani, A. Magni, D. Pizzocri, L. Cognini, P. Van Uffelen, L. Luzzi, and G. Pastore. Modeling and assessment of intra-granular bubble evolution and coarsening in uranium dioxide. In NuMat18: The nuclear materials conference. Seattle, USA, 2018.[BibTeX]
  4. T. Barani, G. Pastore, A. Magni, D. Pizzocri, P. Van Uffelen, and L. Luzzi. Modeling intra-granular fission gas bubble evolution and coarsening in uranium dioxide during in-pile transients. Journal of Nuclear Materials, 538:152195, 2020.[BibTeX]
  5. T. Barani, G. Pastore, D. Pizzocri, D. A. Andersson, C. Matthews, A. Alfonsi, K. A. Gamble, P. Van Uffelen, L. Luzzi, and J. D. Hales. Multiscale modeling of fission gas behavior in u3si2 under LWR conditions. Journal of Nuclear Materials, 522:97–110, 8 2019. doi:10.1016/J.JNUCMAT.2019.04.037.[BibTeX]
  6. T. Barani, D. Pizzocri, F. Cappia, L. Luzzi, G. Pastore, and P. Van Uffelen. Modeling high burnup structure in oxide fuels for application to fuel performance codes. part I: High burnup structure formation. Journal of Nuclear Materials, 539:152296, 2020. doi:10.1016/j.jnucmat.2020.152296.[BibTeX]
  7. Tommaso Barani, Davide Pizzocri, Fabiola Cappia, Giovanni Pastore, Lelio Luzzi, and Paul Van Uffelen. Modeling high burnup structure in oxide fuels for application to fuel performance codes. Part II: Porosity evolution. Journal of Nuclear Materials, 563:153627, 5 2022. doi:10.1016/J.JNUCMAT.2022.153627.[BibTeX]
  8. S. Brémier and C. T. Walker. Radiation-enhanced diffusion and fission gas release from recrystallised grains in high burn-up \hbox \UO\_\2\ nuclear fuel. Radiation Effects and Defects in Solids, 157:311–322, 2002. URL: https://www.tandfonline.com/doi/abs/10.1080/10420150213000, doi:10.1080/10420150213000.[BibTeX]
  9. F. Cappia, D. Pizzocri, A. Schubert, P. Van Uffelen, G. Paperini, D. Pellottiero, R. Macián-Juan, and V. V. Rondinella. Critical assessment of the pore size distribution in the rim region of high burnup UO$_2$ fuels. Journal of Nuclear Materials, 480:138–149, 11 2016. doi:10.1016/J.JNUCMAT.2016.08.010.[BibTeX]
  10. Nathan Capps, Larry Aagesen, David Andersson, Oliver Baldwin, W. Cade Brinkley, Michael W.D. Cooper, Jason Harp, Stephen Novascone, Pierre-Clément A. Simon, Christopher Matthews, and Brian D. Wirth. Empirical and mechanistic transient fission gas release model for high-burnup loca conditions. Journal of Nuclear Materials, 584:154557, 2023. URL: https://www.sciencedirect.com/science/article/pii/S0022311523003240, doi:https://doi.org/10.1016/j.jnucmat.2023.154557.[BibTeX]
  11. M.W.D. Cooper, C. Matthews, and D.A. Andersson. The role of irradiation-enhanced interstitial diffusion in over-pressurizing fission gas bubbles in UO$_2$. Journal of Nuclear Materials, 2024.[BibTeX]
  12. G. Ducros, Y . Pontillon, and P.P. Malgouyres. Synthesis of the VERCORS experimental program: separate-effect experiments on fission product release, in support of the PHEBUS-FP programme. Annals of Nuclear Energy, 61:75–87, 2013.[BibTeX]
  13. U. Gösele. Concentration dependence of rate constants for diffusion- or reaction-controlled void-point-defect reactions. Journal of Nuclear Materials, 78:83–95, 11 1978. doi:10.1016/0022-3115(78)90507-X.[BibTeX]
  14. R. O.A. Hall, M. J. Mortimer, and D. A. Mortimer. Surface energy measurements on UO$_2$ — a critical review. Journal of Nuclear Materials, 148:237–256, 5 1987. doi:10.1016/0022-3115(87)90017-1.[BibTeX]
  15. P. Hermansonn and A.R. Massih. An effective method for calculation of diffusive flow in spherical grains. Journal of Nuclear Materials, 304:204–211, 2002.[BibTeX]
  16. S. Kashibe, K. Une, and K. Nogita. Formation and growth of intragranular fission gas bubbles in UO$_2$ fuels with burnup of 6-83 GWd/t. Journal of Nuclear Materials, 206:22–34, 1969.[BibTeX]
  17. T. Kogai. Modelling of fission gas release and gaseous swelling of light water reactor fuels. Journal of Nuclear Materials, 244:131–140, 1997.[BibTeX]
  18. Y.-H. Koo, B.-H. Lee, and D.-S. Sohn. Analysis of fission gas release and gaseous swelling in UO$_2$ fuel under the effect of external restraint. Journal of Nuclear Materials, 280:86–98, 2000.[BibTeX]
  19. K. Lassmann, C.T. Walker, J. van de Laar, and F. Lindström. Modelling the high burnup UO$_2$ structure in LWR fuel. Journal of Nuclear Materials, 226:1–8, 1995.[BibTeX]
  20. B.J. Lewis. Fission product release from nuclear fuel by recoil and knockout. Journal of Nuclear Materials, 148:28–42, 1987.[BibTeX]
  21. P. Lösönen. On the effect of irradiation-induced resolution in modelling fission gas release in UO\textsubscript 2 LWR fuel. Journal of Nuclear Materials, 496:140–156, 2017.[BibTeX]
  22. A.R. Massih and K. Forsberg. Calculation of grain boundary gaseous swelling in UO$_2$. Journal of Nuclear Materials, 377:406–408, 2008.[BibTeX]
  23. C. Matthews, R. Perriot, M.W.D. Cooper, C.R. Stanek, and D.A. Andersson. Cluster dynamics simulations of xenon diffusion during irradiation in UO$_2$. Journal of Nuclear Materials, 540:152326, 2020. URL: https://www.sciencedirect.com/science/article/pii/S0022311520301884, doi:https://doi.org/10.1016/j.jnucmat.2020.152326.[BibTeX]
  24. L. Noirot. MARGARET: a comprehensive code for the description of fission gas behavior. Nuclear Engineering and Design, 241:2099–2118, 6 2011. doi:10.1016/J.NUCENGDES.2011.03.044.[BibTeX]
  25. M. Oguma. Cracking and relocation behavior of nuclear-fuel pellets during rise to power. Nuclear Engineering and Design, 76(1):35–45, 1983. doi:10.1016/0029-5493(83)90045-6.[BibTeX]
  26. D. R. Olander. Fundamental aspects of nuclear reactor fuel elements. Technical Information Center, Energy Research and Development Administration, 1976.[BibTeX]
  27. D. R. Olander and D. Wongsawaeng. Re-solution of fission gas - A review: Part I. Intragranular bubbles. Journal of Nuclear Materials, 354:94–109, 2006.[BibTeX]
  28. D.R. Olander and P. Van Uffelen. On the role of grain boundary diffusion in fission gas release. Journal of Nuclear Materials, 288:137–147, 2001.[BibTeX]
  29. 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, 256:75–86, 2013.[BibTeX]
  30. G. Pastore, D. Pizzocri, S. R. Novascone, D. M. Perez, B. W. Spencer, R.L. Williamson, P. Van Uffelen, and L. Luzzi. Modelling of transient fission gas behaviour in oxide fuel and application to the BISON code. In Enlarged Halden Programme Group Meeting, Røros, Norway, September 7-12, volume. 2014.[BibTeX]
  31. D Pizzocri, G Pastore, T Barani, A Magni, L Luzzi, P Van Uffelen, SA Pitts, A Alfonsi, and JD Hales. A model describing intra-granular fission gas behaviour in oxide fuel for advanced engineering tools. Journal of Nuclear Materials, 2018.[BibTeX]
  32. D. Pizzocri, T. Barani, and L. Luzzi. SCIANTIX: a new open source multi-scale code for fission gas behaviour modelling designed for nuclear fuel performance codes. Journal of Nuclear Materials, 532:152042, 4 2020. doi:10.1016/J.JNUCMAT.2020.152042.[BibTeX]
  33. D. Pizzocri, G. Pastore, T. Barani, E. Bruschi, L. Luzzi, and P. Van Uffelen. Modelling of Burst Release in Oxide Fuel and Application to the \mbox TRANSURANUS Code. In $\mathrm 11^th$ International Conference on WWER Fuel Performance, Modelling and Experimental Support, Varna, Bulgaria, September 26-October 3. 2015.[BibTeX]
  34. 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, 478:333–342, 2016.[BibTeX]
  35. J. Rest. The effect of irradiation-induced gas-atom re-solution on grain-boundary bubble growth. Journal of Nuclear Materials, 321:305–312, 2003.[BibTeX]
  36. E. Rothwell. The release of Kr$^85$ from irradiated uranium dioxide on post-irradiation annealing. Journal of Nuclear Materials, 5():241–249, 1962.[BibTeX]
  37. E. Sartori, J. Killeen, and J. A. Turnbull. International Fuel Performance Experiments (IFPE) Database. OECD-NEA, 2010, available at http://www.oecd-nea.org/science/fuel/ifpelst.html.[BibTeX]
  38. W. Setyawan, M. W. D. Cooper, K. J. Roche, R. J. Kurtz, B. P. Uberuaga, D. A. Andersson, and B. D. Wirth. Atomistic model of xenon gas bubble re-solution rate due to thermal spike in uranium oxide. Journal of Applied Physics, 124(7):075107 (11 pages), August 2018. \url https://doi.org/10.1063/1.5042770.[BibTeX]
  39. Pierre-Clément A. Simon, Larry Kenneth Aagesen, Jr., Nathan Capps, Michael W. D. Cooper, Kyle A. Gamble, Logan H. Harbour, Christopher Matthews, Stephen R. Novascone, Daniel Schwen, and Brian Wirth. Compare predictions of transient fission gas release by empirical and mechanistic models to experiments in high burnup UO$_2$ fuel. Tech. Rep. INL/RPT-23-75026, Idaho National Laboratory, Idaho Falls, ID United States, 9 2023. URL: https://www.osti.gov/biblio/2203701, doi:10.2172/2203701.[BibTeX]
  40. Pierre-Clément A. Simon, Kyle A. Gamble, Arianna Pagani, Ian T. Ferguson, Daniel Schwen, Logan H. Harbour, Larry Kenneth Aagesen Jr, Stephen R. Novascone, Nathan Capps, Michael W. D. Cooper, Christopher Matthews, and David Andersson. Deployment of BISON models of fuel restructuring at high burnup and related fission gas behavior in UO$_2$. Technical Report INL/RPT-24-05038, Idaho National Laboratory, 9 2024. URL: https://www.osti.gov/servlets/purl/2472822/, doi:10.2172/2472822.[BibTeX]
  41. M.V. Speight. A calculation on the migration of fission gas in material exhibiting precipitation and re-solution of gas atoms under irradiation. Nuclear Science and Engineering, 37:180–185, 1969.[BibTeX]
  42. M.V. Speight and W. Beere. Vacancy potential and void growth on grain boundaries. Metal Science, 9:190–191, 1975.[BibTeX]
  43. J Spino, A D Stalios, H Santa Cruz, and D Baron. Stereological evolution of the rim structure in PWR-fuels at prolonged irradiation: Dependencies with burn-up and temperature. Journal of Nuclear Materials, 354:66–84, 2006. doi:10.1016/j.jnucmat.2006.02.095.[BibTeX]
  44. S. Torquato. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Volume 82. Springer, 2013. ISBN 4420767936. arXiv:1406.6401, doi:10.1016/j.camwa.2013.03.019.[BibTeX]
  45. J.A. Turnbull. The distribution of intragranular fission gas bubbles in uo2 during irradiation. Journal of Nuclear Materials, 38(2):203 – 212, 1971. URL: http://www.sciencedirect.com/science/article/pii/0022311571900444, doi:10.1016/0022-3115(71)90044-4.[BibTeX]
  46. K. Une and S. Kashibe. Fission gas release during post irradiation annealing of BWR fuels. Journal of Nuclear Science and Technology, 27:1002–1016, 1990.[BibTeX]
  47. M. S. Veshchunov and V. I. Tarasov. Modelling of irradiated UO\textsubscript 2 fuel behaviour under transient conditions. Journal of Nuclear Materials, 437(1-3):250–260, 2013. doi:10.1016/j.jnucmat.2013.02.011.[BibTeX]
  48. M.S. Veshchunov. Modelling of grain face bubbles coalescence in irradiated UO$_2$ fuel. Journal of Nuclear Materials, 374:44–53, 2008.[BibTeX]
  49. R.J. White. The development of grain-face porosity in irradiated oxide fuel. Journal of Nuclear Materials, 325:61–77, 2004.[BibTeX]
  50. R.J. White, R.C. Corcoran, and J.P. Barnes. A Summary of Swelling Data Obtained from the AGR/Halden Ramp Test Programme. Technical Report R&T/NG/EXT/REP/0206/02, British Nuclear Fuels Ltd., 2006.[BibTeX]
  51. R.J. White and M.O. Tucker. A new fission-gas release model. Journal of Nuclear Materials, 118(1):1–38, 1983. doi:10.1016/0022-3115(83)90176-9.[BibTeX]
  52. L. Yang and B. Wirth. An improved xenon equation of state for nanobubbles in UO$_2$. Journal of Nuclear Materials, 572:154089, 2022. URL: https://www.sciencedirect.com/science/article/pii/S0022311522005700, doi:https://doi.org/10.1016/j.jnucmat.2022.154089.[BibTeX]
  53. G. Zullo, D. Pizzocri, and L. Luzzi. The SCIANTIX code for fission gas behaviour: status, upgrades, separate-effect validation, and future developments. Journal of Nuclear Materials, 587:154744, 12 2023. doi:10.1016/J.JNUCMAT.2023.154744.[BibTeX]