A model for fission gas release and gaseous swelling of the uranium dioxide fuel coupled with the FALCON code

A comprehensive model GRSW-A was developed to analyse the processes of fission gas release, gaseous swelling and microstructural evolutions in the uranium dioxide fuel during base irradiation and under transient conditions. The GRSW-A analysis incorporates a number of models published in open literature, as well as some original models that were already published by the authors elsewhere. Consequently, only the most prominent aspects of GRSW-A and its coupling with the FALCON fuel behaviour analysis and licensing code are described in this paper. The analysis of fuel behaviour in the REGATE experiment is presented, which includes the base irradiation of the fuel segment in a PWR to a burn-up of about 50 MWd/kgU, which was followed by a power ramp in the SILOE research reactor. Besides, the generalized data on fission gas release (FGR) in PWR fuel during the base irradiation up to a burn-up of about 70 MWd/kgU is interpreted using coupled FALCON and GRSW-A. Moreover, a mechanistic interpretation of the published data for pellet swelling during the base irradiation up to a burn-up of 100 MWd/kgU is put forward. In all the cases, the coupled FALCON/GRSW-A analysis has shown the improved prediction capability compared to the original FALCON MOD01, which is achieved due to the account for the mutual effect of thermal and, in particular, high-burn-up-assisted mechanisms of fission gas release and swelling under steady-state and transient conditions. 1 Introduction An accurate prediction of Fission Gas Release (FGR) and gaseous swelling in the uranium dioxide fuel is proven to be an important part of the analysis of fuel reliability and safety. Evidently, one of the possible ways to account for the corresponding phenomena is to couple an integral fuel behaviour code with appropriate models. In spite of that this topic has been addressed by many researchers since long ago ( Olander, 1976 ), the interest to it still remains in force ( Tonks et al., 2010 ). The FALCON code ( Rashid et al., 2004 ) is currently the main computational tool used at the Paul Scherrer Institut (PSI), Switzerland, for the numerical analysis of Light-Water Reactor (LWR) fuel behaviour. The algorithms used in FALCON make it possible to analyse LWR fuel behaviour under both steady-state and transient conditions. The combination of these capabilities is important for predicting the fuel rod behaviour during a Design Basis Accident (DBA) taking into account the fuel rod conditions after the base irradiation. However, until recently a shortcoming of the code was the absence of a comprehensive model of microstructural fuel behaviour with the particular emphasis on the fission gas kinetics and gaseous fuel swelling in the normal and restructured fuel under steady-state irradiation as well as in slow and fast thermal transients. A Gas Release and gaseous SWelling Advanced (GRSW-A) model by Khvostov (2009) was developed as an attempt to combine the detailed dynamic rate theory-based approach of the GRASS-SST model ( Rest, 1978 ) and the insights of the model by White and Tucker (1983) with the capability to account for the high-burn-up related phenomena for the base irradiation, power ramps and fast thermal transients. The main attention was paid to the prediction of the impact of the corresponding processes on the thermo-physical and mechanical behaviour of the fuel-stack and cladding of the fuel rods, as well as efficiency of the numerical solutions allowing for the application of the fully coupled finite-element FALCON code and GRSW-A model to the problems with detailed presentation of the fuel rod geometry and irradiation history. Consequently, a research activity has been performed at PSI aiming at the strong coupling of the FALCON code with the GRSW-A model ( Khvostov, 2009 ). After coupling, the modified code has been subjected to the verification and validation. As a part of this work, the essentials of the GRSW-A model and approach used for its incorporation into the integral analysis of the FALCON code are described below. Furthermore, the important examples of application of coupled FALCON/GRSW-A to the analysis of LWR high-burn-up fuel rod behaviour are presented. The results of the FALCON/GRSW-A application to the analysis of boiling water reactor (BWR) fuel behaviour during the reactivity-initiated accident (RIA) have been recently reported in another paper ( Khvostov et al., 2009 ). 2 GRSW-A model outline The GRSW-A model predicts the macroscopic characteristics of the fuel state by the analysis of mesoscopic and microscopic processes occurring in fuel material. The main variables define the size distribution of local porosity, pellet swelling and fission gas release into the fuel rod free volume. The main object of the model is a fuel grain approximated by a sphere and its surface approximated by 14 round faces. The dynamic balance of fission gas, generated in the process of fission, is analysed in consideration of a number of the diffusive and irradiation-induced processes in the fuel grains (intragranular processes) and on the grain boundaries (intergranular processes). The intragranular processes include generation and dissolution of the fission gas mono-atoms; their diffusion inside fuel grain; coalescence of mono-atoms resulting in formation of bubbles; growth of bubbles due to their coalescence as well as due to the flux of the irradiation-induced point defects (vacancies and interstitials) from the fuel matrix towards the bubbles; and finally the arrival of the fission gas bubbles at the grain boundary. Using this gas arrival flux as a boundary condition, the intergranular part of the analysis considers formation of the gas clusters on the grain surface; their evolution resulting in the formation and growth of the closed gas pores; and finally the conversion of the closed gas pores to the open (vented) pores resulting in the fission gas release into the fuel rod free volume. In addition to the calculation of FGR, the growth of the intragranular bubbles as well as the intergranular pores determines the corresponding component of the total fuel swelling. It is to be noted that GRSW-A includes a number of models taken from open literature as well as some original models developed by the authors and published in other papers. Therefore, only the most important aspects of the GRSW-A model as well as the main issues of the computational realization and coupling with the FALCON code are addresses in the following two chapters of this paper, while the whole set of equations is summarized in Appendices A and B , presenting the rate equations and closure relations of the models for intragranular and intergranular processes, respectively. 2.1 Fuel restructuring Two types of the fuel restructuring are considered in the model, namely: (1) equiaxed-grain growth (aggregation of grains) under high temperature and (2) intragranular polygonisation (subdivision of grains into sub-grains) taking place under a long enough irradiation at relatively low temperature. The equiaxed-grain growth generally takes place in the central region of the pellet, while the polygonisation is typical for the fuel at a pellet outer rim and known also as an appearance of the High-Burn-up Structure (HBS). These two models deal basically with the response of the grain structure to different regimes of irradiation temperature (e.g., base irradiation, power ramps and fast thermal transients in the pellet centre, mid-radius and periphery), and strongly influence other parts of the GRSW-A model. Therefore, there is a merit in starting the model presentation with a description of this part of the analysis. The fuel grain and sub-grain dimensions as well as the fraction of the restructured fuel are the main output of the fuel restructuring model. The approach used by the GRSW-A model for description of the fuel grains and sub-grains is schematically shown in Fig. 1 . To describe the mutual effects of the polygonisation and equiaxed-grain growth the two independent arrays of boundaries are introduced: boundaries of the grains and of the sub-grains. At the beginning of irradiation, these boundaries are geometrically the same. The intragranular polygonisation results in the formation of more and more sub-grains, i.e. small fuel domains where all the types of intragranular processes and objects occur. The boundaries of these domains serve as the perfect sinks for the fission gas, which is generated in the fuel bulk and, further, migrates by a number of mechanisms (see the outline of the intragranular part of the model in Section 2.2 ). GRSW-A employs also a strong simplification that the sub-grain boundaries, formed during the polygonisation, have no capacity to accumulate fission gas that arrives at them from the fuel bulk. Therefore, an assumption is accepted about instantaneous transition of the gas from the sub-grain boundaries (the dashed lines in Fig. 1 ) to the grain boundaries (the solid lines in Fig. 1 ). The grain polygonisation model by Khvostov et al. (2005) , which is utilized in GRSW-A, impacts mainly the intragranular processes (in particular, intragranular gas loss), through the decrease in the average sub-grain size, d s-gr (see Eqs. (17–20) ). The fraction of the polygonised fuel, ɛ s-gr , is calculated using an expression based on the generic equation by Kolmogorov (1937) : (1) ε s − g r ( t ) = 1 − exp − k d b x ( t ) b 0 3 where ɛ s-gr ( t ) is the local volumetric fraction of the polygonised fuel at time t ; b x is the effective local burn-up; b 0 is the reference burn-up value; k d is the factor accounting for the effects of the as-fabricated fuel structure. Conventionally, the formulation by Kolmogorov (1937) just mentioned is applicable to any restructuring phenomenon, and was already used by other researchers to describe the polygonisation process in the UO 2 fuels ( Kinoshita, 1999 ). The value of ɛ s-gr ( t ) is used to evaluate the effective sub-grain size d s-gr (see Table A.1 ). The equiaxed-grain growth is included into the intergranular part of the GRSW-A analysis. This process results in the decrease of both the specific grain-boundary surface area (the solid lines in Fig. 1 ) and the volumetric concentration of the grains. The model assumes that the equiaxed-grain growth has no effect on the characteristics of the network of the sub-grains which are perfect sinks for the intragranular gas. The equiaxed-grain growth at high temperature is interpreted in the GRSW-A model as the aggregation of the fuel grains, which results in the additional FGR solely due to the assume release of the gas from the disappearing grain boundaries (see the dashed lines in the part of Fig. 1 showing the fuel state after high-temperature restructuring). The dynamics of the equiaxed-grain growth is analysed using the classical approach ( Olander, 1976; Ainscough et al., 1974 ), with a modification to account for the blockage of the grain-merging on the open surfaces: (2) R g • = k 0 exp − E g . g r o w t h k T ⥄ ⥄ 1 R g − 1 R g . max 1 − S F V o p e n S F V t o t − 1 (3) R g . max = k 1 exp − k 2 T where R g is the grain radius; k 0 , k 1 , k 2 are the empirical coefficients; E g.growth is the activation energy for the equiaxed-grain growth; R g.max is the maximum grain radius for the current temperature (i.e. the grain radius rate is set equal to zero if R g > R g.max ); k is the Boltzmann constant; ( S F / V ) open and ( S F / V ) tot are the open and total specific areas of the grain boundaries, respectively. The expression used for the fractional volume of the fuel affected by the high-temperature equiaxed-grain growth, ɛ g.growth , is as follows: (4) ε g . g r o w t h = Δ V g V g where Δ V g is the increase in the grain volume due to the equiaxed-grain growth calculated using Eqs. (2 and 3) for the grain radius growth rate; V g is the current grain volume. Both the equiaxed-grain growth and polygonisation are supposed to entail transformation of the intergranular pores from the lenticular-shaped to spherical ones, as described in Section 2.3 . It is important that the model takes into account the mutual effect of the two types of restructuring, after they have affected the same material element. For example, a fuel element at the mid-radius of the pellet can build up a considerable amount of the polygonised fuel during base irradiation and then be subjected to high temperature during a power ramp. The GRSW-A model copes with such situations by using a variable volumetric fraction, ɛ , to characterize an overall degree of fuel restructuring ( Khvostov et al., 2005 ): (5) 1 − ε = ∏ i ( 1 − ε i ) = ( 1 − ε s − g r )   ( 1 − ε g . g r o w t h ) where i is the index determining the fuel restructuring type, which currently can be either the low-temperature polygonisation ( ɛ 1 ≡ ɛ s-gr ), or the high-temperature equiaxed-grain growth ( ɛ 2 ≡ ɛ g.growth ). 2.2 Main models for intragranular processes The part of the GRSW-A model dealing with the group of intragranular processes considers generation of the fission gas atoms in the process of fission, diffusion of the gas atoms in the fuel lattice, nucleation of the diatomic bubbles, formation and growth of the larger (gaseous) bubbles, resolution of gas atoms from the bubbles caused by their interaction with fission fragments, coalescence of the bubbles due to the migration by different mechanisms, as well as the intragranular gas loss due to arrival at the grain boundaries. Below, in this and next sections, we describe the main elements of the model for intragranular fuel behaviour and its relations with other parts of GRSW-A. A set of the main equations is presented in Appendix A . The dynamic analysis of the intragranular processes is closely linked to the calculation of the smeared concentrations of vacancies and interstitials of the fuel lattice. This calculation is necessary for evaluating the growth of both intragranular bubbles and intergranular pores, because this growth is controlled by the flux balance of the uranium vacancies and interstitials towards the bubbles and pores. The thermo-dynamical equilibrium concentrations of Schottky-type vacancies and interstitials in the uranium and oxygen sub-lattices of the UO 2 + x , as function of temperature and O/U-ratio (2 + x ), are calculated using the published model ( Griesmeyer and Ghoniem, 1979 ). The concentrations of the irradiation-induced point defects (vacancies and interstitials) are obtained from integration of the corresponding rate equations (see Table A.1 ) with respect to time, using the homogeneous dynamic rate theory ( Griesmeyer et al., 1979 ). In these equations, the irradiation-induced point defects are treated as the Frenkel pairs remaining after the displacement spikes caused by the interaction of the fission fragments with the atoms of fuel lattice. These equations also include vacancy-interstitial recombination, trapping of vacancies by the intragranular gaseous bubbles, as well as precipitation at grain boundaries and dislocations. The vacancy resolution from the intragranular as-fabricated pores – due to interaction with the fission fragments – is also taken into account, which allows the model to predict the macroscopic effect of fuel densification. Finally, the concentration of uranium vacancies (which are the slowest of the two types of the point defects considered, and thus kinetically control the bubble growth) is calculated as the sum of the concentration of the vacancies that are generated as uranium-Frenkel-pairs, vacancies ejected from the pores by fission fragments, and Schottky-type thermally induced vacancies. It should be noted that the irradiation-induced uranium vacancies and interstitials are calculated independently on the thermal ones, which is a simplification of the model. The bubble-size distribution in GRSW-A is sampled, as suggested by Griesmeyer et al. (1979) . A bubble-size class is determined by the number of the gas atoms in the bubble. The radial distribution of the gas mono-atoms in the grain is calculated in the model, while only one grain-smeared concentration is evaluated for the diatomic bubbles and all other classes of the gas bubbles. The state of the diatomic and small gaseous bubbles is defined by their smeared concentrations only (no size evolution), since they are considered as ‘solid’ spheres, the volumes of which are defined by the number of atoms in the bubble. For the larger intragranular gaseous bubbles, the current state of each bubble-size class is specified by the two independent variables: the smeared concentration, B i , and the first-order momentum of the statistical bubble-size distribution: M i = N i = c o n s t ∫ 0 ∞ r ∂ B i / ∂ r d r , where i is the bubble-size class; r is the bubble radius and B i is the concentration. Each bubble-size class is considered for the fixed number of gas atoms in the bubbles, N i . The current effective bubble radius is calculated as: 〈 R i 〉 ≡ R i = M i / B i . The concentrations of the intragranular bubbles are calculated using the model that considers the nucleation of the new bubbles, gas mono-atoms trapping, coalescence of the bubbles due to the random and biased motion as well as irradiation-induced resolution. The equation of Chandrasekhar (1943) for the rate of the bubbles coalescence due to the random motion has been slightly modified on the basis of the conclusions of Garcia et al. (2006) to take into account the restrained interactions between the intragranular bubbles due to the bubble over-pressure. The tuning of the empirical factors, included into the modified equation, has been implemented using the experimental data for the swelling and FGR during the high-temperature annealing of the pre-irradiated fuel samples ( White, 2000a; Zacharie et al., 1998a,b ), which is discussed in Section 4.1.3 . For the coalescence caused by the random motion of the bubbles, the expression used by GRSW-A is: (6) K i j = 4 π ( R i + R j )   ( δ i D i + δ j D j ) B i B j where K ij is the smeared volumetric rate of the bubble coalescence for the classes i and j ; B i and B j are the concentrations of the intragranular bubbles for the classes i and j ; R i and R j are the effective radii of the interacting species; D i and D j are the diffusion coefficients of the interacting bubbles; δ i and δ j are the empirical coefficients set equal to unity for the small ‘solid’ bubbles and to an essentially small value (see Table 1 ) for the larger gaseous bubbles. The coalescence of two bubbles of classes i and j leads to the formation of a bubble containing N g = N i + N j gas atoms. For the condensed sampling, the number of atoms in the new bubble, N g , can belong to neither of the existing bubble classes. Given N g falls in the range between the two adjacent classes, k and k + 1 (i.e. N k ≤ N g ≤ N k + 1 ), the bubble concentrations for the involved classes are calculated by the following expressions: (7) B k • K i j = f k ( N g ) K i j (8) B k + 1 • K i j = f k + 1 ( N g ) K i j (9) B i • K i j = − K i j (10) B j • K i j = − K i j where B l • K i j is the generalized designation used hereinafter for the bubble concentration rate of class l , caused by the generalized process K (e.g., coalescence); K ij is the rate of the process in question; f k and f k + 1 are the weighting factors providing compliance with the overall balance condition. The weighting factors in Eqs. (7–10) are calculated as recommended by Griesmeyer et al. (1979) : (11) f k ( N g ) N k + f k + 1 ( N g ) N k + 1 = N g (12) f k ( N g ) N k 2 + f k + 1 ( N g ) N k + 1 2 = N g 2 The rate equation used for the momentum of the bubble-size distribution, M i , is: (13) M i • = B i R • + ∑ K R i ( K ) B i • K where R i ( K ) is the radius to be adopted by a bubble after its formation resulting from an intragranular process, denoted here by the lower index K . On the right-hand side of Eq. (13) , the first partial derivative with respect to time refers to the processes resulting in a bubble growth (or shrinkage) due to the diffusion of the point defects, which does not affect the bubble concentrations. The second term on the right-hand side of Eq. (13) refers to the different processes affecting the bubble concentration, namely: (1) trapping of the gas mono-atom by the bubble, (2) precipitation of the gas mono-atom on the grain boundary, (3) coalescence of two bubbles and (4) irradiation-induced resolution, the index K denoting all these processes. The bubble radius rate due to the point-defect diffusion is calculated using the corresponding equations of the fundamental diffusion-controlled reaction rate theory ( Olander, 1976 ): (14) R i • = 1 R i D V Δ C V − D I Δ C I (15a) Δ C I = C I ( i r r ) + C I U 1 − exp + Δ P i Ω k T (15b) Δ C V = C V ( i r r ) + C V U 1 − exp − Δ P i Ω k T (16) Δ P i = P i − 2 γ R i − P e x t where D V and D I are the diffusion coefficients of vacancies and interstitials, respectively; Ω is the atomic volume of UO 2 ; C V ( i r r ) and C I ( i r r ) are the smeared fractional concentrations of irradiation-induced Frenkel type vacancies and interstitials, respectively; C VU and C IU are the fractional thermo-dynamic equilibrium concentrations of Schottky-type vacancies and interstitials, respectively; Δ P i is the pressure balance across the surface of the bubble; P i is the gas pressure in the bubble; γ is the specific fuel–gas surface energy; 2 γ / R i is the capillarity pressure; P ext the external pressure exerted on the bubble. The intragranular gas loss rate L • 1 , i.e. the number of the gas atoms arriving at the (sub-) grain boundaries per unit of fuel volume and time (1/cm 3 s), is calculated as a sum of three terms, describing the gas loss rates due to: (1) gas mono-atom diffusion in the gradient of gas-atom concentration, L • 1   m − a ; (2) the bubble random migration, L • 1 r b ; and (3) the bubble biased motion in the temperature gradient, L • 1   b b : (17) L • 1 = L • 1 ⥄ m − a + L • 1 ⥄ r b + L • 1 ⥄ b b (18) L • 1 ⁡ m − a   = ⥄ − π d s − g r 2 D g ∇ C 1 ρ = d s − g r / 2 B s − g r (19) L • 1 ⁡ r b = ∑ i = 2 n b λ s − g r ( i ) ( b ) D i B i N i (20) L • 1 ⁡ b b = ∑ i = 2 n b S s − g r ( i ) ( b ) V i B i N i where d s-gr is the current sub-grain size; D g is the diffusion coefficient of gas mono-atoms; C 1 is the gas mono-atom concentration; B s-gr is the (sub-)grain concentration; λ s − g r ( i ) ( b ) is the (sub-)grain-boundary sink-strength for intragranular bubbles; D i is the diffusion coefficient of intragranular bubbles of class i ; B i is the concentration of intragranular bubbles of class i ; N i is the number of gas atoms in a bubble of class i ; V i is the bubble drift velocity calculated using the temperature gradient; S s − g r ( i ) ( b ) is the (sub-)grain cross-sectional area; n b is the number of the bubble-size classes. 2.3 Main models for intergranular processes The mechanistic model used in GRSW-A for the analysis of intergranular processes was described in details by Khvostov et al. (2003) . A set of the main equations is presented in Appendix B . To some degree, this model had originated from the approach of White (2000b) , but was adopted to simulate both normal and restructured fuel, meaning both the equiaxed-grain growth and intragranular polygonisation. A similar approach was successfully employed for the interpretation of the data on gaseous-pore behaviour and FGR in the HBS under conditions of the out-of-pile annealing ( Blair et al., 2008 ). The elementary domain used by the model, i.e. the structural unit where all modelled processes occur, is set up on the basis of the so-called rationalized TKD-geometry of the grain ( White and Tucker, 1983 ). This geometric presentation assumes that a grain consists of 14 identical right circular cones ( Fig. 2 ). The base of the cone – generally belonging to the two adjacent grains – is called a grain face, and the circumference around it is referred to as a grain edge. According to the model, the arrival of fission gas from the grain interior to the grain-boundary results in either instantaneous formation of a new gas cluster, or trapping of the gas by already existing gaseous pores. An intergranular gas cluster is a hypothesized fixed-size object allowing to probabilistically evaluate the rate of the pores formation through an implicit accounting for the fast migration of the intergranular gas, just after its arriving at the closed grain boundary (see Appendix B ). After the gas cluster formation, a number of types of intergranular interactions are considered. The trapping of the gas by the existing intergranular closed pores, generally, leads to the pore over-pressure and growth. When growing, the closed pore may geometrically get in touch with the grain edge, becoming thus an open (vented) pore. The gas contained in the open pore, is assumed to be instantaneously released into the free volume of the fuel rod. The further evolution of the vented-pore size is controlled by the effects of capillarity and external pressure and leads to the pore shrinkage. The closed pore in the normal (not restructured) fuel is assumed to have a lens shape. During the base irradiation the pore shape may evolve from the lens shape to the sphere shape, as shown in Fig. 3 , due to the assumed higher isotropy of the fuel material surrounding a pore in the restructured fuel than in the normal fuel. In high-burn-up structure, the well-known ‘stretching’ effect of the grain-boundary tension – turning the grain face pore to lens – is supposed to weaken, or disappear, after the fuel around is polygonised into small sub-grains, which are generally smaller than the pores and allowing the pore to change its form to sphere – according to the idealized approach used in GRSW-A. Similarly, after the equiaxed-grain growth, the majority of the remaining intergranular pores are assumed to be located in the grain corners, i.e. the points where four grains meet, which are more isotropic positions than the grain faces and thus again allowing the pore to change its form to sphere. It is worthwhile noting that, in the mathematical sense, the fuel restructuring is assumed to have a direct effect on the effective pore shape only, whereas the curvature radius, r f , and the projected radius, r x , remain unchanged. Another straightforward assumption of the present model is that the HBS-specific pores are predominantly formed on the grain boundary, and further stay in their close vicinity. Only a displacement of the centres of the pores from the original grain-boundary planes is allowed in the course of restructuring, as shown in Fig. 3 (C). For the closed pores, the generalized equation for the pore growth rate is given by: (21) V • n v = ( 1 − ε ) V • l e n s + ε V • s p h e r e where V nv is the closed-pore volume; ɛ is the parameter characterizing overall degree of fuel restructuring, which includes the mutual effect of the polygonisation and equiaxed-grain growth (Eq. (5) ). The pore growth is simulated as a result of the diffusion of the point defects: the bases of the corresponding theory can be found in Section “Diffusion-limited reaction” by Olander (1976) . The diffusion of vacancies and interstitials from the grain bulk and along the grain boundaries in the un-restructured and restructured fuel is presented by the first and second terms of the right-hand side of Eq. (21) , respectively. The interactions between grain-boundary objects, caused by the formation and growth of the intergranular pores, are described using the probabilistic theory. The corresponding dynamic analysis ( Khvostov et al., 2003 ) is based on the use of geometric probabilities. Specifically, a Venn-diagram is used in this calculation, as shown in Fig. 4 , which presents the zones corresponding to different simple and complex interactions considered in the model. The details of the model, in particular, cross-sections of the different regions defining probabilities of the interactions associated with these regions, could be found in Appendix B . It is to be noted that the present model does not include any mechanism resulting in a gas return from the grain boundaries back into the grain bulk, as opposite to the intragranular model, according to which the account for irradiation-induced resolution plays an important role. Such analysis would have essentially complicated the model, while it seems to be inconsistent with the model structure. Specifically, the re-coil range for the gas mono-atoms knocked out from the grain boundaries (about 1–2 × 10 −8 m, according to White and Tucker, 1983 ) is, according to the GRSW-A model assumptions, essentially less than the effective thickness of the grain boundary determined by Eq. (B.10) : ∼10 −7 m – see Fig. 11 a. After such resolution, a gas mono-atom, knocked out by the fission fragment, would not leave the effective boundary region. Moreover, an assumption of the gas resolution would make no sense for the boundaries of the sub-grains in the restructured fuel (see Fig. 1 ), since they are assumed to be the ideal conductors of the precipitating intragranular gas to the initial grain boundaries and the HBS-specific pores. Besides, according to the Nelson model for the irradiation-induced gas resolution ( Griesmeyer et al., 1979 ), used by GRSW-A, the predicted gas resolution from the relatively large intergranular pores has a reduced efficiency because of a lower density of gas atoms in the intergranular pores compared to the intragranular bubbles. 2.4 As-fabricated pores behaviour The GRSW-A model provides the analysis of the small as-fabricated pores behaviour, which is responsible for the early-in-life irradiation-induced densification and high-temperature sintering of the pellets. Specifically, the model includes the rate equations for the as-fabricated pore radii, R pi , of the pore classes considered (see Table A.1 ), using as input the data for the initial pore-size distribution in the analysed fuel. The sinterable as-fabricated pores are assumed to be immobile and homogeneously distributed in the grains. The following processes are considered: (1) The irradiation-induced resolution (injection) of vacancies from the pores due to their interaction with the fission fragments, which is described using the Turnbull model ( Griesmeyer et al., 1979 ); (2) the diffusive flow of the injected vacancies just mentioned back into the pores, along with the precipitation of these vacancies on the grain boundaries; (3) Vacancy- and interstitial-diffusion-controlled shrinkage of the sub-pressurized pores at high temperature. The large as-fabricated intergranular pores are assumed to be dimensionally stable. 2.5 Computational aspects As far as the basic numerical algorithm of the model is concerned, each integration point of a fuel element is characterized by the set of independent variables, describing the microstructural state in this point. The model receives a number of external time-dependent variables affecting the current rate equations, namely: temperature, temperature gradient, external and hydrostatic pressure, and fission rate. The impacts of all the considered processes on the rates of all the variables are calculated and the mutual effects are evaluated. The same model, without exceptions, is applied to all the integration points of the fuel elements under all possible regimes of the external variables, e.g., low- and high-burn-up in low- and high-temperature regions of the pellet. Each set of rate equations is integrated with respect to time using the portable Livermore Solver for Ordinary Differential Equations ( Hindmash and Sherman, 1983 ) with general sparse Jacobian matrices (LSODES). 3 Implementation in the FALCON code The GRSW-A model has been integrated into the FALCON fuel analysis and licensing code. The general approach to coupling of the model with the integral fuel behaviour analysis is illustrated by flowchart in Fig. 5 . The GRSW-A analysis uses variables calculated by FALCON, namely: temperature, temperature gradient, fission rate and either hydrostatic pressure (i.e., the average of the normal stress components in the integration point of the fuel element in question taken with the opposite sign), or gas pressure in the rod free volume. The present version of the coupled FALCON/GRSW-A analysis uses the gas plenum pressure as the pressure exerted on the pores and bubbles in the fuel (e.g., in Eq. (16) ), although the hydrostatic pressure can be used in the research analysis, as well. This assumption is quite adequate for the small intragranular bubbles, about 1 nm in size, for which the capillarity pressure is in the order of few GPa and largely exceeds the mechanical stresses predicted in the fuel. For the larger intergranular pores, these values may be comparable, and some constraint of the pore growth could be expected due to the compressive mechanical stresses around the pores. However, the corresponding analysis methods essentially vary from code to code, and the result of each method usually depends on a number of code-specific assumptions, let alone applicability of the calculated macroscopic stresses to the material zones around a microscopic pore. It is to be noted that all the verification and validation of the FALCON code coupled with the GRSW-A model have been performed using the mentioned simplification (i.e., using the gas plenum pressure only) and the results of the verification – partly presented in Section 4 – are basically very satisfactory. There is a number of explicit and implicit feedback effects of the GRSW-A analysis on the FALCON analysis through the following variables returned by GRSW-A to FALCON: fuel porosity distribution in the pellet affecting fuel thermal conductivity; fuel pellet swelling due to the solid and gaseous fission products (i.e., the gas mono-atoms, intragranular gaseous bubbles and intergranular pores); fuel densification and sintering, caused by the evolution of small as-fabricated pores; as well as the effects of FGR into the fuel rod free volume resulting in reduction of the fuel-cladding heat conductance and increase in the internal gas pressure. When the coupled FALCON/GRSW-A analysis is activated by the user through a command in the input file, the results of the program sub-routine corresponding to the GRSW-A model replace the ones obtained from the correlations and semi-empirical models built in the original FALCON code. Therefore, new mechanisms of the pellet swelling have been included into the FALCON code, which are caused by the build-up of fission products and the evolution of gaseous and as-fabricated porosity. The porosity is also largely associated with the analysis of heat conduction in the pellets. The local fuel swelling is expected to be isotropic. However, the impact of this swelling on macroscopic pellet expansion (or shrinkage) – let alone the fuel-stack elongation – may be anisotropic, due to the presence of the cracks, inter-pellet gaps and technological dishes that are able to accommodate some fuel when it swells. The coupled FALCON/GRSW-A analysis of the experimental data of the SCIP project ( Herranz et al., 2009 ) showed that the best agreement of the calculation with the measured residual hoop strain of the cladding after the power ramps was obtained when setting the radial and tangential components of the pellet strain exactly equal to the one-third of the volumetric swelling calculated by GRSW-A – in line with the assumption on the isotropic effect of fuel swelling on the corresponding components of the pellet strain. At the same time, no or small impact of the transient gaseous-bubble swelling had to be assumed to interpret the data for the dynamics of the cladding elongation during the same power ramps, whatever the value of fuel-cladding friction coefficient was assumed in the parametric study. For this reason, the reference model of the FALCON code ( Rashid et al., 2004 ) for steady-state irradiation-induced swelling is used for calculating the contribution to the axial fuel-stack growth in the analysis of base irradiation and transients. After optimization of the coupling algorithm, the equations for the macroscopic fuel swelling – used in the coupled FALCON/GRSW-A analysis – are as follows: (22) ε i i f = ε v f 3 (23) ε v f = P F + B 0 n 0 ν g + ε int r a (24) ε int r a = α s o l i d b ⥄ + ⥄ v g ( B 1 + ∑ i = 2 s N i B i ) ⥄ + ⥄ 4 π 3 ∑ i = s + 1 n b R i 3 B i + Δ P G . a s − f a b r i c a t e d while the equations that define the integral value of fuel porosity, P , and its components are as follows: (25) P = P G + P F , (26) P G = 4 π 3 ∑ s + 1 n b R i 3 B i + P G . a s − f a b r i c a t e d (27) P F = B n v 4 3 π r f 3 k v 0 ( 1 − ε ) + ε ⥄ + ⥄ B v n 4 3 π r f ( v n ) 3 k v 0 ⥄ + ⥄ P F . a s − f a b r i c a t e d where ε i i f are the tangential ( i = θ ) and radial ( i = r ) components of the fuel material element strain caused by swelling and densification; ε v f is the volumetric swelling assumed to be isotropic; P F is the calculated porosity due to the grain-boundary pores; B 0 is the volumetric (smeared) concentration of the grain-boundary gas clusters; n 0 is the model parameter that denotes the number of gas atoms in a grain-boundary gas cluster; v g is the volume increase due to the addition of a gas atom in the lattice; ɛ intra is the volumetric intragranular fuel swelling; α solid is the coefficient defining the rate of swelling due to the solid fission products; b is the local burn-up; B 1 is the volume-averaged concentration of gas mono-atoms in the grain; s is the number of bubble classes simulated as solid spheres; N i is the number of gas atoms in an intragranular bubble of the i -th class; B i is the concentration of intragranular bubbles of the i -th class; R i is the intragranular bubble radius; n b is the number of gaseous-bubble classes considered; Δ P G.as-fabricated is the resulting densification- and sintering-induced reduction of the as-fabricated porosity; P G is the calculated porosity caused by the intragranular gaseous pores; B nv is the volumetric concentration of the closed grain-boundary gaseous pores; r f is the curvature radius of the closed grain-boundary pore; B vn is the volumetric concentration of the vented grain-boundary pores; r f ( vn ) the curvature radius of the vented grain-boundary pore; k v 0 is the ratio of the lenticular- to spherical-pore volumes; ɛ is the model parameter characterizing the overall effect of the fuel restructuring (Eq. (1) ). Finally, the generalized rate equation for the local FGR into the free volume contains the terms linked with the intergranular processes, as described in Section 2.3 and Appendix B , namely: (28) d L 2 d t = L 2 • d i r e c t + L 2 • n 0 + L 2 • b + L 2 • b . g r o w t h + L 2 • g . g r o w t h where L 2 is the absolute gas release into the fuel rod free volume per unit of fuel volume. The lower index direct refers to the direct release of fission gas through the open surface of the grains; n 0 is used for the interactions resulting from the gas cluster formation on the closed surface of the grains; b refers to the consequences of formation of the closed pores; b.growth to the processes resulting from the pores growth/shrinkage; g.growth refers to the equiaxed-grain growth. Usually, sufficient computational efficiency of the advanced code is reached by simultaneous running of multiple instances of single-rod calculation, with FALCON/GRSW-A, on the cluster of relatively powerful computers. So far, this approach makes it possible to extensively use the coupled FALCON and GRSW-A codes at PSI for the fuel reliability justification and safety analysis, despite a high computational cost that essentially reduces the calculation speed in comparison to FALCON with the GRSW-A model switched off: typically by a factor of 2–10, depending on the required accuracy of physical solution (e.g. number of groups used for describing bubble-size distribution, characteristics of spatial mesh, etc.). At the same time, the improvement of computational efficiency of the coupled FALCON/GRSW-A code has been continuously addressed by the optimization of the numerical methods and the use of more and more powerful computers. 4 Results The GRSW-A model was subjected to comprehensive validation and verification with respect to its numerical behaviour (e.g., algorithm stability and robustness) and simulation capability for separate physical processes. After coupling of the GRSW-A model to the FALCON code, the integral analysis has been extensively verified in the part of results essentially affected by the new model, i.e. FGR, pellet swelling and cladding strain, distribution of fission gases and gaseous porosity in the pellet, etc. The current status of the validation work mentioned is presented in Table 2 . In this chapter, the main results of the model validation are outlined, first, using the analytical tests and the data of single-process experiments (Section 4.1 ). Then, the improved capabilities of the FALCON code after coupling with GRSW-A are verified by selected examples of the analysis and interpretation of the integral fuel rod behaviour (Section 4.2 ). 4.1 Validation of GRSW-A as a stand-alone model 4.1.1 Model algorithms robustness and numerical stability The extensive testing of the model had been performed before it was coupled to an integral fuel analysis code to ensure that the used algorithms are numerically stable, robust and the results are qualitatively and quantitatively sound (items 1 and 2 in Table 2 ). Specifically, a number of hypothetical cases were analysed, e.g. ones considering long-term irradiation under constant power conditions, irradiation with numerous cycling of the parameters using different periods and levels of parameter variation, as well as single transients, such as RIA ( Bibilashvili et al., 2000 ), etc. The numerical solutions for the concentration of irradiation-induced point defects and intragranular bubble radii were verified by the comparison with the analytical solutions under the assumption of quasi-steady-state. 4.1.2 Analysis of intragranular gas loss in the fuel of normal structure The published data ( Griesmeyer et al., 1979 ) for intragranular FGR from the UO 2 -based fuel, after the isothermal irradiation to a burn-up of 4.04% FIMA, was used to verify the corresponding results of GRSW-A ( Fig. 6 ). A comment is to be made that the fuel consisting of 75 wt.% of highly enriched UO 2 , mechanically mixed with 25 wt.% PuO 2 , was used in the HEDL experiment. No effect of Pu dioxide has been included in the present analysis. The model has interpreted the experimentally observed sharp increase in the intragranular FGR at a temperature exceeding 1250–1300 K, although showing a shift of the threshold, by about 80 K, to the higher values in comparison with the measurement. It should be noted that the use of the activation energies for gas-atom diffusion and self-diffusion equal to 4.6 and 4.5 eV ( Matzke, 1980; Mikhlin and Chkuaseli, 1982 ), respectively, has allowed to predict as sharp increase in intragranular FGR with temperature as measured. The discrepancy with the data of the predicted threshold temperature could have resulted from the uncertainty in the definition of the irradiation temperature, which could have had a strong impact on the fission gas behaviour, and would have been fairly expected in consideration of the peculiarity of the fuel samples used in the tests in question. 4.1.3 Analysis of fuel annealing after irradiation An important part of the model verification was based on the data from out-of-pile annealing of the pre-irradiated fuel samples ( White, 2000a; Zacharie et al., 1998a,b ). The annealing tests are quite useful for evaluation of the model credibility due to the precise knowledge of the test conditions, which ensures a modeling quality of the corresponding tests. Besides, the annealing tests simulate conditions that are fairly representative of ones taking place in the pellet centre during the post-impulse phase of the RIA, which is characterized by an essentially high level of temperature, low-temperature gradient and a relatively small effect of the irradiation-induced resolution. In that case, a classical model would have predicted a very high fuel swelling due to the intragranular porosity increase and, correspondingly, a small intragranular FGR, which would contradict to the experimental data available. The GRSW-A model copes with the modeling problem just mentioned, by introducing a semi-empirical modification into the classical equation of Chandrasekhar for the bubble coalescence rate (empirical coefficients δ i and δ j in Eq. (6) ). Qualitatively, this modification is consistent with the important experimental and theoretical findings with respect to a repulsive ability of the gaseous bubbles due to the bubble over-pressure ( Garcia et al., 2006 ). A number of tests were analysed aiming at the study of microstructural evolutions and FGR in the standard UO 2 fuel after low-temperature irradiation in an Advanced Gas Cooled Reactor to a burn-up of 9.8 and 20.8 MWd/kgU ( White, 2000a ). Different conditions were utilized in the 20 tests considered, namely: the slow and fast heat-up phase (0.5 and 25.0 °C/s, respectively), the peak temperatures varying from 1600 to 1900 °C, and the conditioning times varying form 0 to 60 min. The predicted intragranular porosity in the fuel samples after the tests of subseries K1–K4 and K5–K8 is shown in Fig. 7 compared to the corresponding measured values. The two subseries of these tests were conducted using the fuel of lower (9.8 MWd/kgU) and higher burn-up (20.8 MWd/kgU), respectively. Slow heat-up and no high-temperature conditioning were applied in the analysed cases. As seen from Fig. 7 , the calculated porosity is in a reasonable agreement with the measurement. Specifically, the calculation has simulated the experimental trend of saturation of the intragranular porosity increase with the target temperature increase, basically, due to the account for the mutual effects of the restrained coalescence and intragranular bubble precipitation on the grain boundaries (Eqs. (6 and 19) , respectively). All the results of the model verification by the Scanning Electron Microscopy (SEM) data for intragranular gaseous porosity after the out-of-pile annealing are presented in Fig. 8 . Besides, the prediction versus data for a few available experimental points obtained from the Transmission Electron Microscopy (TEM) of the high-burn-up fuel samples ( Kashibe et al., 1993 ) are also shown in Fig. 8 . The calculation seems to be in a reasonable agreement with the data, although the points corresponding to the out-of-pile annealing tests are lying, basically, on the ‘conservative’ side of the calculation-versus-measurement diagram. The possible explanation of this fact can be that the calculation accounts for the bubbles of all sizes, including very small ones (less than about 1 nm), while the experimental methods (such as SEM) usually have certain resolution limits. The calculated and measured FGR during the two annealing tests within the experimental series reported by White (2000a) – using the two different scenarios of temperature – is presented in Fig. 9 , showing very good description of these data by the calculation. Furthermore, the GRSW-A model was verified using the results of the annealing tests ( Zacharie et al., 1998a,b ), from which the dynamics of not only FGR, but also intergranular fuel swelling and average size of the intergranular gaseous pores were known. The fuel samples used in the tests were pre-irradiated in a PWR to a burn-up of 25 MWd/kgU. As seen in Fig. 10 a, the predicted rate of FGR is close to the measured one only after about the first 2 h of the transients, whereas during the initial period the predicted dynamics is considerably slower than in the measurement, when assuming the uniform temperature distribution in the specimens throughout the tests. Moreover, for temperature 1715 °C, the measured dynamics of FGR displays in the two different sub-sets of tests the two distinct modes during the primary transient phase. The differences just mentioned were assumed to be caused by the effect of temperature gradients in the fuel samples. As shown in Fig. 10 b–d – which are zooms of the first 2 h of the tests for annealing temperature 1715, 1630 and 1545 °C, respectively, – the assumption of temperature gradient within the fuel samples during the first 15 mins into the transients allows for reconciling the overall calculated and measured dynamics of FGR. The higher rates of the measured FGR during the primary transient phase in the considered range of annealing temperature can be attributed to the effect of a temperature gradient on FGR due to the intragranular gas loss caused by the biased bubble motion (Eq. (20) ). It is possible that the temperature gradient as high as one assumed in the sensitivity study ( Fig. 10 b–d) could have occurred in the samples during the initial heat-up phase, which complements a possible uncertainty in the measured temperature put forward by Zacharie et al. (1998a) as a hypothesized cause of the two different rates of FGR for annealing temperature 1715 °C. The model prediction for the increase in the intergranular-pore radius (the projected radius, r x , as defined in Fig. 3 ) and the corresponding fraction of the fuel swelling with the annealing time for the considered temperatures ( Fig. 11 ) is in a good agreement with the data ( Zacharie et al., 1998b ). The predicted pore growth, which is, probably, the main reason for the increase in the intergranular swelling, is caused by the increase in the pore gas pressure due to the increase in temperature and gas arrival from the grain interior to the pores on the grain boundaries. It is to be noted that the dynamics of the intergranular swelling ( Fig. 11 b) is not completely the same as the dynamics of the closed pore size ( Fig. 11 a), because beside of the single-pore growth the swelling is supposed to be affected by a number of complex processes, namely: formation of the new pores; coalescence of the pores resulting in the decrease of their number density; transformation of the closed pores into the vented pores and shrinkage of the vented pores at high temperature. 4.1.4 Analysis of high-burn-up structure A particular attention has been paid to verification of the model for the HBS in LWR fuels. The following published data was used for model verification: (1) data on the intragranular xenon concentration and intragranular porosity in the rim-zone of the standard LWR fuel pellets as function of local burn-up ( Lassmann et al., 1995; Manzel and Walker, 2000 ); (2) few experimental points for the total concentration of Krypton-85 in fuel disks irradiated to high burn-up at constant temperature in the range of 670–800 K ( Kinoshita et al., 2004 ); (3) data related to the effect of a grain size on the rate of the intragranular xenon depletion in the HBS published by Une et al. (2000) ; and (4) the specific fuel porosity in the pellet rim-zone ( Khvostov et al., 2005 ). A good agreement with all these data has been shown for the threshold burn-up, at which the intragranular gas depletion starts in the HBS as well as for the ‘plateau’ level that is reached by intragranular xenon concentration after the depletion process has been completed, which is shown in Fig. 12 by the solid and dashed curves compared to the corresponding experimental points. As seen from Fig. 12 , the scatter in the measured intragranular gas concentration – particularly, during the depletion phase – can be attributed to the effect of the local fuel temperature on the dynamics of fuel polygonisation, taken into account by the GRSW-A model. In summary, the model seems to describe the well-known features characterizing the behaviour of HBS in the LWR pellet rim-zone, such as the reduction of intragranular gas concentration and growth of the rim-specific porosity. Yet, the prediction of the relatively high local FGR from the HBS (the dashed curves in Fig. 12 ) into the free volume has not been directly confirmed by specific data. Anyway, the FALCON capability to predict the integral FGR in the high-burn-up fuel rods of Light Water Reactors has been improved after coupling with the GRSW-A model, which was illustrated by a number of calculations including the ones presented in Section 4.2 . To a large degree this improvement has been achieved by accounting for the fission gas release from the pellet periphery. At the same time, the HBS-assisted FGR is expected to essentially depend on the fuel design and irradiation conditions. Of particular importance can be the presence of a pellet central hole, conditions of fuel-cladding bonding and contact pressure, which can be crucial for the ‘liberation’ of the gas locked in the closed gap after it has being released by the HBS. In respect to the predicted FGR from the HBS in a pellet rim, the GRSW-A model seems to stand on a rather conservative side so far, since the predicted local FGR in this region may exceed 50%. 4.2 Validation of coupled FALCON and GRSW-A 4.2.1 Analysis of steady-state FGR in the AREVA case of FUMEX II The data on FGR in a peak-power fuel rod of PWR 15 × 15 design, during the normal operation in a power reactor was made available in the framework of the IAEA research programme on Fuel Modeling at Extended Burn-up FUMEX-II ( OECD/NEA Database, FUMEX-II Cases, Simplified case 2d ). The calculation versus measurement is presented in Fig. 13 . All the FGR models available in the coupled FALCON/GRSW-A code have been validated against these data. The considerable improvement can be seen in the results provided by the new model compared to all the other models of the original FALCON code (MOD01 Update 29). The calculated radial distribution of the retained Xe in the pellet at the fuel-stack mid-plane is shown in Fig. 14 compared to the total Xe generation for different burn-up levels. The distance between the curves for generation and retention – along the vertical axis – indicates the absolute intragranular FGR ( Fig. 14 a) and total FGR ( Fig. 14 b) for the corresponding radial positions. The presented diagrams, thus, allow for finding out how the FGR was predicted using the GRSW-A model. Specifically, it can be seen that the thermal FGR from the pellet central region is mainly responsible for the predicted dynamics of FGR at a pellet burn-up below about 45 MWd/kgU. For higher burn-up the predicted contribution of the HBS-assisted release from the outer part of the pellet gradually increases. It is to be noted that the calculated intragranular retention in the rim-zone is saturated at a level of about 0.2 wt.% ( Fig. 14 a), which is consistent with the experimental observations ( Fig. 12 ). Obviously, both mechanisms are crucial for modeling of the sharp increase in the integral FGR at high burn-up observed experimentally. 4.2.2 Analysis of pellet swelling during normal operation The burn-up-dependent fuel swelling during the base irradiation up to a high burn-up has been studied. The specification and typical irradiation history of the BWR fuel ( Ledergerber et al., 2006 ) were used for modeling. This typical irradiation history was composed by adding five hypothetical low-power cycles to the real 7-year BWR fuel history to extend the end-of-life burn-up to a value of about 100 MWd/kgU ( Fig. 15 ). The swelling calculated for this typical irradiation history by the GRSW-A model coupled to the FALCON code is in a good agreement with the data ( Fig. 16 ), which was obtained by the hydrostatic weighing 1 1 The presented values, approximating the total swelling, S tot , have been defined directly from the published experimental points for pellet-average density as follows: S tot ≈ (1 − ρ / ρ TD ) × 100%, where ρ is the measured density; ρ TD the theoretical density, which was set equal to 10.98 g/cm 3 . and quantitative ceramography of the pellets after the normal operation in different power reactors ( Spino et al., 2005 ). The calculation shows up the decrease in the intragranular fraction of the pellet-average swelling rate with burn-up, which is predicted due to the account for the HBS-related depletion of the intragranular fission gas on the pellet periphery (the fuel polygonisation model). As analysis shows, the intragranular gas depletion in the HBS of the pellet-rim entails the onset and growth of the intergranular gaseous swelling due to the formation of the HBS-specific pores (the change of the pore shapes from lens to sphere). The increase in total pellet swelling rate is eventually predicted, which is consistent with the measurement by the immersion method. 4.2.3 Analysis of fuel behaviour in the integral experiment REGATE The FALCON code coupled to the GRSW-A model has been validated using the experimental data for FGR, fission product distribution in the pellets and residual cladding strain in a short rodlet, originally belonging to a specially designed segmented PWR fuel rod. The sample rod used in this analysis was, first, base-irradiated in the Gravlines 5 PWR to a pellet-average burn-up of 50 MWd/kgU, which was distributed almost uniformly over the short fuel-stack height. Afterwards, the discharged rodlet was subjected to an intensive power ramp in the SILOE research reactor. The test was carried out within the REGATE experimental programme ( Salot et al., 1998 ) and the corresponding data was made available in the framework of the IAEA research programme on Fuel Modeling at Extended Burn-up FUMEX-II ( OECD/NEA Database, FUMEX-II Cases, REGATE Case ). Specification of the important parameters used for the present calculation is given in Table 3 . The history of the Linear Heat Generation Rate (LHGR) for the tested segment is presented in Fig. 17 , indicating a linear power density during the base irradiation and the intensive power ramp performed in the research reactor. The calculated dynamics of FGR in the fuel rod during these two stages is presented in Fig. 18 compared to the experimental data for FGR after the base irradiation and after the power ramp. As seen from Fig. 18 , the FALCON code coupled with the GRSW-A model shows a good result for FGR both after the normal operation and after the slow thermal transient. Similar to the analysis presented in Section 4.2.1 , the account for the mutual effects of the thermal FGR in the pellet centre and a-thermal one on the periphery has resulted in a reasonable agreement with the data. The obtained results are illustrated by the comparison of the calculation and data from the Electron Probe Micro Analysis (EPMA) for the intragranular xenon concentration as a function of radial position in the pellet after the power ramp, see Fig. 19 . Evidently, the good agreement of the calculation with the experimental data for FGR, fission product distribution and cladding strain (see the appropriate measurement against prediction in Fig. 20 ) in the test fuel rod after the base irradiation have ensured reliable initial conditions for the code application to the power ramp analysis. Beside FGR, the analysis has been addressing an important question regarding the impact of transient gaseous swelling on fuel rod mechanical behaviour during the thermal transients. The code application to the REGATE experiment shows that a considerable impact of gaseous swelling on the cladding strain during the ramp is predicted. Therefore, transient gaseous swelling must be taken into account by the FALCON code (e.g., via coupling it with GRSW-A) in order to explain the available experimental data on the residual cladding deformation resulting from the power ramp ( Fig. 20 ). The calculated radial distribution of the gaseous porosity in the pellet before and after the power ramp in the REGATE experiment, presented in Fig. 21 , shows up the main microstructural reason for such a significant increase in the cladding strain predicted. The calculation has revealed a possibility of the formation and development of the grain-boundary pores in the pellet central region during the high-temperature operation in the research reactor, additionally to the HBS-porosity in the rim-layer formed during the long-term base irradiation, which has resulted in the predicted transient pellet swelling and more intensive Pellet-Cladding Mechanical Interaction (PCMI). 5 Conclusions A GRSW-A computer model was developed to analyse the fission gas release (FGR), gaseous swelling and microstructural evolutions in uranium dioxide fuel under the base-irradiation conditions up to high burn-up and subsequent slow or fast transients (e.g. power ramp or power pulse). The developed model predicts macroscopic characteristics of the fuel state, such as porosity, swelling and FGR, by analysing of the meso- and microscopic processes occurring in the fuel material. Specifically, the following processes are represented in the model: (1) The group of the intragranular processes, including the kinetics of point defects in the lattice and the diffusion of the gas mono-atoms, as well as nucleation, migration, coalescence, trapping, irradiation-induced resolution and growth of the gaseous bubbles. (2) The group of the grain-boundary related (intergranular) processes, namely: formation and growth of the gaseous pores resulting in the intergranular swelling and FGR into the free volume of the fuel rod. Besides, both the intra- and intergranular behaviour are modelled as closely dependent on the phenomena of intragranular fuel polygonisation in the high-burn-up structure (HBS), occurring under low-temperature irradiation, as well as on the process of the equiaxed-grain growth under higher temperature. On top of it, a special model is devoted to the behaviour of the as-fabricated intragranular pores, which results in the macroscopic effects of low-temperature irradiation-induced densification and high-temperature sintering. A research activity has been performed at PSI aiming at the coupling of the GRSW-A model to the FALCON fuel analysis and licensing code. The code modifications affect different parts of the integral fuel rod analysis, including the treatments of thermo-physical and mechanical behaviour. An improvement of the FALCON code results after coupling with the GRSW-A analysis has been shown by comparing the experimental data with the calculation of FGR in the PWR fuel rod during the irradiation up to a burn-up of about 70 MWd/kgU. The account for both mechanisms of FGR – the thermal release from the pellet bulk and a-thermal HBS-assisted release from the pellet periphery – seems to be important for the improved result obtained using FALCON/GRSW-A. The coupled FALCON/GRSW-A calculation has been successfully used for interpretation of the published data on the characteristics of the fuel pellet swelling during the base irradiation under typical LWR conditions. In particular, the calculated pellet swelling is found to be consistent with the data for a pellet burn-up up to 100 MWd/kgU. A good agreement of the FALCON/GRSW-A calculation with the fundamental experimental findings has been shown for the decrease with burn-up in the intragranular pellet swelling rate due to depletion of the intragranular fission gas on the pellet periphery subjected to the intragranular polygonisation, which entails the onset and further growth of grain-boundary bubble swelling due to the formation of the HBS-pores. The resulting tendency towards the increase in the rate of the total pellet swelling with burn-up, as predicted by the calculation, is consistent with the available experimental data for the pellet density measured by the immersion method. The comparison of the calculation with the available data of the integral experiment REGATE has shown good prediction capability of the FALCON code coupled with the GRSW-A model and adequacy of the proposed approach as applied to the sequence of base irradiation and a power ramp. The significant impact of the gaseous swelling on the residual cladding strain caused by the power ramp is clearly demonstrated. Acknowledgments The activity on coupling of the FALCON code with the GRSW-A model was partly funded by the Swiss Federal Nuclear Safety Inspectorate (ENSI) in the framework of the STARS project. Some specific studies have been supported by swiss nuclear through its expert group on fuel safety. The authors gratefully acknowledge the work of Dr. Younsuk Yun (PSI) on application of coupled FALCON/GRSW-A in fuel behaviour analysis. Appendix A See Table A.1 . Appendix B The model for intergranular processes used in the GRSW-A analysis ( Khvostov et al., 2003 ) (1) The grain face radius, according to the rationalized-TKD model presentation used for the fuel grains ( Fig. 2 ), is given by: (B.1) R g F ( t ) = 4 M t g arccos 1 − 2 M 1 / 3 R g ( t ) ⥄ ⥄ ⥄ = M = 14 ⥄ ⥄ 0.5558 R g ( t ) where M = 14 is the accepted number of the faces per a single grain, R g ( t ) the mean grain radius for the spherical presentation. (2) The surface concentration of grain faces on the notional surface simulating the total grain boundary is calculated by the following expression: (B.2) C F = 1 π R g F 2 (3) For the transition from the specific surface characteristics of the grain boundary, C , to the corresponding volumetric characteristics, B , the following general expression is used: (B.3) B = C S F V where S F / V is the specific surface area of the grain boundary. (4) The specific surface area of the total grain boundary is given by: (B.4) S F V t o t = 14 2 π R g F 2 3 4 π R g 3 The calculated area of the 14 round faces is divided by 2 because each face generally belongs to the two adjacent grains. At the same time, the specific surface area of the total grain boundary is assumed to be equal to the sum of two terms: (B.5) S F V t o t = S F V o p e n + S F V c l o s e d where the former term on the right-hand side is the open surface-to-volume ratio for the grain boundary in question, the second term equals to the specific closed surface area of the grain boundary. Note that the latter term only defines the working area for the grain-boundary processes described in the model. (5) The default correlation used in the model for the open-surface-to-volume ratio is: (B.6) S F V o p e n = 3.25 P 2.8 ⁡ P o p e n 0.5 0.65 where P is the total as-fabricated porosity of the fuel in question (vol.%); P open the volumetric open porosity (vol.%). Note that Eq. (B.6) yields the open surface-to-volume ratio in cm 2 /cm 3 . (6) The radius of the grain-boundary gas cluster used in the equations of the model is defined from the following equation: (B.7) 4 3 π r x n 0 3 = n 0 ν g where r x n 0 is the grain-boundary gas cluster radius, v g the volume ascribed to one fission gas atom, n 0 the model parameter denoting the number of gas atoms per one elementary cluster (set equal to 5 × 10 3 ). (7) The semi-dihedral angle of the grain-boundary lens-shape pores in the normal structure ( Fig. 3 ) is calculated using the following expression: (B.8) θ = arccos γ g b 2 γ where γ gb is the fuel–fuel surface energy; γ is the fuel–gas surface energy; θ is the semi-dihedral angle for intergranular pores in normal structure. (8) The projected radius of the grain-boundary pore ( Fig. 3 ), used in the equations of the model that describe pore interactions is calculated as follows: (B.9) r x = r f sin   θ where r x is the projected radius of the pore; r f is the curvature radius of the pore. (9) The lateral dimension (i.e., along the direction normal to the grain face plane) of the grain-boundary lens-shape pore in the normal structure, see Fig. 3 (A), which characterizes also the effective thickness of the grain boundary, is calculated by the following expression: (B.10) w F = 2 r f ( 1 − cos   θ ) where w F is the effective grain-boundary thickness. (10) The condition of static equilibrium of the force exerted on the pore/bubble surface is used in the model only once, namely when defining their initial sizes. The corresponding equation reads: (B.11) P g = 2 γ r f + P e x t where P ext is the current external pressure in the fuel rod free volume, as calculated by the FALCON code. Note that in the part of the model including Eq. (B.11) , there is an option to replace the external pressure by the hydrostatic one, which is calculated as the average of the normal stress components in the integration point of the fuel element in question taken with the opposite sign. (11) In line with Eq. (B.3) , the expression for the rate of gas arrival at the boundary surface due to the intragranular gas loss reads: (B.12) J g = L 1 • S F V t o t − 1 where J g is the specific surface gas generation density (atoms/cm 2 /s); L 1 • is the volumetric intragranular gas loss rate (atoms/cm 3 /s). The rate of fission gas release into the fuel rod free volume due to the direct arrival of the intragranular fission gas at the open surface of the fuel is calculated by the following expression: (B.13) L • 2 d i r e c t = J g S F V o p e n where L • 2 is the volumetric absolute gas release rate into the fuel rod free volume (atoms/cm 3 /s). (12) The following equations are derived for the process relating to the gas cluster and closed pore generation on the grain faces: (B.14) R n = J g n 0 ( 1 − S 01 − S 02 − S 03 ) 1 − ∑ i = 1 n max ( i + 1 ) S n 0 i + ∑ i = 1 n max w n ( i ) S n 0 i (B.15) R b = J g n 0 ( 1 − S 01 − S 02 − S 03 ) ∑ i = 1 n max w b ( i ) S n 0 i (B.16) S n 0 = C 0 π ( 2 r x n 0 ) 2 (B.17) S 01 = C n v π ( r x + r x n 0 ) 2 (B.18) S 02 = C F π ( R g F 2 − ( R g F − r x n 0 ) 2 ) (B.19) S 03 = C v n π ( r x ( v n ) + r x n 0 ) 2 with the weighting factors, w n ( i ) and w b ( i ) , calculated from the following conservation equations, similar to Eqs. (11 and 12) ( Griesmeyer et al., 1979 ): (B.20) w n ( i ) n 0 + w b ( i ) N a . n v = n 0 ( i + 1 ) w n ( i ) n 0 2 + w b ( i ) N a . n v 2 = n 0 2 i + 1 2 where R n is the resulting surface rate density of the gas cluster generation; R b is the surface rate density of the closed pore generation; S n 0 is the probability of a new cluster instantaneously meeting another one already existing on the surface; C 0 is the surface concentration of the gas clusters; C nv is the surface concentration of the closed pores; C vn the surface concentration of the vented pores; r x and r x ( vn ) are the projected radii of the closed and vented pores, respectively; S n 0 i is the probability ascribed to the event of instantaneous meeting of i already existing clusters (assuming that i runs over the whole logically feasible range of integers from 2 to n max = int [ N a.nv / n 0 − 1]); S 01 , S 02 and S 03 are the probabilities of new cluster to be instantaneously trapped by a closed pore, grain edge or vented pore ( Fig. 2 ), respectively; N a.nv is the current number of gas atoms per a closed pore. Finally, after a few identical substitutions of the power series on the right-hand sides of Eqs. (B.14 and B.15) , one can express the resulting rate density of gas cluster generation and the rate density of closed pore generation in the following way: (B.21) R n = J g − N a . n v R b n 0 (B.22) R b = J g ( 1 − S 01 − S 02 − S 03 ) n 0 N a . n v ( N a . n v − n 0 ) 2 S n 0 ( 1 − S n 0 ) 3 − S n 0 n max + 1 ( 1 − S n 0 ) 2 ( 1 − S n 0 ) 2 + 2 n max ( 1 − S n 0 ) + n max ( n max + 1 ) Note that the latter value does not yet include the instant interactions taking place with the closed pores just after their generation, which are subtracted from the final effective rate of the closed pore generation. Terms of the total set of rate equations related to the generation of the gas clusters are as follows: (B.23) C 0 • n 0 = R n (B.24) C n v • n 0 = R b (B.25) F n v • n 0 = R b π r x 2 (B.26) N a . n v • n 0 = J g S 01 C n v (B.27) L 2 • n 0 = J g ( S 02 + S 03 ) S F V c l o s e d where F nv is the fraction of the grain-boundary surface area covered by the closed pores. (13) The following macroscopic cross-sections, characterizing the probabilities of the events assumed to occur on the closed surface of the grain boundaries ( Fig. 4 ), are as follows: (B.28) S 1 * = C n v π ( 2 r x ) 2 , (B.29) S 2 * = C F π ( R g F 2 − ( R g F − r x ) 2 ) (B.30) S 3 * = C v n π r x + r x ( v n ) 2 (B.31) S 123 = S 1 * S 2 * S 3 * (B.32) S 12 = S 1 * ⋅ S 2 * − S 123 (B.33) S 13 = S 1 * ⋅ S 3 * − S 123 (B.34) S 23 = S 2 * ⋅ S 3 * − S 123 (B.35) S 1 = S 1 * − S 12 − S 13 − S 123 (B.36) S 2 = S 2 * − S 12 − S 23 − S 123 (B.37) S 3 = S 3 * − S 13 − S 23 − S 123 where S 1 * ,   S 2 * ,   S 3 * are the probabilities for the instantaneous interactions of the newly generated closed pore with another already existing closed pore, a grain edge and a vented pore, respectively; S 123 , S 12 , S 13 , S 23 , S 1 , S 2 , S 3 are the true probabilities of the simple and complex interactions shown in Fig. 4 . Using the above-determined macroscopic cross-sections, the following expressions are developed for the terms of the model rate equations accounting for the consequences of closed pores generation on the grain boundaries: (B.38) C n v • b = − R b ( S 1 + S 2 + S 3 + S 123 + S 12 + S 13 + S 23 ) (B.39) F n v • b = − R b π r x 2 ( S 1 + S 2 + S 3 + S 123 + S 12 + S 13 + S 23 ) (B.40) N a . n v • b = R b N a . n v S 1 C n v (B.41) L 2 • b = R b N a . n v ( S 2 + S 3 + S 123 + S 12 + S 13 + S 23 ) S F V c l o s e d (14) The rate of the projected radius of the pore, r x • , is calculated using the following equations: (B.42) V n v • = 4 3 π k v 3 r x 2 r x • = ( 1 − ε ) d V d t l e n s + ε d V d t s p h e r e (B.43) V • l e n s = 2 π r f 2 sin 2 θ 1 r f D V Δ C V − D I Δ C I + 2 π w F D s 0 Δ C V U 1 F ( λ ) (B.44) V s p h e r e • = 4 π r f 2 1 r f D V ( H B S ) Δ C V − D I Δ C I (B.45) Δ C V , I = Ω C V , I ( i r r ) + C V U , I U 1 − exp ∓ Δ P Ω k T (B.46) Δ C V U , I U = Ω C V U , I U 1 − exp ∓ Δ P Ω k T (B.47) Δ P = P g ( V , N a . n v , T ) − 2 γ f r f − P e x t where Δ P is the pressure balance across the surface of the pore; r f is the radius of pore surface curvature; θ is the semi-dihedral angle ( Fig. 3 ); C V ( i r r ) and C I ( i r r ) are the concentration of irradiation-induced vacancies and interstitials; C VU and C IU are the thermo-dynamic equilibrium concentrations of vacancies and interstitials; w F = 2 r f (1 − cos θ ) is the effective thickness of the grain boundary assumed in the model (it is set equal to the current dimension of the pores normal to the grain face plane, as shown in Fig. 3 ); F ( λ ) is the function of boundary conditions 2 2 The appearance of function F ( λ ) depends on the set of assumptions and boundary conditions that are used for the derivation of the equations describing the transport of point defects from the corresponding capture volume to the sink in question. Generally, it can be expressed in terms of the ratio of the size of the sink to the radius of the capture volume, denoted here as λ . For example, Eqs. (13.79, 82, 83 and 86) are suggested by Olander (1976) as the corresponding boundary conditions for the description of vacancy diffusion to the dislocation, which results in the simple solution: F = − ln ( λ ). Also, a sophisticated example was presented by White (2004). adopted for the vacancy surface diffusion to the pore; D V , I is the volumetric diffusivity of vacancies and interstitials; D s 0 is the surface diffusivity of vacancies (for the flat surface); D V ( HBS ) = f ( D s 0 , D v , ɛ s ) >> D v is the effective volumetric diffusivity of vacancies, assuming its significant enhancement in HBS compared to the normal structure; Ω is the atomic volume for UO 2 ; ɛ s is the relative volume of HBS (Eq. (5) ). Model equations dealing with the consequences of the diffusion-controlled growth of the grain-boundary pores are found by differentiating with respect to time of the corresponding macroscopic cross-sections (Eqs. (B.17 and B.28–B.37) ): (B.48) d S 01 d t = C n v 2 π ( r x + r x n 0 ) d r x d t (B.49) d S 1 * d t = C n v 8 π r x d r x d t (B.50) d S 2 * d t = C F 2 π ( R g F − r x ) d r x d t (B.51) d S 3 * d t = C v n 2 π ( r x + r x ( v n ) ) d r x d t (B.52) d S 123 d t = d S 1 * d t S 2 * S 3 * + S 1 * d S 2 * d t S 3 + S 1 * S 2 * d S 3 * d t (B.53) d S 12 d t = d S 1 * d t S 2 * + S 1 * d S 2 * d t − d S 123 d t (B.54) d S 13 d t = d S 1 * d t S 3 * + S 1 * d S 3 * d t - d S 123 d t (B.55) d S 23 d t = d S 2 * d t S 3 * + S 2 * d S 3 * d t - d S 123 d t (B.56) d S 1 d t = d S 1 * d t − d S 12 d t − d S 13 d t − d S 123 d t (B.57) d S 2 d t = d S 2 * d t − d S 12 d t − d S 23 d t − d S 123 d t (B.58) d S 3 d t = d S 3 * d t − d S 13 d t − d S 23 d t − d S 123 d t Using the above derivatives, one can write the terms of rate equations that account for the diffusion growth of the closed pores on the grain boundaries in the following way: (B.59) C 0 • b . g r o w t h = − C 0 d S 01 d t (B.60) C n v • b . g r o w t h = − C n v 1 2 d S 1 d t + d S 12 d t + d S 13 d t + d S 123 d t + d S 2 d t + d S 3 d t + d S 23 d t (B.61) F n v • b . g r o w t h = 1 4 d S 1 * d t + π r x 2 C n v • b . g r o w t h (B.62) C v n • b . g r o w t h = C n v d S 2 d t + 1 2 d S 12 d t (B.63) F v n • b . g r o w t h = C v n 2 π r x ( v n ) d r x ( v n ) d t + π r x ( v n ) 2 C v n • b . g r o w t h + C n v π r x 2 1 2 d S 13 d t + 1 2 d S 123 d t (B.64) L 2 • b . g r o w t h = C n v N a . n v d S 12 d t + d S 13 d t + d S 123 d t + d S 2 d t + d S 3 d t + d S 23 d t S F V c l o s e d (B.65) N a . n v • b . g r o w t h = n 0 C 0 C n v d S 01 d t ⥄ − ⥄ L 2 • b . g r o w t h + N a . n v C n v • b . g r o w t h C n v where F vn is the fraction of the grain-boundary surface area covered by the vented pores. (15) The rate of grain radius growth caused by elevated temperature is calculated using the following expressions: (B.66) R • g = k 0 exp − E g . g r o w t h k T ⥄ 1 R g − 1 R g . max 1 − S F V o p e n S F V t o t − 1 (B.67) R g . max = k 1 exp − k 2 T where k 0 , k 1 , k 2 , are the constant coefficients; E g.growth is the activation energy for the equiaxed-grain growth; R g.max is the limit grain radius for the current temperature (i.e. the grain radius rate is forcibly set equal to zero at R g > R g.max ); k is Boltzmann constant. Consequently, the terms of the model rate equations arising from the equiaxed-grain growth consideration – in terms of the grain-boundary gas- and pore-surface concentrations – are as follows: (B.68) B 0 • g . g r o w t h = C 0 d d t S F V c l o s e d = − C 0 S F V t o t R • g R g (B.69) B n v • g . g r o w t h = C n v d d t S F V c l o s e d = − C n v S F V t o t R • g R g (B.70) F n v ( B ) • g . g r o w t h = F n v d d t S F V c l o s e d = − F n v S F V t o t R • g R g (B.71) B v n • g . g r o w t h = C v n d d t S F V c l o s e d = − C v n S F V t o t R • g R g (B.72) F v n ( B ) • g . g r o w t h = F n v d d t S F V c l o s e d = − F n v S F V t o t R • g R g (B.73) L 2 • g . g r o w t h = − ( C 0 n 0 + C n v N a . n v ) d d t S F V c l o s e d = ( C 0 n 0 + C n v N a . n v ) S F V t o t R • g R g (16) The generalized rate equations of the intergranular model of GRSW-A, using the above defined terms, are as follows: (B.74) d B 0 d t = C 0 • n 0 + C 0 • b . g r o w t h S F V c l o s e d + B 0 • g . g r o w t h (B.75) d B n v d t = C n v • n 0 + C n v • b + C n v • b . g r o w t h S F V c l o s e d + B n v • g . g r o w t h (B.76) d F n v ( B ) d t = F n v • n 0 + F n v • b + F n v • b . g r o w t h S F V c l o s e d + F n v ( B ) • g . g r o w t h (B.77) d N a . n v d t = N a . n v • n 0 + N a . n v • b + N a . n v • b . g r o w t h (B.78) d B v n d t = C v n • b . g r o w t h S F V c l o s e d + B v n • g . g r o w t h (B.79) d F v n ( B ) d t = F v n • b . g r o w t h S F V c l o s e d + F v n ( B ) • g . g r o w t h (B.80) d L 2 d t = L 2 • d i r e c t + L 2 • n 0 + L 2 • b + L 2 • b . g r o w t h + L 2 • g . g r o w t h where subscript n 0 refers to the processes resulting from cluster generation on the closed surface having the specific area, denoted here as ( S F / V ) closed ; b deals with the consequences of closed pore formation; b.growth corresponds to the processes resulting from pore growth/shrinkage; g.growth relates to equiaxed-grain growth; direct means the direct release of fission gas through the open surface with the specific area amounting to ( S F / V ) open . References Ainscough et al., 1974 J.B. Ainscough B.W. Oldfield J.O. Ware Isothermal grain growth kinetics in sintered UO 2 pellets J. Nucl. Mater. 49 1974 117 128 Bibilashvili et al., 2000 Yu.K. Bibilashvili A. Medvedev G. Khvostov S. Bogatyr L. Korystine Development of the fission gas behaviour model in the START-3 code and its experimental support Proc. OECD/NEA International Seminar on Fission Gas Behaviour in Water Reactor Fuels Cadarache, France 2000 Blair et al., 2008 P. Blair G. Khvostov A. Romano C. Hellwig R. Chawla Interpretation of high-burnup fuel annealing tests J. Nucl. Sci. Technol. 45 7 2008 639 646 Chandrasekhar, 1943 S. Chandrasekhar Stochastic problems in physics and astronomy Rev. Mod. Phys. 15 1943 1 Garcia et al., 2006 P. Garcia P. Martin G. Carlot E. Castelier M. Ripert C. Sabathier C. Valot F. D’Acapito J.-L. Hazemann O. Proux V. Nassif A study of xenon aggregates in uranium dioxide using X-ray absorption spectroscopy J. Nucl. Mater. 352 2006 136 143 Griesmeyer and Ghoniem, 1979 J.M. Griesmeyer N.M. Ghoniem The response of fission gas bubbles to the dynamic behavior of point defects J. Nucl. Mater. 80 1979 88 Griesmeyer et al., 1979 J.M. Griesmeyer B.W. Ghoniem D. Okrent A dynamic intragranular fission gas behaviour model Nucl. Eng. Des. 55 1979 69 Herranz et al., 2009 L.E. Herranz I. Vallejo G. Khvostov J. Sercombe G. Zhou Insights into fuel rod performance codes during ramps: results of a code benchmark based on the SCIP project Proc. TOP Fuel 2009 (CD ROM, Paper 2188) Paris, France 2009 Hindmash and Sherman, 1983 Hindmash, C.A., Sherman, H.A., 1983. LSODES: Livermore Solver for Ordinary Differential Equations with General Sparse Jacobian Matrices ( http://www.netlib.org , 13.07.2010). Kashibe et al., 1993 S. Kashibe K. Une K. Nogita Formation and growth of intragranular fission gas bubbles in UO 2 fuels with burnup of 6–83 GWd/t J. Nucl. Mater. 206 1993 22 Khvostov et al., 2003 G. Khvostov A. Medvedev S. Bogatyr The dynamic model of grain-boundary processes in high burn-up LWR fuel and its application in analysis by the START-3 code Proc. International Conference on WWER Fuel Performance, Modeling and Experimental Support Albena-Varna, Bulgaria 2003 392 407 Khvostov et al., 2005 G. Khvostov V. Novikov A. Medvedev S. Bogatyr Approaches to modeling of high burn-up structure and analysis of its effects on the behaviour of light water reactor fuels in the START-3 fuel performance code Proc. WRFPM-2005 (CD ROM, Paper 1104) Kyoto, Japan 2005 Khvostov et al., 2008 G. Khvostov M.A.T. Zimmermann T. Sugiyama T. Fuketa On the use of the FALCON code for modeling the behaviour of high burn-up BWR fuel during the LS-1 pulse-irradiation test Proc. Int. Conf. on the Physics of Reactors (CD Rom, Paper 213) PHYSOR’08, Interlaken, Switzerland 2008 Khvostov, 2009 G. Khvostov A dynamic model for fission gas release and gaseous swelling integrated into the FALCON fuel analysis and licensing code In Proc. TOP Fuel 2009 (CD Rom, Paper 2085) Paris, France 2009 Khvostov et al., 2009 G. Khvostov M.A. Zimmermann G. Ledergerber Parametric study of fuel rod behaviour during the RIA using the modified FALCON code Proc. OECD/NEA Workshop on Nuclear Fuel Behaviour during Reactivity Initiated Accidents (CD Rom) Paris, France 2009 Kinoshita, 1999 Kinoshita, M., 1999. Mesoscopic approach to describe high burn-up fuel behaviour Enlarged Halden Project Group Meeting EHPGM-1999. Loen, Norway. Kinoshita et al., 2004 M. Kinoshita T. Sonoda S. Kitajima High burnup rim project: (III) properties of rim-structured fuel Proc. 2004 International Meeting on LWR Fuel Performance (WRFPM 2004) Orlando 2004 Kolmogorov, 1937 A.N. Kolmogorov Statistical theory of crystallization of metals Izv. Akad. Nauk SSSR Ser. Math. (Izv. Akad. Nauk SSSR, Ser. Mat; Bull. Acad. Sci. USSR. Ser. Math.) 1 1937 355 359 (in Russian) Lassmann et al., 1995 K. Lassmann C.T. Walker J. Van De Laar F. Lindstrom Modelling the high burnup UO2 structure in LWR fuel J. Nucl. Mater. 226 1995 1 Ledergerber et al., 2006 G. Ledergerber S. Abolhassani M. Limback R.J. Lundmark K.-A. Magnusson Characterization of high burnup fuel for safety related fuel testing JNST 43 2006 1006 1014 Manzel and Walker, 2000 R. Manzel C.T. Walker High burnup fuel microstructure and its effect on fuel rod performance Proc. Water Reactor Fuel Performance Meeting 2000 (WRFPM 2000) Park City 2000 Matzke, 1980 Hj Matzke Radiat. Eff. 53 1980 219 Mikhlin and Chkuaseli, 1982 E.Ya. Mikhlin V.F. Chkuaseli Gas release and swelling in oxide fuel; modeling of the kinetics of gas porosity development J. Nucl. Mater. 105 1982 223 OECD/NEA Database, 2010a OECD/NEA Database, 2010. FUMEX-II Cases. REGATE Case; http://www.nea.fr/tools/abstract/detail/nea-1696 (14.07.2010). OECD/NEA Database, 2010b OECD/NEA Database, 2010. FUMEX-II Cases. Simplified case 2d. http://www.nea.fr/tools/abstract/detail/nea-1720 (14.07.2010). Olander, 1976 Olander, D.R., 1976. Fundamental aspects of the nuclear fuel reactor elements. US Energy Research and Development Administration. Rashid et al., 2004 Rashid, Y.R., Dunham, R.S., Montgomery, R.O., 2004. FALCON MOD01: Fuel Analysis and Licensing Code – New, Technical Report ANA-04-0666 vol. 1. ANATECH Corp. Rest, 1978 Rest, J., 1978. GRASS-SSET: A Comprehensive, Mechanistic Model for the Prediction of Fission-gas Behavior in UO2-base Fuels during Steady-state and Transient Conditions NUREG/CR-0202, ANL-78-53. Salot et al., 1998 R. Salot L. Caillot P. Blanpain L.C. Bernard M.C. Grandjean Fission gas release from high burn-up fuel during power transients: experimental data for modelling IAEA TCM on High Burn-up Fuel Specially Oriented to Fuel Chemistry and Pellet Clad Interaction Nyköping 1998 Spino et al., 2005 J. Spino J. Rest W. Goll C.T. Walker Matrix swelling rate and cavity volume balance of UO 2 fuels at high burn-up J. Nucl. Mater. 346 2005 131 Tonks et al., 2010 M. Tonks D. Gaston C. Permann P. Millet G. Hansen D. Wolf A coupling methodology for mesoscale-informed nuclear fuel performance codes Nucl. Eng. Des. 2010 10.1016/j.nucengdes.2010.06.005 Turnbull and Kolstad, 2000 J.A. Turnbull E. Kolstad Investigation of radioactive and stable fission gas release behavior at the HALDEN reactor Proc. of OECD/NES International Seminar on Fission Gas Behaviour in Water Reactor Fuels Cadarache, 2000 2000 Une et al., 2000 K. Une M. Hirai K. Nogita T. Hosokawa Y. Suzawa S. Shimizu Y. Etoh Rim structure formation and high burnup fuel behavior of large-grained UO 2 fuels J. Nucl. Mater. 278 2000 54 White and Tucker, 1983 R.J. White M.O. Tucker A new fission-gas release model J. Nucl. Mater. 118 1983 1 38 White, 2000a R.J. White The growth of intra-granular bubbles in post-irradiation annealed UO 2 fuel Proc. of a Technical Committee meeting held in Windermere, United Kingdom, IAEA-TECDOC-1233 2000 91 104 White, 2000b R. White The development of grain-face porosity in irradiated oxide fuel Proc. of OECD/NEA International Seminar on Fission Gas Behaviour in Water Reactor Fuels Cadarache, France 2000 Zacharie et al., 1998a I. Zacharie S. Lansiart P. Combette M. Trotabas M. Coster M. Groos Thermal treatment of uranium oxide irradiated in pressurized water reactor: swelling and release of fission gases J. Nucl. Mater. 255 1998 85 Zacharie et al., 1998b I. Zacharie S. Lansiart P. Combette M. Trotabas M. Coster M. Groos Microstructural analysis and modeling of intergranular swelling of an irradiated UO 2 fuel at high temperature J. Nucl. Mater. 255 1998 92