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Towards the inclusion of open fabrication porosity in a fission gas release model
Journal of Nuclear Materials 466 (2015) 351e356
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Towards the inclusion of open fabrication porosity in a fission gas release model Antoine Claisse a, *, Paul Van Uffelen b a b
KTH Royal Institute of Technology, Reactor Physics, AlbaNova University Centre, 106 91, Stockholm, Sweden European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, D-76125, Karlsruhe, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 January 2015 Received in revised form 19 July 2015 Accepted 13 August 2015 Available online 18 August 2015
A model is proposed for fission product release in oxide fuels that takes into account the open porosity in a mechanistic manner. Its mathematical framework, assumptions and limitations are presented. It is based on the model for open porosity in the sintering process of crystalline solids. More precisely, a grain is represented by a tetrakaidecahedron and the open porosity is represented by a continuous cylinder along the grain edges. It has been integrated in the TRANSURANUS fuel performance code and applied to the first case of the first FUMEX project as well as to neptunium and americium containing pins irradiated during the SUPERFACT experiment and in the JOYO reactor. The results for LWR and FBR fuels are consistent with the experimental data and the predictions of previous empirical models when the thermal mechanisms are the main drivers of the release, even without using a fitting parameter. They also show a different but somewhat expected behaviour when very high porosity fuels are irradiated at a very low burn-up and at low temperature. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Fuel performance codes are an important tool for safety studies of nuclear reactors and can also be envisaged for final repository studies. One of the key points of any nuclear fuel performance code is the prediction of fission gas release (FGR). Indeed, the release of noble gases such as Xe or Kr can greatly degrade the thermal conductance of the initially He-filled gap. In addition, a high pressure can be exerted on the cladding, which can eventually lead to failure and the release of radioactive iodine and cesium. On the other hand, the gas that is not released can form bubbles, which contributes to fuel swelling. An early review of the main effects involved in fission gas behaviour can be found in Ref. [1]. The FGR is usually treated in two separate steps in fuel performance codes. First, the fission gas creation is estimated from the number of fissions (that is the linear power rating) and the fraction reaching the grain boundaries is determined by means of an effective diffusion coefficient applied in simplified models, abundantly described in the literature [2e5]. The effective diffusion coefficient can be divided into three temperature ranges [6]. At high
* Corresponding author. E-mail address: [email protected] (A. Claisse). http://dx.doi.org/10.1016/j.jnucmat.2015.08.022 0022-3115/© 2015 Elsevier B.V. All rights reserved.
temperature, the thermal diffusion is dominating. At moderate temperature, the irradiation enhanced defect concentrations drive the fission gas diffusion in addition to the thermally activated defects, whereas at low temperature the athermal part is the most important. This corresponds to radiation enhanced diffusion that is usually associated with the fission spikes [7] and is called athermal or irradiation-induced diffusion. The athermal diffusion is of prime importance for fast neutron reactors using fuels with high thermal conductivity, such as nitride fuels, where the release may not be as strong as at higher temperatures in oxide fuels but still enough to degrade the gap conductance [8e10]. In the second step, the gas at the grain boundaries is taken into consideration and can be released or stored depending on its concentration and on a release threshold that can vary with, among other properties, the temperature or the temperature gradient. The models for FGR have to remain simple enough to ensure short computational times and easy convergence while being sufficiently complex to include most of the relevant phenomena driving the fission gas, reviewed for instance in Refs. [1,7,11]. Another reason to keep the models relatively simple are the uncertainties pertaining to the model parameters such as the diffusion coefficient. The porosity has also been shown to be important [12] and several attempts have been made to try to account for it in FGR models. Several models have adopted a pure empirical expression
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for an additional release term as a function of the initial open porosity fraction [13e17]. Others (e.g. Ref. [18]) have considered the effect of open porosity resulting from the fabrication process via an empirical reduction of the effective sphere radius of the Booth sphere for diffusional release from the grains. More recently, Feng et al. have used an additional fitting parameter in the effective diffusion coefficient [19] for the intragranular part of the fission gas release model, and they considered the total fabrication porosity rather than only the open porosity fraction that is in direct contact with the free volume in the rod. Other groups have explored the percolation phenomenon [20,21] along grain boundaries, which is intuitively more appropriate when considering open porosity. Nevertheless, this seems rather hard to implement in a fuel performance code since the grains and therefore the boundaries are usually not modelled individually. Instead, a model considering the porosity as a mean-field should be developed if one wishes to use a percolation model. The current version of the TRANSURANUS code does not allow to consider the effect of the open porosity on the fission gas release fraction. In addition to the conventional so-called thermal release modelling based on diffusion theory that also can include an athermal diffusion term, it has an additional empirical term for athermal release that is only burnup dependent, and which corresponds to release due to knock-out and recoil. In this short communication, an approach taking into consideration the open porosity in the inter-granular fission gas release model of TRANSURANUS is presented. The motivations, mathematical derivation, implementation and limitations are briefly presented. In order to illustrate the range of potential application for nuclear fuels with large amounts of open porosity under various irradiation conditions, a comparison with previous models is made for the first case of the FUMEX project of the IAEA, which is an LWR rod, in addition to fast breeder reactor oxide fuel pins of the SUPERFACT experiment and fuel rods irradiated in the JOYO reactor. 2. Model Before describing the model, the nomenclature has to be defined. Porosity (P) will classically refer to the ratio of voided volume over the total volume. The fraction of this voided volume that is in contact with the open space is Open Porosity. The other fraction is Closed Porosity. For the needs of this study, the grain surface (ST) and the surface of open porosity are introduced (SP). The latter is defined as the fraction of a grain surface occupied by the open pores along the grain edges of a grain that is represented by a tetrakaidecahedra (TKD, see Appendix and Fig. 1 for details). In accordance with these definitions, any fission gas atom reaching the open porosity at the grain boundary is deemed released to the open volume (gap and plena). The model considers the conventional bubble formation and interconnection at grain boundaries that lead to tunnel formation through which gas atoms can be released when the local temperature and burnup values are high enough. The model also considers an open porosity fraction due to the fabrication process that forms an independent escape pathway for fission gas. Assuming that the flux of atoms arriving at the grain boundaries is known and given by the intragranular module of the fission gas release model that is essentially relying on effective diffusion (in a spherical grain), one needs thus to determine the fraction of the grain boundaries that is occupied by open porosity in order to estimate the direct release due to open pores. This is done in two steps. In a first step, it is necessary to derive the volume fraction of open-porosity from the total porosity that is usually provided. This presently relies on an empirical correlation [22]. In a second step, the open (fabrication) porosity is converted to a surface fraction (SP/Sgrain). For this purpose, the grains are
modelled as TKD. Coble [23] and White et al. [24] suggested this, advocating that it is the highest order polyhedron that can be used to completely fill the space, and is also used in more recent studies of fission gas release (e.g. Ref. [25]). Its surface is composed of six squares and eight regular hexagons with edges of the same length. The open porosity is represented as cylinders along the grain edges. It should be underlined that in doing so, the following additional assumptions are made: a) all the grains have the same size within a finite element, b) all porosity is localized on the grain edges, and c) all the cylinders representing the porosity have the same radius within a finite element. These assumptions limit the validity of the model for instance to fuel densities above 70%. The open porosity fraction should change during the irradiation. More precisely the fuel can undergo densification, while the creation and precipitation of fission gas along the grain boundaries can act to increase the total porosity along grain boundaries. The creation and destruction of an interconnected tunnel network of bubbles will temporarily turn closed porosity into open porosity, until the excess gas is released and the network is closed again. The burn-up, temperature and fission gas production vary at different radial positions of the fuel, making the prediction of the open porosity based upon the porosity an intricate problem. A percolation study [20,21] could address this. However, since a fuel performance code cannot model each grain, but only finite elements representing a fraction of the fuel much bigger than a grain, such an approach would have to be somehow averaged. In this study, the empirical correlation coming from the work of Song et al. [22] has been adopted for estimating the open pore fraction on the basis of the total fabrication porosity. It is valid only for fresh oxide fuels. The fraction of porosity remaining open at the end of life is sometimes known from experiments to be very low since pellets have densified and tunnel networks have closed [10], but to the best of the authors' knowledge, a better correlation does not yet exist. One might apply an empirical densification model to the open porosity resulting from the fabrication process, but this also requires more experimental data. 3. Implementation The model for open porosity has been implemented in the latest version of the TRANSURANUS fuel performance code [26]. The parameters for the conventional fission gas release model are the diffusion coefficient and a threshold at the grain boundaries to determine when the (thermal) release occurs. The as-manufactured porosity was not taken into account for the FGR. Nevertheless, an athermal release term was added either as an empirical percentage of the total created gas atoms or through an athermal term in the (effective) diffusion coefficient, dominating at low temperatures. This athermal part of the FGR model is now a combination of the latter and of the new model described above, in which the dependence on the porosity is now taken into account in a nonempirical way thanks to geometric considerations. The conversion from total porosity to open porosity has been inferred from the work of Song [22], which is reproduced in Fig. 2. Since information on the manufacturing process is not always available and the differences are anyway quite limited according to the same reference, only one correlation has been used. This expression is separated in three linear parts that can be found in Eqs. (1)e(3). Under 5% of porosity, with P() being the porosity and Popen() being the open porosity
Popen ¼ P=20 Under 5.8% of porosity
(1)
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nearly 30% of porosity are very similar and the simpler expression has therefore been chosen to ensure shorter calculation times without loss of precision. The model is considered to break down above 30% of porosity, as assumptions such as the porosity taking a cylindrical shape along the grain edges are not valid anymore. For a TKD with an edge length l, a cylindrical porosity with a radius r and P representing either the porosity or the open porosity, the ratio of surface connected to the open porosity to the total surface of the TKD is shown below.
pffiffiffi SP 12 r pffiffiffi ¼ 1:54 P ¼ ST 1þ2 3 l
(4)
4. Results and discussion Fig. 1. Representation of a TKD where l is the side-length.
4 UO2 powder 5 wt% milled UO2 10 wt% milled UO2 20 wt% milled UO2 Linear fitting
Open porosity (vol %)
3.5 3 2.5 2 1.5 1 0.5 0
92
93
94
95
Pellet density (% TD)
96
97
Fig. 2. Open porosity as function of the density in % of theoretical density (TD) [22]. Linear fitting following Eqs (1)e(3).
Popen ¼ 3:10P 0:1525
(2)
Above 5.8% of porosity
Popen ¼ P=2:1 3:2,104
(3)
The justification for this approximate correlation is that the values of the porosity and its fraction that will remain open are prone to large uncertainties. Furthermore, in the work of Song, both the porosity and open porosity values are averaged whereas fuel performance codes such as TRANSURANUS would need values and therefore correlations that are dependent on the radial position in the pellet. Finally, the large uncertainties on the diffusion coefficients should also be mentioned. A higher precision in the description of the open porosity fraction is only relevant when the main part of the fission gas release occurs through athermal processes, as pointed out in the simulation of fuel in the JOYO reactor in the next section, and is usually less important than the uncertainties pertaining to the diffusion coefficient. The conversion from the volume fraction of open porosity to the surface fraction of open porosity has been calculated with two different levels of approximations: one is rigorous, although keeping the assumptions made for the model, while the other one presents geometric simplifications of second order considerations. Both derivations are presented in Appendix A. The results up until
In order to test the model that allows to account for the open porosity resulting from fabrication in a mechanistic manner, we have analysed two types of oxide fuels where this fraction is large but that have undergone very different irradiation conditions. The first type is a LWR rod of the first FUMEX co-ordinated research project of the IAEA [27]. The second type concerns minor-actinidecontaining FBR fuels: two fuel rods irradiated in the JOYO reactor for a very short time with a very high porosity [28,29] and two cases of the SUPERFACT irradiation experiment, with a low porosity [30]. Important data pertaining to these experiments are presented in Table 2. The power history of the irradiations conducted in JOYO are reproduced in Ref. [31]. The burn-up and FGR is not given for these two rods. They are however used to show the impact of the athermal release at the beginning of the irradiation, when the temperature is low, and which gradually becomes less important when the temperature increases and stays high for a long time (Tables 3 and 4). In order to test the applicability of the new model for open porosity, a small sensitivity study was carried out. More precisely, each case was run in four different configurations, namely where the empirical athermal release is considered as a fixed percentage (RTþA) or as a low-temperature term of the diffusion coefficient (RT), where the empirical athermal release module is replaced by the new model considering the porosity (RTþP), and where the empirical athermal release module is replaced by the new model considering the open porosity (RTþOP). All the results are summarised in Table 1. As mentioned earlier, the previous athermal model is releasing a given (empirical) fraction of the gas created and is therefore not depending on the chosen diffusion coefficient. In the FUMEX case analysed it adds a release fraction of slightly more than 0.23%. The mechanistic model presented here leads to a release fraction depending on the amount of gas reaching the open portion of the grain boundaries, hence on the (athermal) diffusion coefficient, in contrast with the previous (empirical) athermal release model. The max linear power of the irradiations Am1-2-1 and PTM001 are reached after respectively 22.3 and 69.07 h. Then, this maximum is kept for some time (which was not the case in the irradiation experiment, to witness the fact that the thermally driven release is indeed starting to have and influence and even to Table 1 Fission gas release (%) for the first case of the FUMEX project [27] of the IAEA (case 1) and for the two types of rods considered in the SUPERFACT irradiation experiment [32] (case 2: rods 4 and 16, case 3: rods 7 and 13). Case
Exp
RTþA
RT
RTþP
RTþOP
1 2 3
1.8 68 72
0.53 57.06 73.47
0.30 56.86 73.34
0.39 59.99 76.11
0.32 57.50 73.89
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Table 2 Fuel and irradiation properties of the first case of the FUMEX project of the IAEA, of the SUPERFACT irradiation experiment and of the JOYO irradiation.
Fuel type PueAm/Np fraction Density Grain size Gap filling Fuel radius Clad inner radius Clad outer radius Exp FGR
FUMEX
SUPERFACT 4 and 16
SUPERFACT 7 and 13
Am1-2-1
PTM001
UO2
(UPuAm)O2 0.235e0.0215 96.73%TD 0.01 mm He 2.709 mm 2.825 mm 3.275 mm 68%
(UPuNp)O2 0.241e0.0195 97.46%TD 0.01 mm He 2.682 mm 2.825 mm 3.275 mm 72%
(UPuAm)O2 0.305e0.005 93.00%TD 0.01 mm 91%Hee9%Kr 2.71 mm 2.78 mm 3.25 mm NC
(UPuAm)O2 0.31e0.024 85.00%TD 0.01 mm 91%Hee9%Kr 2.7 mm 2.78 mm 3.25 mm NC
94.1%TD 0.01 mm He 4.045 mm 4.11 mm 4.75 mm 1.8%
Table 3 Fission gas produced [cm3] and released (%) for the first case of the FUMEX project for the fuel rod Am1-2-1. Time [h]
Produced
RTþA
RTþP
RTþOP
7.11 12.25 22.3 46.3 70.3 94.3 118.3 142.3 166.3 502.3
0.01 0.02 0.07 0.28 0.49 0.69 0.90 1.11 1.32 4.22
0 0 0 0 0 0.07 1.62 4.16 6.8 29.17
0 0.01 7.78 7.31 6.87 6.72 7.19 9.06 11.29 31.59
0 0.01 4.76 3.38 2.95 2.8 3.96 6.29 8.82 30.67
Table 4 Fission gas produced [cm3] and released (%) for the first case of the FUMEX project for the fuel rod PTM001. Time [h]
Produced
RTþA
RTþP
RTþOP
5 20 44 68 69.07 93.07
0.00 0.01 0.40 0.74 0.76 1.17
0 0 0 0 0 1.65
0 11.99 13.01 13.76 14.05 15.19
0 7.13 7.20 7.37 7.49 8.69
dominate. Due to the very small amounts of fission gas produced, RTþA and RT are very similar and only the first one is reported. As expected, what can be seen is that when the linear power is small, the previous model was not predicting any release, whereas the new one based on the high open-porosity is now anticipating some gas escaping to the open volume. When the high linear power (respectively 43 and 47 kW/m) is kept for some time, some fission gas is thermally released, and as is displayed for the case Am1-2-1, this effect becomes dominant after a few days. This is also observed for the first FUMEX case and the two cases from the SUPERFACT irradiation experiment. Comparing the results for the LWR rod and the FBR rods points out the dominant role played by the diffusion coefficient (via the temperature) and its uncertainties in the release model. In addition, they show that the fraction released when considering the open porosity instead of the (total) porosity is smaller, as expected. The results also point out that when the density is high, the fraction of open porosity is smaller, and therefore the difference between the model considering the porosity and the model considering the open porosity is bigger. Indeed, considering the open porosity instead of the porosity in the SUPERFACT cases can lead to differences in the release of almost 2.5%, but this is small in comparison with the total release fraction observed because most of the release is still dominated by thermally activated processes.
It should be underlined that the model decribed here is a first step towards the mehanistic description of open porosity in a fission gas behaviour model based on a physical value that can be linked to measured microstructural features instead of a fitting parameter. This can be important in fuels with a large fabrication porosity fraction and subjected to low temperatures so that athermal release mechanisms dominate. This could be the case in nitride fuels. Nevertheless, the uncertainties pertaining to the diffusion coefficients are so important that further model refinements for describing the open porosity should be carefully considered. Such model improvements could include the evolution (e.g. densification) of the open porosity resulting from the fabrication process and the interaction between this porosity and that resulting from the gas filled bubbles. In any case, more data about open porosity are needed for fresh as well as irradiated fuels.
5. Conclusions A new model feature to treat the athermal release of fission gas in oxide fuels with open porosity resulting from the fabrication process has been developed and implemented into the TRANSURANUS fuel performance code. It relies on the geometrical description of fine tunnels along the edges of tetrakaihedra grains as suggested for models for sintering of materials. The simulation of the first case of the FUMEX project of the IAEA, of two rod types from the SUPERFACT irradiation experiment and of two rods irradiated in the JOYO reactor was carried out in order to test the modified model for oxide fuels with large amounts of open (fabrication) porosity under different conditions. Predicting the release by the physical description of the open porosity rather than by means of a pure empirical contribution leads to similar results when the thermally activated diffusion is dominating and to dramatically different results when only the athermal release is activated. This is therefore encouraging, especially since no particular fitting was needed for predicting the release. Nevertheless, more experimental data about the fraction of porosity and the corresponding fraction of open porosity in irradiated fuels are required. In particular, the open porosity has been observed to decrease much faster than the porosity [10] and the correlation should reflect it. Other important parameters for fission gas release such as the lattice diffusion coefficient were out of the scope of this study but should also be considered carefully as they are subject to large uncertainties and may have a corresponding strong impact on the predicted release fractions by a fuel performance code. Finally, it should be pointed out that the model for open porosity is not restricted to oxide fuels. It can also be considered for nitride and carbide fuel, and it is generally expected to be relevant for fuels with large fractions of open pores and for irradiation conditions where fuel temperatures are low so that athermal release dominates over thermal release.
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Acknowledgements The authors wish to thank Arndt Schubert and Jacques van de Laar for the stimulating discussions and their support for the implementation in the TRANSURANUS fuel performance code. P€ ar Olsson, Janne Wallenius and Rudy Konings are also acknowledged for reviewing the manuscript. Financial support from SKB is acknowledged.
VP 3 r2 r ¼ P ¼ pffiffiffi p 2 0 ¼ l VTKD 2 2 l
VP ¼
36 lpr 2 3
(A.1)
2pr 3
(A.2)
SP ¼ 36l
The volume of the TKD is reproduced in Eq (A.3). For a complete derivation, see for instance [23,33].
pffiffiffi VTKD ¼ 8 2l3
(A.3)
SP STKD
SP STKD
(A.4)
Appendix A.2. Approximations Not considering the surface alteration by the porosity cylinders leads to the much easier equations, already derived in Ref. [23].
pffiffiffi STKD ¼ 6l2 1 þ 2 3
SP STKD
2p rl
13 3
4p þ rl 2p 13 3
(A.8)
SP 12 r pffiffiffi ¼ STKD 1 þ 2 3 l sffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffi 12 2 2pffiffiffi pffiffiffi P ¼ 1:54 P ¼ 3p 1þ2 3
(A.9)
The surface porosity is then simply the first order term of the previous more exact one and both of them are represented in Fig. A.3.
Surface porosity [-]
0.8 0.6 0.4 No approx all approx
0.2 0 0
The volume of the TKD does not have to undergo a similar treatment since the volume used to calculate the porosity is the one of the TKD and of its associated porosity. The surface porosity is defined as the ratio between the surface of the porosity and the surface of the TKD.
pffiffiffi r2 2 þ 3 þ l2
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi 2 2P 2p 3p sffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi pffiffiffi ffi 1 pffiffiffi 13 2 2 13 2 2P þ 3þ 4p P þ 2p 2 3 3p 3 3p pffiffiffi pffiffiffi 3:44 P 1:54 P pffiffiffi pffiffiffi ¼ ¼ 2:23 þ 1:07 P 2:47P 1 þ 0:48 P 1:11P
1
The surface of the TKD is slightly harder to find if one wishes to consider the reduction of the surface when the porosity cylinders are getting thicker. For each flat surface of the TKD, the edges are shortened due to the bigger porosity cylinders and a fraction of this porosity cylinder is added to the surface (Eq (A.4)).
SP ¼ STKD 1
(A.6)
(A.7)
Appendix A.1. Exact derivation
STKD ¼ 8 hexagons þ 6 squares # " pffiffiffi 3 3 2pr ðl 2rÞ2 þ 6ðl 2rÞ STKD ¼ 8 2 6 2Pr þ6 ðl 2rÞ2 þ 4ðl 2rÞ 6 p ffiffiffiffiffiffi ffi STKD ¼ l2 6 þ 12 ð3Þ þ r 2 ð56 48PÞ þ rlð24P 56Þ
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi 2 2P 3p
Inserting this relation in the surface porosity leads to the following results.
Appendix A The edges of the TKD have a length l and the radius of the cylindrical porosity along the edges is r. Each TKD has 36 edges and 14 faces, of which 8 are regular hexagons and 6 are squares. The volume and the surface of the porosity are therefore found as expressed below. They are corresponding to 36 cylinders shared among 3 TKD. The volume of the porosity normalized to one grain (VP) and the surface of the grain connected to the porosity are then as follows.
355
(A.5)
Using the conventional volume porosity definition, one can get a relationship between the measured or calculated porosity and the geometric variables of the problem.
0.1
0.2 Volume porosity [-]
0.3
Fig. A.3. Comparison of the exact solution A.7 and of the approximation A.9.
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nitride fuels in frapcon-ep, J. Nucl. Mater. 427 (1) (2012) 30e38. [20] P.C. Millett, M.R. Tonks, S. Biner, Grain boundary percolation modeling of fission gas release in oxide fuels, J. Nucl. Mater. 424 (1) (2012) 176e182. [21] P.C. Millett, Percolation on grain boundary networks: application to fission gas release in nuclear fuels, Comput. Mater. Sci. 53 (1) (2012) 31e36. [22] K.W. Song, K.S. Kim, Y.M. Kim, K.W. Kang, Y.H. Jung, Reduction of the open porosity of UO2 pellets through pore structure control, J. Nucl. Mater. 279 (2e3) (2000) 253e258. [23] R.L. Coble, Sintering crystalline solids. i. intermediate and final state diffusion models, J. Appl. Phys. 32 (5) (1961) 787e792. [24] R. White, M. Tucker, A new fission-gas release model, J. Nucl. Mater. 118 (1) (1983) 1e38. [25] Y.-s. Kim, Theoretical analysis of two-stage fission gas release processes: grain lattice and grain boundary diffusion, J. Nucl. Mater. 326 (2) (2004) 97e105. [26] K. Lassmann, Transuranus: a fuel rod analysis code ready for use, J. Nucl. Mater. 188 (0) (1992) 295e302. [27] Fuel Modelling at Extended Burnup”, Report of the Co-ordinated Research Programme on Fuel Modelling at Extended Burnup - FUMEX 1993-1996,” Tech. Rep. IAEA-TECDOC-998, IAEA. [28] K. Maeda, S. Sasaki, M. Kato, Y. Kihara, Radial redistribution of actinides in irradiated FR-MOX fuels, J. Nucl. Mater. 389 (1) (2009) 78e84. [29] K. Maeda, K. Katsuyama, Y. Ikusawa, S. Maeda, Short-term irradiation behavior of low-density americium-doped uranium-plutonium mixed oxide fuels irradiated in a fast reactor, J. Nucl. Mater. 416 (1) (2011) 158e165. [30] J. Babelot, N. Chauvin, Joint CEA/ITU Synthesis Report of the Experiment SUPERFACT 1, ITU Report, JRC-ITU-TN-99/03, 1999. [31] V. Di Marcello, V. Rondinella, A. Schubert, J. van de Laar, P. Van Uffelen, Modelling actinide redistribution in mixed oxide fuel for sodium fast reactors, Prog. Nucl. Energy 72 (2014) 83e90. [32] C. Walker, G. Nicolaou, Transmutation of neptunium and americium in a fast neutron flux: EPMA results and KORIGEN predictions for the superfact fuels, J. Nucl. Mater. 218 (2) (1995) 129e138. [33] B. Wong, J.A. Pask, Models for kinetics of solid state sintering, J. Am. Ceram. Soc. 62 (3e4) (1979) 138e141.
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Towards the inclusion of open fabrication porosity in a fission gas release model Antoine Claisse a, *, Paul Van Uffelen b a b
KTH Royal Institute of Technology, Reactor Physics, AlbaNova University Centre, 106 91, Stockholm, Sweden European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, D-76125, Karlsruhe, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 January 2015 Received in revised form 19 July 2015 Accepted 13 August 2015 Available online 18 August 2015
A model is proposed for fission product release in oxide fuels that takes into account the open porosity in a mechanistic manner. Its mathematical framework, assumptions and limitations are presented. It is based on the model for open porosity in the sintering process of crystalline solids. More precisely, a grain is represented by a tetrakaidecahedron and the open porosity is represented by a continuous cylinder along the grain edges. It has been integrated in the TRANSURANUS fuel performance code and applied to the first case of the first FUMEX project as well as to neptunium and americium containing pins irradiated during the SUPERFACT experiment and in the JOYO reactor. The results for LWR and FBR fuels are consistent with the experimental data and the predictions of previous empirical models when the thermal mechanisms are the main drivers of the release, even without using a fitting parameter. They also show a different but somewhat expected behaviour when very high porosity fuels are irradiated at a very low burn-up and at low temperature. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Fuel performance codes are an important tool for safety studies of nuclear reactors and can also be envisaged for final repository studies. One of the key points of any nuclear fuel performance code is the prediction of fission gas release (FGR). Indeed, the release of noble gases such as Xe or Kr can greatly degrade the thermal conductance of the initially He-filled gap. In addition, a high pressure can be exerted on the cladding, which can eventually lead to failure and the release of radioactive iodine and cesium. On the other hand, the gas that is not released can form bubbles, which contributes to fuel swelling. An early review of the main effects involved in fission gas behaviour can be found in Ref. [1]. The FGR is usually treated in two separate steps in fuel performance codes. First, the fission gas creation is estimated from the number of fissions (that is the linear power rating) and the fraction reaching the grain boundaries is determined by means of an effective diffusion coefficient applied in simplified models, abundantly described in the literature [2e5]. The effective diffusion coefficient can be divided into three temperature ranges [6]. At high
* Corresponding author. E-mail address: [email protected] (A. Claisse). http://dx.doi.org/10.1016/j.jnucmat.2015.08.022 0022-3115/© 2015 Elsevier B.V. All rights reserved.
temperature, the thermal diffusion is dominating. At moderate temperature, the irradiation enhanced defect concentrations drive the fission gas diffusion in addition to the thermally activated defects, whereas at low temperature the athermal part is the most important. This corresponds to radiation enhanced diffusion that is usually associated with the fission spikes [7] and is called athermal or irradiation-induced diffusion. The athermal diffusion is of prime importance for fast neutron reactors using fuels with high thermal conductivity, such as nitride fuels, where the release may not be as strong as at higher temperatures in oxide fuels but still enough to degrade the gap conductance [8e10]. In the second step, the gas at the grain boundaries is taken into consideration and can be released or stored depending on its concentration and on a release threshold that can vary with, among other properties, the temperature or the temperature gradient. The models for FGR have to remain simple enough to ensure short computational times and easy convergence while being sufficiently complex to include most of the relevant phenomena driving the fission gas, reviewed for instance in Refs. [1,7,11]. Another reason to keep the models relatively simple are the uncertainties pertaining to the model parameters such as the diffusion coefficient. The porosity has also been shown to be important [12] and several attempts have been made to try to account for it in FGR models. Several models have adopted a pure empirical expression
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for an additional release term as a function of the initial open porosity fraction [13e17]. Others (e.g. Ref. [18]) have considered the effect of open porosity resulting from the fabrication process via an empirical reduction of the effective sphere radius of the Booth sphere for diffusional release from the grains. More recently, Feng et al. have used an additional fitting parameter in the effective diffusion coefficient [19] for the intragranular part of the fission gas release model, and they considered the total fabrication porosity rather than only the open porosity fraction that is in direct contact with the free volume in the rod. Other groups have explored the percolation phenomenon [20,21] along grain boundaries, which is intuitively more appropriate when considering open porosity. Nevertheless, this seems rather hard to implement in a fuel performance code since the grains and therefore the boundaries are usually not modelled individually. Instead, a model considering the porosity as a mean-field should be developed if one wishes to use a percolation model. The current version of the TRANSURANUS code does not allow to consider the effect of the open porosity on the fission gas release fraction. In addition to the conventional so-called thermal release modelling based on diffusion theory that also can include an athermal diffusion term, it has an additional empirical term for athermal release that is only burnup dependent, and which corresponds to release due to knock-out and recoil. In this short communication, an approach taking into consideration the open porosity in the inter-granular fission gas release model of TRANSURANUS is presented. The motivations, mathematical derivation, implementation and limitations are briefly presented. In order to illustrate the range of potential application for nuclear fuels with large amounts of open porosity under various irradiation conditions, a comparison with previous models is made for the first case of the FUMEX project of the IAEA, which is an LWR rod, in addition to fast breeder reactor oxide fuel pins of the SUPERFACT experiment and fuel rods irradiated in the JOYO reactor. 2. Model Before describing the model, the nomenclature has to be defined. Porosity (P) will classically refer to the ratio of voided volume over the total volume. The fraction of this voided volume that is in contact with the open space is Open Porosity. The other fraction is Closed Porosity. For the needs of this study, the grain surface (ST) and the surface of open porosity are introduced (SP). The latter is defined as the fraction of a grain surface occupied by the open pores along the grain edges of a grain that is represented by a tetrakaidecahedra (TKD, see Appendix and Fig. 1 for details). In accordance with these definitions, any fission gas atom reaching the open porosity at the grain boundary is deemed released to the open volume (gap and plena). The model considers the conventional bubble formation and interconnection at grain boundaries that lead to tunnel formation through which gas atoms can be released when the local temperature and burnup values are high enough. The model also considers an open porosity fraction due to the fabrication process that forms an independent escape pathway for fission gas. Assuming that the flux of atoms arriving at the grain boundaries is known and given by the intragranular module of the fission gas release model that is essentially relying on effective diffusion (in a spherical grain), one needs thus to determine the fraction of the grain boundaries that is occupied by open porosity in order to estimate the direct release due to open pores. This is done in two steps. In a first step, it is necessary to derive the volume fraction of open-porosity from the total porosity that is usually provided. This presently relies on an empirical correlation [22]. In a second step, the open (fabrication) porosity is converted to a surface fraction (SP/Sgrain). For this purpose, the grains are
modelled as TKD. Coble [23] and White et al. [24] suggested this, advocating that it is the highest order polyhedron that can be used to completely fill the space, and is also used in more recent studies of fission gas release (e.g. Ref. [25]). Its surface is composed of six squares and eight regular hexagons with edges of the same length. The open porosity is represented as cylinders along the grain edges. It should be underlined that in doing so, the following additional assumptions are made: a) all the grains have the same size within a finite element, b) all porosity is localized on the grain edges, and c) all the cylinders representing the porosity have the same radius within a finite element. These assumptions limit the validity of the model for instance to fuel densities above 70%. The open porosity fraction should change during the irradiation. More precisely the fuel can undergo densification, while the creation and precipitation of fission gas along the grain boundaries can act to increase the total porosity along grain boundaries. The creation and destruction of an interconnected tunnel network of bubbles will temporarily turn closed porosity into open porosity, until the excess gas is released and the network is closed again. The burn-up, temperature and fission gas production vary at different radial positions of the fuel, making the prediction of the open porosity based upon the porosity an intricate problem. A percolation study [20,21] could address this. However, since a fuel performance code cannot model each grain, but only finite elements representing a fraction of the fuel much bigger than a grain, such an approach would have to be somehow averaged. In this study, the empirical correlation coming from the work of Song et al. [22] has been adopted for estimating the open pore fraction on the basis of the total fabrication porosity. It is valid only for fresh oxide fuels. The fraction of porosity remaining open at the end of life is sometimes known from experiments to be very low since pellets have densified and tunnel networks have closed [10], but to the best of the authors' knowledge, a better correlation does not yet exist. One might apply an empirical densification model to the open porosity resulting from the fabrication process, but this also requires more experimental data. 3. Implementation The model for open porosity has been implemented in the latest version of the TRANSURANUS fuel performance code [26]. The parameters for the conventional fission gas release model are the diffusion coefficient and a threshold at the grain boundaries to determine when the (thermal) release occurs. The as-manufactured porosity was not taken into account for the FGR. Nevertheless, an athermal release term was added either as an empirical percentage of the total created gas atoms or through an athermal term in the (effective) diffusion coefficient, dominating at low temperatures. This athermal part of the FGR model is now a combination of the latter and of the new model described above, in which the dependence on the porosity is now taken into account in a nonempirical way thanks to geometric considerations. The conversion from total porosity to open porosity has been inferred from the work of Song [22], which is reproduced in Fig. 2. Since information on the manufacturing process is not always available and the differences are anyway quite limited according to the same reference, only one correlation has been used. This expression is separated in three linear parts that can be found in Eqs. (1)e(3). Under 5% of porosity, with P() being the porosity and Popen() being the open porosity
Popen ¼ P=20 Under 5.8% of porosity
(1)
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353
nearly 30% of porosity are very similar and the simpler expression has therefore been chosen to ensure shorter calculation times without loss of precision. The model is considered to break down above 30% of porosity, as assumptions such as the porosity taking a cylindrical shape along the grain edges are not valid anymore. For a TKD with an edge length l, a cylindrical porosity with a radius r and P representing either the porosity or the open porosity, the ratio of surface connected to the open porosity to the total surface of the TKD is shown below.
pffiffiffi SP 12 r pffiffiffi ¼ 1:54 P ¼ ST 1þ2 3 l
(4)
4. Results and discussion Fig. 1. Representation of a TKD where l is the side-length.
4 UO2 powder 5 wt% milled UO2 10 wt% milled UO2 20 wt% milled UO2 Linear fitting
Open porosity (vol %)
3.5 3 2.5 2 1.5 1 0.5 0
92
93
94
95
Pellet density (% TD)
96
97
Fig. 2. Open porosity as function of the density in % of theoretical density (TD) [22]. Linear fitting following Eqs (1)e(3).
Popen ¼ 3:10P 0:1525
(2)
Above 5.8% of porosity
Popen ¼ P=2:1 3:2,104
(3)
The justification for this approximate correlation is that the values of the porosity and its fraction that will remain open are prone to large uncertainties. Furthermore, in the work of Song, both the porosity and open porosity values are averaged whereas fuel performance codes such as TRANSURANUS would need values and therefore correlations that are dependent on the radial position in the pellet. Finally, the large uncertainties on the diffusion coefficients should also be mentioned. A higher precision in the description of the open porosity fraction is only relevant when the main part of the fission gas release occurs through athermal processes, as pointed out in the simulation of fuel in the JOYO reactor in the next section, and is usually less important than the uncertainties pertaining to the diffusion coefficient. The conversion from the volume fraction of open porosity to the surface fraction of open porosity has been calculated with two different levels of approximations: one is rigorous, although keeping the assumptions made for the model, while the other one presents geometric simplifications of second order considerations. Both derivations are presented in Appendix A. The results up until
In order to test the model that allows to account for the open porosity resulting from fabrication in a mechanistic manner, we have analysed two types of oxide fuels where this fraction is large but that have undergone very different irradiation conditions. The first type is a LWR rod of the first FUMEX co-ordinated research project of the IAEA [27]. The second type concerns minor-actinidecontaining FBR fuels: two fuel rods irradiated in the JOYO reactor for a very short time with a very high porosity [28,29] and two cases of the SUPERFACT irradiation experiment, with a low porosity [30]. Important data pertaining to these experiments are presented in Table 2. The power history of the irradiations conducted in JOYO are reproduced in Ref. [31]. The burn-up and FGR is not given for these two rods. They are however used to show the impact of the athermal release at the beginning of the irradiation, when the temperature is low, and which gradually becomes less important when the temperature increases and stays high for a long time (Tables 3 and 4). In order to test the applicability of the new model for open porosity, a small sensitivity study was carried out. More precisely, each case was run in four different configurations, namely where the empirical athermal release is considered as a fixed percentage (RTþA) or as a low-temperature term of the diffusion coefficient (RT), where the empirical athermal release module is replaced by the new model considering the porosity (RTþP), and where the empirical athermal release module is replaced by the new model considering the open porosity (RTþOP). All the results are summarised in Table 1. As mentioned earlier, the previous athermal model is releasing a given (empirical) fraction of the gas created and is therefore not depending on the chosen diffusion coefficient. In the FUMEX case analysed it adds a release fraction of slightly more than 0.23%. The mechanistic model presented here leads to a release fraction depending on the amount of gas reaching the open portion of the grain boundaries, hence on the (athermal) diffusion coefficient, in contrast with the previous (empirical) athermal release model. The max linear power of the irradiations Am1-2-1 and PTM001 are reached after respectively 22.3 and 69.07 h. Then, this maximum is kept for some time (which was not the case in the irradiation experiment, to witness the fact that the thermally driven release is indeed starting to have and influence and even to Table 1 Fission gas release (%) for the first case of the FUMEX project [27] of the IAEA (case 1) and for the two types of rods considered in the SUPERFACT irradiation experiment [32] (case 2: rods 4 and 16, case 3: rods 7 and 13). Case
Exp
RTþA
RT
RTþP
RTþOP
1 2 3
1.8 68 72
0.53 57.06 73.47
0.30 56.86 73.34
0.39 59.99 76.11
0.32 57.50 73.89
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Table 2 Fuel and irradiation properties of the first case of the FUMEX project of the IAEA, of the SUPERFACT irradiation experiment and of the JOYO irradiation.
Fuel type PueAm/Np fraction Density Grain size Gap filling Fuel radius Clad inner radius Clad outer radius Exp FGR
FUMEX
SUPERFACT 4 and 16
SUPERFACT 7 and 13
Am1-2-1
PTM001
UO2
(UPuAm)O2 0.235e0.0215 96.73%TD 0.01 mm He 2.709 mm 2.825 mm 3.275 mm 68%
(UPuNp)O2 0.241e0.0195 97.46%TD 0.01 mm He 2.682 mm 2.825 mm 3.275 mm 72%
(UPuAm)O2 0.305e0.005 93.00%TD 0.01 mm 91%Hee9%Kr 2.71 mm 2.78 mm 3.25 mm NC
(UPuAm)O2 0.31e0.024 85.00%TD 0.01 mm 91%Hee9%Kr 2.7 mm 2.78 mm 3.25 mm NC
94.1%TD 0.01 mm He 4.045 mm 4.11 mm 4.75 mm 1.8%
Table 3 Fission gas produced [cm3] and released (%) for the first case of the FUMEX project for the fuel rod Am1-2-1. Time [h]
Produced
RTþA
RTþP
RTþOP
7.11 12.25 22.3 46.3 70.3 94.3 118.3 142.3 166.3 502.3
0.01 0.02 0.07 0.28 0.49 0.69 0.90 1.11 1.32 4.22
0 0 0 0 0 0.07 1.62 4.16 6.8 29.17
0 0.01 7.78 7.31 6.87 6.72 7.19 9.06 11.29 31.59
0 0.01 4.76 3.38 2.95 2.8 3.96 6.29 8.82 30.67
Table 4 Fission gas produced [cm3] and released (%) for the first case of the FUMEX project for the fuel rod PTM001. Time [h]
Produced
RTþA
RTþP
RTþOP
5 20 44 68 69.07 93.07
0.00 0.01 0.40 0.74 0.76 1.17
0 0 0 0 0 1.65
0 11.99 13.01 13.76 14.05 15.19
0 7.13 7.20 7.37 7.49 8.69
dominate. Due to the very small amounts of fission gas produced, RTþA and RT are very similar and only the first one is reported. As expected, what can be seen is that when the linear power is small, the previous model was not predicting any release, whereas the new one based on the high open-porosity is now anticipating some gas escaping to the open volume. When the high linear power (respectively 43 and 47 kW/m) is kept for some time, some fission gas is thermally released, and as is displayed for the case Am1-2-1, this effect becomes dominant after a few days. This is also observed for the first FUMEX case and the two cases from the SUPERFACT irradiation experiment. Comparing the results for the LWR rod and the FBR rods points out the dominant role played by the diffusion coefficient (via the temperature) and its uncertainties in the release model. In addition, they show that the fraction released when considering the open porosity instead of the (total) porosity is smaller, as expected. The results also point out that when the density is high, the fraction of open porosity is smaller, and therefore the difference between the model considering the porosity and the model considering the open porosity is bigger. Indeed, considering the open porosity instead of the porosity in the SUPERFACT cases can lead to differences in the release of almost 2.5%, but this is small in comparison with the total release fraction observed because most of the release is still dominated by thermally activated processes.
It should be underlined that the model decribed here is a first step towards the mehanistic description of open porosity in a fission gas behaviour model based on a physical value that can be linked to measured microstructural features instead of a fitting parameter. This can be important in fuels with a large fabrication porosity fraction and subjected to low temperatures so that athermal release mechanisms dominate. This could be the case in nitride fuels. Nevertheless, the uncertainties pertaining to the diffusion coefficients are so important that further model refinements for describing the open porosity should be carefully considered. Such model improvements could include the evolution (e.g. densification) of the open porosity resulting from the fabrication process and the interaction between this porosity and that resulting from the gas filled bubbles. In any case, more data about open porosity are needed for fresh as well as irradiated fuels.
5. Conclusions A new model feature to treat the athermal release of fission gas in oxide fuels with open porosity resulting from the fabrication process has been developed and implemented into the TRANSURANUS fuel performance code. It relies on the geometrical description of fine tunnels along the edges of tetrakaihedra grains as suggested for models for sintering of materials. The simulation of the first case of the FUMEX project of the IAEA, of two rod types from the SUPERFACT irradiation experiment and of two rods irradiated in the JOYO reactor was carried out in order to test the modified model for oxide fuels with large amounts of open (fabrication) porosity under different conditions. Predicting the release by the physical description of the open porosity rather than by means of a pure empirical contribution leads to similar results when the thermally activated diffusion is dominating and to dramatically different results when only the athermal release is activated. This is therefore encouraging, especially since no particular fitting was needed for predicting the release. Nevertheless, more experimental data about the fraction of porosity and the corresponding fraction of open porosity in irradiated fuels are required. In particular, the open porosity has been observed to decrease much faster than the porosity [10] and the correlation should reflect it. Other important parameters for fission gas release such as the lattice diffusion coefficient were out of the scope of this study but should also be considered carefully as they are subject to large uncertainties and may have a corresponding strong impact on the predicted release fractions by a fuel performance code. Finally, it should be pointed out that the model for open porosity is not restricted to oxide fuels. It can also be considered for nitride and carbide fuel, and it is generally expected to be relevant for fuels with large fractions of open pores and for irradiation conditions where fuel temperatures are low so that athermal release dominates over thermal release.
A. Claisse, P. Van Uffelen / Journal of Nuclear Materials 466 (2015) 351e356
Acknowledgements The authors wish to thank Arndt Schubert and Jacques van de Laar for the stimulating discussions and their support for the implementation in the TRANSURANUS fuel performance code. P€ ar Olsson, Janne Wallenius and Rudy Konings are also acknowledged for reviewing the manuscript. Financial support from SKB is acknowledged.
VP 3 r2 r ¼ P ¼ pffiffiffi p 2 0 ¼ l VTKD 2 2 l
VP ¼
36 lpr 2 3
(A.1)
2pr 3
(A.2)
SP ¼ 36l
The volume of the TKD is reproduced in Eq (A.3). For a complete derivation, see for instance [23,33].
pffiffiffi VTKD ¼ 8 2l3
(A.3)
SP STKD
SP STKD
(A.4)
Appendix A.2. Approximations Not considering the surface alteration by the porosity cylinders leads to the much easier equations, already derived in Ref. [23].
pffiffiffi STKD ¼ 6l2 1 þ 2 3
SP STKD
2p rl
13 3
4p þ rl 2p 13 3
(A.8)
SP 12 r pffiffiffi ¼ STKD 1 þ 2 3 l sffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffi 12 2 2pffiffiffi pffiffiffi P ¼ 1:54 P ¼ 3p 1þ2 3
(A.9)
The surface porosity is then simply the first order term of the previous more exact one and both of them are represented in Fig. A.3.
Surface porosity [-]
0.8 0.6 0.4 No approx all approx
0.2 0 0
The volume of the TKD does not have to undergo a similar treatment since the volume used to calculate the porosity is the one of the TKD and of its associated porosity. The surface porosity is defined as the ratio between the surface of the porosity and the surface of the TKD.
pffiffiffi r2 2 þ 3 þ l2
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi 2 2P 2p 3p sffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi pffiffiffi ffi 1 pffiffiffi 13 2 2 13 2 2P þ 3þ 4p P þ 2p 2 3 3p 3 3p pffiffiffi pffiffiffi 3:44 P 1:54 P pffiffiffi pffiffiffi ¼ ¼ 2:23 þ 1:07 P 2:47P 1 þ 0:48 P 1:11P
1
The surface of the TKD is slightly harder to find if one wishes to consider the reduction of the surface when the porosity cylinders are getting thicker. For each flat surface of the TKD, the edges are shortened due to the bigger porosity cylinders and a fraction of this porosity cylinder is added to the surface (Eq (A.4)).
SP ¼ STKD 1
(A.6)
(A.7)
Appendix A.1. Exact derivation
STKD ¼ 8 hexagons þ 6 squares # " pffiffiffi 3 3 2pr ðl 2rÞ2 þ 6ðl 2rÞ STKD ¼ 8 2 6 2Pr þ6 ðl 2rÞ2 þ 4ðl 2rÞ 6 p ffiffiffiffiffiffi ffi STKD ¼ l2 6 þ 12 ð3Þ þ r 2 ð56 48PÞ þ rlð24P 56Þ
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi 2 2P 3p
Inserting this relation in the surface porosity leads to the following results.
Appendix A The edges of the TKD have a length l and the radius of the cylindrical porosity along the edges is r. Each TKD has 36 edges and 14 faces, of which 8 are regular hexagons and 6 are squares. The volume and the surface of the porosity are therefore found as expressed below. They are corresponding to 36 cylinders shared among 3 TKD. The volume of the porosity normalized to one grain (VP) and the surface of the grain connected to the porosity are then as follows.
355
(A.5)
Using the conventional volume porosity definition, one can get a relationship between the measured or calculated porosity and the geometric variables of the problem.
0.1
0.2 Volume porosity [-]
0.3
Fig. A.3. Comparison of the exact solution A.7 and of the approximation A.9.
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