https://doi.org/10.1051/epjn/e201550036x
Regular Article
Thermalhydraulics/thermalmechanics temporal coupling for unprotected loss of flow accidents simulations on a SFR
^{1}
CEA, DEN, DM2S, SERMA, 91191 GifsurYvette, France
^{2}
CEA, DEN, DER, CPA, 13108 SaintPaulLezDurance cedex, France
^{*} email: cyril.patricot@cea.fr
Received:
12
May
2015
Accepted:
25
November
2015
Published online:
11
January
2016
In the frame of ASTRID designing, unprotected loss of flow (ULOF) accidents are considered. As the reactor is not scrammed, power evolution is driven by neutronic feedbacks, among which Doppler effect, linked to fuel temperature, is prominent. Fuel temperature is calculated using thermal properties of fuel pins (we will focus on heat transfer coefficient between fuel pellet and cladding, H_{gap}, and on fuel thermal conductivity, λ_{fuel}) which vary with irradiation conditions (neutronic flux, mass flow and history for instance) and during transient (mainly because of dilatation of materials with temperature). In this paper, we propose an analysis of the impact of spatial variation and temporal evolution of thermal properties of fuel pins on a CFVlike core [M.S. Chenaud et al., Status of the ASTRID core at the end of the preconceptual design phase 1, in Proceedings of ICAPP 2013, Jeju Island, Korea (2013)] behavior during an ULOF accident. These effects are usually neglected under some a priori conservative assumptions. The vocation of our work is not to provide a bestestimate calculation of ULOF transient, but to discuss some of its physical aspects. To achieve this goal, we used TETAR, a thermalhydraulics system code developed by our team to calculate ULOF transients, GERMINAL V1.5, a CEA code dedicated to SFR pin thermalmechanics calculations and APOLLO3^{®}, a neutronic code in development at CEA.
© C. Patricot et al., published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The CFV (Cœur Faible Vidange, low void coefficient core) concept [1], which includes several innovations, is viewed as a way to improve the sodium void effect (reactivity effect of a core voiding) and the accidental behavior of large sodium fast reactors (SFRs). A scheme of this kind of core is given in Figure 1. A sodium plenum, with an upper absorbing protection, is positioned just above the core in order to increase the neutrons leakage in case of voiding. This effect is enhanced by the heterogeneities of the inner core, and by the height difference between the outer core and the inner core. These particularities increase the flux at the top of the core, and therefore in the plenum.
Loss of flow accidents are especially difficult for large SFRs and are therefore studied in depth in the frame of their designing. A detailed analysis of these accidents can be found in reference [2]. In order to clarify the explanations, our paper focuses on the unprotected loss of flow accident, during which primary pumps are lost, but not the secondary ones (we will call it ULOF/PP). The reactor is not scrammed, and the power evolution is driven by the neutronic feedbacks (Doppler, sodium dilatation and dilatations of structures). During the accident, the coolant mass flow decreases until it reaches the natural convection equilibrium. It results in sodium heating in the upper part of the core, making the power decrease, thanks to CFV design. As a consequence, fuel temperature decreases and the Doppler effect is positive. Thus, the stabilization effect of the Doppler is, in this case, an obstacle to the power decrease.
An accurate evaluation of fuel temperature evolution during the transient is therefore necessary. It is usually derived from diffusion equation with given thermal properties. These properties are often homogenized over core zones and are usually constant in time. However, in reality, their spatial variations (mainly due to the heterogeneity of the core and to the mixing of subassemblies of different ages) and temporal evolutions (mainly due to differential thermal dilatations) can be quite important.
In this work, we propose an analysis of the impact of spatial variation and temporal evolution of thermal properties of fuel pins on a CFVlike core behavior during an ULOF/PP accident. Section 2 presents the evolution of the core under irradiation, calculated with APOLLO3^{®} [3] and GERMINAL V1.5 [4]. In Section 3, ULOF/PP accidents are calculated with TETAR (developed in the frame of TRIAD [5]) and different spatial descriptions of thermal properties. In Section 4, we show the results of the temporal coupling. Section 5 provides some general conclusions.
Note that TETAR is not ASTRID reference tool and that the CFVlike core used is an academic model. As a consequence, the numerical results of this paper should not be considered as reference ones. They are given for the physical analysis of the phenomena.
Fig. 1 Scheme of the CFV core concept. 
2 Core evolution under irradiation
2.1 Neutronic evolution
We used APOLLO3^{®} for the neutronic calculations with 33 energy groups. Crosssections were computed by the module ECCO of ERANOS [6]. Control rods are withdrawn in every calculation.
The chosen reloading procedure uses four batches. As the subassemblies are not moved during the reloading, the core is a mixing of subassemblies with different burnup. The resulting power distribution is quite heterogeneous, as shown in Figures 2 and 3. In Figure 2, the power distribution is given, for a cut in the center of the core, at beginning of cycle. Fresh subassemblies have high fissile content and have therefore a high linear power. At end of cycle, in Figure 3, the power distribution becomes more homogeneous. The color ranges are the same for both figures.
The same kind of flux and power redistributions occurs axially because of the combination of consumption of Pu in fissile zones and breeding in fertile ones (located at the bottom of the core).
Fig. 2 Linear power distribution (W/cm by pin) in the center of the core at beginning of cycle. 
Fig. 3 Linear power distribution (W/cm by pin) in the center of the core at end of cycle. 
2.2 Thermomechanical evolution
The evolution of thermomechanical properties of fuel pins is evaluated with GERMINAL V1.5. It uses simplify fuel description model based on monogroup neutron flux, linear power and irradiation damage distributions calculated by APOLLO3^{®}. It also needs sodium inlet temperature and mass flow per pin.
The heat transfer coefficient between fuel pellet and cladding, called H_{gap}, has strong nonlinear variations with irradiation. H_{gap} and gap size evolutions are given in Figures 4 and 5 respectively, at a fixed position (in fissile) of chosen subassemblies. One can see that the initial thermal dilatation of the pellet makes the H_{gap} increase, at the very beginning. A peak is then observed when the pellet comes in contact with the cladding (it does not occur here for the external subcore subassembly). A quite linear phase follows, with constant decrease of the H_{gap} due to the degradation of the contact surface. Finally, threshold effects occur, swelling of the cladding, creation of an oxide layer on its surface and strong gaseous fission products release. The discontinuities at 400, 800 and 1200 EFPD (equivalent full power days) are due to the reloading of a quarter of the core, which changes the linear power and flux in the studied subassemblies.
This nonlinear behavior, together with the positioning of subassemblies in the core, and the axial heterogeneity of the fuel produce quite heterogeneous 3D maps of H_{gap}, as one can see in Figure 6. To build this 3D map, one mean pin per subassembly has been calculated. A 3D map of thermal conductivity of fuel (called λ_{fuel}) is given in Figure 7. The evolution of this quantity is much more linear: the irradiation degrades the ceramics and thus its conductivity. As a consequence, λ_{fuel} is maximal where the irradiation damages are minimal.
Fig. 4 Typical heat transfer coefficient evolution for chosen subassemblies (in top fissile zone). 
Fig. 5 Typical gap size evolution for chosen subassemblies (in top fissile zone). 
Fig. 6 Typical 3D map of H_{gap} (W/cm^{2}/K). 
Fig. 7 Typical 3D map of λ_{fuel} (W/cm/K). 
3 Impact of spatial descriptions of thermal properties and of neutronic feedbacks on the ULOF/PP accident
3.1 Calculations comparison with integrated neutronic feedback coefficients
We used TETAR (Transients Estimation Tool for nAcooled Reactors) to calculate the ULOF/PP accident. It solves 1D thermohydraulic equations in each subassembly. We emphasize that each subassembly is calculated separately by a dedicated 1D thermohydraulic channel in all calculations presented in this paper. This ability of TETAR allowed us to perform our studies on spatial descriptions impacts. Mass flow in each subassembly is calculated to create a given pressure drop. Pin temperature is calculated through 1D diffusion. Point kinetic, fed with feedback coefficients (integrated or local) from APOLLO3^{®}, is used for the power estimation. The system is closed with sodium collectors and sodiumsodium heat exchangers simple models. The accident is driven by a given decrease of the pumps pressure. The overall pressure drop due to gravity (this term leads to natural convection) is calculated precisely.
In this section, the thermal properties are constant during the transient. Four models were used to estimate their initial value:

Exact: one mean pin per subassembly is calculated by GERMINAL V1.5, and the results feed directly the TETAR calculation;

Global average: we calculate, from the exact core calculation, the mean H_{gap} and λ_{fuel} of the core and use them everywhere in the TETAR calculation;

Zones average: we calculate, from the exact core calculation, the mean H_{gap} and λ_{fuel} of the core main five zones (Fig. 1). They are used in the corresponding meshes in TETAR;

Groups: we gather subassemblies in groups and calculate one mean pin per group (subassemblies of the same ring, from the same batch). In comparison with the exact model, the number of GERMINAL V1.5 calculations is reduced by almost a factor 10.
Sodium maximal temperature and power evolutions during the ULOF/PP accident are given in Figures 8 and 9 for the models above. One can see that the exact and the group models are indistinguishable and that the maximal temperature they reached is slightly above the zone average model, which is slightly above the global average model.
Sodium maximal temperatures for some other models are given in Table 1. One can see that the zones average model is enough for λ_{fuel}, its results are very close to those of the exact model. In addition, nonlinearities seem to be weak; the effect of a combination of models is the sum of the effects of the models. Finally, the difference between the groups and the exact models is very small in all cases, about 3 °C.
Fig. 8 Sodium maximal temperature during ULOF/PP accident for different thermal properties models. 
Fig. 9 Power during ULOF/PP accident for different thermal properties models. 
Comparison of sodium maximal temperature (°C) during ULOF/PP accident for different thermal properties models.
3.2 Interpretation of the results with integrated neutronic feedback coefficients
All the presented calculations used two integrated Doppler coefficients, one for the fertile zones, and one for the fissile zones. The power is therefore affected by the average fissile and fertile temperatures. One can see their evolution in Figures 10 and 11, for the calculations of the second column of Table 1 (λ_{fuel} model is always global average). Except for the global average model which mixes fertile and fissile meshes, one can see that every H_{gap} averaging leads to a cooler fuel.
This observation can be explained. Let us consider two fuel meshes, i and j, in contact with the cladding. Because of the linearity of the diffusion equation, the temperature of i can be written as: (1)with the temperature of the cladding, h_{i} the H_{gap} coefficient and α_{i} a scalar depending on local power. The same equation can be written for mesh j. We introduce now the temperatures and obtained using average H_{gap} value, that is to say . The difference between the average values with exact and average H_{gap} is equal to: (2)
is the temperature increase between fuel and cladding. This equation means therefore that using average H_{gap} reduces average fuel temperature if the H_{gap} of the hottest mesh is smaller than the one of the coolest mesh. The point is that it may be the reason why the hottest mesh is the hottest. Therefore, without strong positive correlation between power and H_{gap}, using average H_{gap} usually reduces fuel temperature.
In addition, we can prove that, starting with a cooler fuel, for the same power decrease, the Doppler effect is smaller. To show that, let us write the temperature of a given mesh in the situation i like: (3)with T^{Cl} the temperature of the cladding, P the local power and α_{i} a scalar depending on mesh state. The same equation can be written for the same mesh in the situation j by replacing α_{i} by α_{j}.
Now we consider that the power becomes, at time t, P.f with f a given factor (f < 1 in the case of a ULOF). The mesh contribution to the Doppler effect is: (4)with C a given feedback coefficient (usually C < 0). Thanks to the form chosen for equation (4) (this is the usual one), the coefficient C has no dependence on temperature. We assume here that the cladding temperature is constant. The difference between the Doppler contributions of the mesh in both situations is: (5)
This quantity is positive if C < 0, f < 1 and α_{i} > α_{j}. In other words, if the Doppler effect is negative and if the power decreases, we show that the Doppler effect is smaller for an initially cooler mesh. As a consequence, the power decreases a little bit more. These results explain the impact of spatial description of thermal properties of fuel pins we observed in Section 3.1.
Fig. 10 Average fissile temperature during ULOF/PP for different H_{gap} treatments. 
Fig. 11 Average fertile temperature during ULOF/PP for different H_{gap} treatments. 
3.3 Impact of local neutronic feedback coefficients
The previous analysis is based on the use of average fuel temperatures to calculate the Doppler feedback. One could wonder if it still stands if we use local neutronic feedbacks. Because this work is on the impact of the fuel pin thermomechanics on ULOF/PP accident, we focused our study on the Doppler effect. Comparisons of sodium maximal temperatures reached during ULOF/PP accident with and without local Doppler coefficients (the global Doppler effect is the same) are presented in Table 2. 3D maps of Doppler coefficients, derived from the perturbation theory, are given in Figure 12 (fissile) and Figure 13 (fertile).
One can see that, here, using local neutronic feedbacks always increases the sodium maximal temperature. The impact depends on the thermal properties model, but is pretty small (about 5 °C) for exact treatment of H_{gap} and λ_{fuel}. An analysis shows that this difference is mainly due to the heterogeneity of the fertile zones. Indeed, in the inner fertile there are in the same time a much stronger fuel temperature decrease and much stronger Doppler coefficients than in the fertile blanket (this is visible in Fig. 13). These two differences together create a bias when one uses integrated Doppler coefficients.
In Section 3, we saw the impact of spatial description of thermal properties. The more accurate it is, the hotter the sodium becomes during the ULOF/PP accident, whatever is the spatial treatment of the Doppler effect. We will now see the impact of the temporal evolution of the thermal properties.
Comparisons of sodium maximal temperature (°C) during ULOF/PP accident with and without local neutronic feedbacks.
Fig. 12 3D map of Doppler coefficients (pcm) in fissile zones. 
Fig. 13 3D map of Doppler coefficients (pcm) in fertile zones. 
4 Impact of temporal evolution of thermal properties during the ULOF/PP accident
4.1 Calculations comparison with integrated neutronic feedback coefficients
We used the simple explicit coupling scheme illustrated in Figure 14. GERMINAL V1.5 gives local H_{gap} and λ_{fuel} values to TETAR, which returns mass flow per pin and local power (through the global power factor calculated by the point kinetic). The coupling time step is set to 10 s. The already presented groups model for GERMINAL V1.5 is chosen in order to save calculation time.
It would be interesting to enhance this coupling scheme, and it should be done in future work. However, preliminary studies show that this scheme is correct.
The results of the coupled calculation are given in Figures 15 and 16 with equivalent noncoupled case. Here, integrated Doppler coefficients are used. One can see that the temporal coupling has a very strong impact, about –38 °C.
Fig. 14 The temporal coupling scheme used. 
Fig. 15 Sodium maximal temperature during ULOF/PP accident with and without coupling. 
Fig. 16 Power during ULOF/PP accident with and without coupling. 
4.2 Interpretation of the results with integrated neutronic feedback coefficients
This very strong impact of the coupling is due to the opening of the gap during the transient: as the cladding is getting hotter and the fuel cooler, the differential thermal expansion takes them away from each other. As a consequence, the H_{gap} decreases and the fuel temperature drop is reduced, leading to a smaller Doppler effect.
H_{gap} profile evolutions for one subassembly from the inner core is given in Figure 17, and for one subassembly from the outer core in Figure 18. The H_{gap} does decrease everywhere. It is especially important (divided by about 3) where the initial value was high: there was a contact between the cladding and the pellet at these locations. This contact is lost during the transient.
While H_{gap} changes a lot during the transient, λ_{fuel} is found to be almost constant.
One can note that the H_{gap} temporal evolution is rather smooth, and could be approximated by polynomial functions, as suggested in reference [2].
Fig. 17 H_{gap} profile evolutions during ULOF/PP accident for one subassembly from the inner core. 
Fig. 18 H_{gap} profile evolutions during ULOF/PP accident for one subassembly from the outer core. 
4.3 Impact of local neutronic feedback coefficients
We found, in Section 3, that local neutronic feedback coefficients have a small impact on ULOF/PP when used with a good spatial discretization of the thermal properties of the pins (see Tab. 2). The cause of the discrepancy has been identified to be the combined heterogeneities of H_{gap} and Doppler coefficients in the fuel zones. The temporal coupling, because it reduces the H_{gap} preferentially where it is high, that is to say in the center of the core, where the Doppler effect is the strongest, reduces these heterogeneities. As a consequence, the impact of the local feedback coefficients is reduced. This is visible in Figure 19. We used, here again, the groups model for H_{gap} and λ_{fuel}.
The comparison with the noncoupled equivalent calculation with local feedbacks coefficients leads to a reduction of the sodium maximal temperature of about 45 °C.
Fig. 19 Sodium maximal temperature during ULOF/PP accident with coupling and different Doppler effect treatments. 
5 Conclusions
In this paper, we proposed an analysis of the impact of spatial variation and temporal evolution of thermal properties of fuel pins on the behavior of a CFVlike core during an ULOF accident.
Sources of spatial variations and temporal evolution of the main thermal properties of fuel pins were identified. The impact of their spatial variations was found to be about +30 °C on sodium temperature during ULOF/PP transient. It is mainly due to H_{gap}, and simple zones averages seem to be enough for λ_{fuel}. The combined effect of local thermal properties and local Doppler coefficients leads to an impact of about +35 °C. On the other hand, the temporal coupling, because of the opening of the gap, improves the reactor behavior during the ULOF/PP and leads to a decrease of about 45 °C of the sodium temperature. This improvement of the core behavior is very strong and could help greatly to demonstrate the safety of large SFRs.
From the above observations we can make the following comments:

a static estimation of λ_{fuel} in the main zones of the core is sufficient;

for a conservative calculation, the spatial variations of H_{gap} and of the Doppler effect should be taken into account;

the temporal coupling between thermalhydraulics and thermalmechanics of fuel pins brings out substantial margins, because of the H_{gap} evolution.
References
 M.S. Chenaud et al., Status of the ASTRID core at the end of the preconceptual design phase 1, in Proceedings of ICAPP 2013, Jeju Island, Korea (2013) (In the text)
 R. Lavastre et al., State of art of CATHARE model for transient safety analysis of ASTRID SFR, in Proceedings of NUTHOS10 Okinawa, Japan 2014 (2014) (In the text)
 H. Golfier et al., APOLLO3: a common project of CEA, AREVA and EDF for the development of a new deterministic multipurpose code for core physics analysis, in Proceedings of M&C 2009 New York, USA 2009 (2009) (In the text)
 L. Roche, M. Pelletier, Modelling of the thermomechanical and physical process in FR fuel pins using GERMINAL code, in Proceedings of MOX Fuel Cycle Technologies for Medium and Long Term Deployment IAEASM358/25, Vienna, Austria 1999 (1999), 322335 (In the text)
 E. Hourcade et al., SFR core design: a systemdriven multicriteria core optimisation exercise with TRIAD, in Proceedings of FR13 Paris, France 2013 (2013) (In the text)
 G. Rimpault et al., The ERANOS code and data system for fast reactor neutronic analyses, in Proceedings of PHYSOR 2002 Seoul, Korea 2002 (2002) (In the text)
Cite this article as: Cyril Patricot, Grzegorz Kepisty, Karim Ammar, Guillaume Campioni, Edouard Hourcade, Thermalhydraulics/thermalmechanics temporal coupling for unprotected loss of flow accidents simulations on a SFR, EPJ Nuclear Sci. Technol. 2, 2 (2016)
All Tables
Comparison of sodium maximal temperature (°C) during ULOF/PP accident for different thermal properties models.
Comparisons of sodium maximal temperature (°C) during ULOF/PP accident with and without local neutronic feedbacks.
All Figures
Fig. 1 Scheme of the CFV core concept. 

In the text 
Fig. 2 Linear power distribution (W/cm by pin) in the center of the core at beginning of cycle. 

In the text 
Fig. 3 Linear power distribution (W/cm by pin) in the center of the core at end of cycle. 

In the text 
Fig. 4 Typical heat transfer coefficient evolution for chosen subassemblies (in top fissile zone). 

In the text 
Fig. 5 Typical gap size evolution for chosen subassemblies (in top fissile zone). 

In the text 
Fig. 6 Typical 3D map of H_{gap} (W/cm^{2}/K). 

In the text 
Fig. 7 Typical 3D map of λ_{fuel} (W/cm/K). 

In the text 
Fig. 8 Sodium maximal temperature during ULOF/PP accident for different thermal properties models. 

In the text 
Fig. 9 Power during ULOF/PP accident for different thermal properties models. 

In the text 
Fig. 10 Average fissile temperature during ULOF/PP for different H_{gap} treatments. 

In the text 
Fig. 11 Average fertile temperature during ULOF/PP for different H_{gap} treatments. 

In the text 
Fig. 12 3D map of Doppler coefficients (pcm) in fissile zones. 

In the text 
Fig. 13 3D map of Doppler coefficients (pcm) in fertile zones. 

In the text 
Fig. 14 The temporal coupling scheme used. 

In the text 
Fig. 15 Sodium maximal temperature during ULOF/PP accident with and without coupling. 

In the text 
Fig. 16 Power during ULOF/PP accident with and without coupling. 

In the text 
Fig. 17 H_{gap} profile evolutions during ULOF/PP accident for one subassembly from the inner core. 

In the text 
Fig. 18 H_{gap} profile evolutions during ULOF/PP accident for one subassembly from the outer core. 

In the text 
Fig. 19 Sodium maximal temperature during ULOF/PP accident with coupling and different Doppler effect treatments. 

In the text 