ANALYSIS OF THE COOLANT DENSITY REACTIVITY COEFFICIENT IN LFRs AND SFRs VIA MONTE CARLO PERTURBATION/SENSITIVITY

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1 ANALYSIS OF THE COOLANT DENSITY REACTIVITY COEFFICIENT IN LFRs AND SFRs VIA MONTE CARLO PERTURBATION/SENSITIVITY Manuele Aufiero, Michael Martin and Massimiliano Fratoni University of California, Berkeley, Department of Nuclear Engineering Berkeley, CA USA Emil Fridman Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstraße 400, Dresden, Germany Stefano Lorenzi Politecnico di Milano, Department of Energy Via la masa 34, Milano, Italy ABSTRACT The coolant density effect represents one of the main reactivity feedback in LFR and SFR. These reactivity feedback effects are often calculated via direct perturbation in Monte Carlo codes. This work presents a new approach for reactivity coefficient calculations based on Perturbation Theory. New methods were implemented in a extended Serpent version and are here verified against reference results for spatial coolant reactivity maps. In the paper, a detailed investigation of the effect of the calculation parameters and the applicability range of the perturbation calculations is presented. The main advantage of the present approach is the capability of providing, within a single Monte Carlo run, several reactivity coefficients. This feature is demonstrated via the reactivity feedback decomposition in energy, space, isotope and reaction in the ALFRED lead cooled reactor case. The obtained decompositions suggest that the reactivity coefficient is the result of large opposite effect. A preliminary uncertainty quantification analysis shows that 208 Pb cross sections uncertainties have a large impact on k eff estimates in the ALFRED reactor. Key Words: Lead cooled fast reactors; Sodium cooled fast reactors; Coolant void coefficient; Serpent Monte Carlo; Sensitivity/Uncertainty analysis; Perturbation theory 1. INTRODUCTION The coolant density coefficient represents one of the main reactivity feedback in Lead-cooled Fast Reactors (LFRs) and Sodium-cooled Fast Rectors (SFRs), and its accurate calculation is important for a 3493

2 correct evaluation of the dynamics of these systems. Coolant density reactivity maps have been calculated in the past adopting perturbation theory in deterministic codes. Usually, full-core simulations employed multi-group diffusion codes or 2D (r, z) geometrical approximations. Nowadays, Monte Carlo neutron transport simulations are commonly adopted for the study of LFR and SFR [1, 2]. Nonetheless, reactivity feedbacks are usually calculated via direct perturbations, i.e., comparing the effective multiplication factor of two separate Monte Carlo runs [3, 4]. When small effects are to be investigated via the direct perturbation approach, the adoption of either large system perturbations or a large number of simulated particles is required, in order to reduce the statistical errors. Moreover, if spatial maps of coolant density reactivity coefficients are to be generated via direct perturbation, one criticality source Monte Carlo simulation is required for each spatial region. In this view, the sensitivity/perturbation methods offer the advantage of producing a large number of sensitivity coefficients in a single calculation [5]. More important, this approach allows decomposing reactivity effects by energy and reaction for a deeper investigation of the feedback. In this work, two LFR and SFR core designs are considered, focusing on the calculation and analysis of the coolant density reactivity coefficient. The space-dependent lead and sodium density reactivity worth are calculated adopting the sensitivity/perturbation capabilities recently implemented in an extended version [6] of the Serpent-2 code [7], previously adopted for the calculation of coolant void maps [8]. The present work focuses on the verification of the sensitivity/perturbation results against direct perturbation calculations, on the analysis of the optimal parameters to be adopted for the simulations and on the discussion of the peculiar results obtained for the two considered cases. A preliminary analysis of the effect of nuclear data uncertainties in the LFR system is presented as well. The effect of 208 Pb cross-section uncertainties is evaluated by means of both the Total Monte Carlo (TMC) method and by adopting advanced perturbation techniques (XGPT), and is briefly discussed. (a) ALFRED, xy cross-section (b) ALFRED, xz cross-section Figure 1: Schematic illustration of the ALFRED core showing inner fuel elements (yellow), outer fuel elements (light blue) and reflector/dummy elements (green). The control elements positions are shown in red. 3494

3 The Advanced Lead Fast Reactor European Demonstrator (ALFRED), developed within the European FP7 LEADER (Lead-cooled European Advanced Demonstration Reactor) Project, is a small-size (300 MWth) pool-type LFR conceived to be a scaled-down demonstrator fully representative of the industrial scale reference system. The ALFRED core is composed by wrapped hexagonal Fuel Assemblies (FAs) with pins arranged on a triangular lattice. The 171 FAs are subdivided into two radial zones with different plutonium fractions guaranteeing an effective power flattening, and surrounded by two rows of dummy elements serving as reflector [2]. Two different and independent control rod systems are foreseen, namely, Control Rods (CRs) and Safety Rods (SRs). The former performs power regulation and reactivity swing compensation during the cycle, while the simultaneous use of both is foreseen for scram purposes, assuring the required reliability for a safe shutdown. In Fig. 1a, a simplified XY cross section of the reactor core is represented for illustrative purposes. Fig. 1b shows the seven different core zones adopted for the calculation of the coolant void map. Figure 2: Schematic illustration of the XY cross section of the OECD/NEA SFR core showing inner fuel elements (yellow), outer fuel elements (green) and reflector/dummy elements (gray). The control elements positions are shown in red. The chosen SFR core design is a large 3600MWth U-Pu mixed oxide (MOX) core specified by the OECD/NEA SFR Benchmark Task Force [9]. The core is loaded with 225 inner and 228 outer MOX fuel sub-assemblies with axially variable Pu content, and 27 control elements as shown in Fig. 2. In the present work, ten different spatial core zones are considered for the calculation of the sodium density reactivity map. Namely, the inner and outer active fuel regions are divided in 5 axial zones each. 3495

4 Table I: Sodium density k eff sensitivity coefficient in different OECD/NEA SFR core zones. Comparison between sensitivity/perturbation and direct perturbation Serpent results. k eff sodium density sensitivity coefficient Axial Inner fuel region Outer fuel region zone Direct Pert. # Pert. Theory Diff. Direct Pert. # Pert. Theory Diff ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± pcm/(% sodium density) ± 1 sigma # Direct pert. with 20% sodium density variation 2. COOLANT DENSITY REACTIVITY MAPS In the following, the Serpent perturbation theory calculations obtained via the collision history approach are verified against direct perturbation results. The coolant density reactivity effects are decomposed by spatial region, isotope, energy and reaction for a better understating of the feedback phenomena Verification of the Serpent perturbation calculations Adopting the Iterated Fission Probability (IFP) method [10, 11], the coolant density reactivity coefficient can be calculated from the neutron particles collision histories via the following formula [6]: S k eff x dk eff/k eff dx/x = E [ ( γ) ACC x ( γ) REJ x ] (1) where S k eff x represents the relative k eff sensitivity coefficient to the parameter x. ( γ) ACC x and ( γ) REJ x represent the accepted and rejected events x (i.e., events related to the parameter x ) in the history of the particles 1, for the generation γ [see 6, for the details of the method and the implementation]. Here, the considered parameters x are defined as the density of coolant materials (i.e., sodium and lead isotopes) in different spatial core regions. Consequently, for the purposes of the present work, the collision history approach keeps track of accepted and rejected scattering and absorption events in the different spatial zones. γ represents the number of latent generations [11] adopted in the calculations for the convergence of the adjoint IFP estimators. The impact of this parameter on the accuracy of the Monte Carlo results is discussed in the following. 1 E [X] represents the expected value over the whole neutron population of the quantity X associated to each particle. 3496

5 Table II: Lead density k eff sensitivity coefficient in the considered ALFRED reactor core zones. Comparison between sensitivity/perturbation and direct perturbation Serpent results. k eff lead density sensitivity coefficient Fuel region Axial zone Direct Pert. # Pert. Theory Difference Inner Active 7.54 ± ± Outer Active 2.69 ± ± Reflector 4.34 ± ± Inner Lower plenum 4.86 ± ± Outer Lower plenum 7.20 ± ± Inner Upper plenum 5.64 ± ± Outer Upper plenum 6.96 ± ± Total 30.4 ± ± pcm/(% lead density) ± 1 sigma # Direct pert. with 10% lead density variation In Table I and Table II, the comparisons between direct perturbation and sensitivity/perturbation results are presented for the coolant density k eff sensitivity maps in the OECD/NEA SFR and in ALFRED, respectively. Ten latent generations were adopted for the convergence of the IFP adjoint estimators. For better clarity, results are expressed in pcm of reactivity change over % of relative coolant density variation. Differences in the results beyond 2 sigma are highlighted with bold characters. In both cases, fairly good agreement can be appreciated between the two methods, confirming the promising capabilities of the Monte Carlo perturbation theory approach for fast reactor analysis. In the case of the SFR core, all the considered zones present a negative k eff sodium density sensitivity coefficient (which corresponds to a positive void coefficient). This shows that spectral effects are dominant in this case: the reactivity of the system increases upon local reduction of the sodium density over the whole active core region. In the case of ALFRED (LFR), only the central active zone shows a negative sensitivity coefficient. The remainder regions of the considered coolant density map show positive sensitivities due to increased leakage. In both cases, the number of active particles in the direct perturbation Serpent runs was adjusted in order to have statistical errors for the k eff estimated in the order of 1 pcm or below Non-linearity effect and convergence of the IFP estimators The present collision history approach to sensitivity/perturbation calculations adopts first-order perturbation theory. For this reason, its applicability is limited to coolant density perturbations for which 3497

6 second order effects can be neglected. 2 In Fig. 3, non-linearity effects are presented for two selected cases. 3 Preliminary results show that larger non-linearity arises when high leakage components of the coolant reactivity effects are present, as result of second order effects, possibly related to large modifications of the flux distribution. Nonetheless, in all the considered cases, the linear approximation resulted to be suitable for coolant void reactivity analysis up to 10% density perturbation and, in most cases, well beyond this threshold. 300 OECD/NEA SFR - Sodium void reactivity effect - Non-linearity effects - Inner fuel central region ALFRED - Lead void reactivity effect - Outer fuel active region 0 Non-linearity effects Perturbation/sensitivity Direct perturbation -10 Perturbation/sensitivity Direct perturbation ρ [pcm] ρ [pcm] Sodium density perturbation [%] Lead density pertubration [%] (a) SFR, inner fuel region, central axial zone. (b) ALFRED (LFR), outer fuel region, active zone. Figure 3: Evaluation of the non-linearity effects in the coolant void reactivity. Adjoint-weighting in forward criticality source Monte Carlo simulations is obtained via the Iterated Fission Probability (IFP) method [10, 11]. This approach requires that information is passed through the neutron generations to the descendants. Despite the fact that different implementations of the IFP can be adopted, the number of latent generations required to reach the convergence of the adjoint estimators might be a limiting factor to the efficient applicability of this method to full core reactor analysis. A detailed investigation of this issue for LFR and SFR full core analysis has not been performed in previous works. Preliminary encouraging results are presented here. The ongoing investigations show that the correct choice of the number of latent generations adopted in the sensitivity/perturbation Monte Carlo calculations is crucial to the effective production of accurate results. On one side, the Iterated Fission Probability estimators require several latent generations for the convergence of the perturbation effects on the fission source. On the other hand, increasing the number of latent generations introduces larger statistical errors in the estimated parameters, reducing the efficiency of the perturbation/sensitivity method. In Fig. 4, a clear example of this behavior is reported for the ALFRED outer fuel region k eff lead density sensitivity coefficient. The convergence is reached at around ten latent generations, and the sensitivity coefficient is largely underestimated if only few generations are adopted. In this case, a rather unexpected result is obtained: the sensitivity 2 In [8], simple strategies to overcome this limitation are presented and discussed. 3 In Fig. 3a, statistical errors for the direct perturbation calculations are below 1 pcm and were omitted from the plot. 3498

7 estimates at different latent generations have opposite values. This highlights that more generations are required for the convergence of perturbations dominated by leakage effects, in which the deformation of the fission source plays a major role. ALFRED - Lead density k eff sensitivity coefficient Latent generation convergence - Central outer fuel region ALFRED - Lead density k eff sensitivity coefficient std. dev. Effect of latent generations - Central outer fuel region Sens. coeff. [pcm/(% Lead density)] Direct perturbation Sensitivity/perturbation Sens. coeff. std. dev. [pcm/(% Lead density)] Latent generations Latent generations (a) Lead density sensitivity coefficient. (b) Statistical error (std. dev.). Figure 4: Effect of latent generations on the convergence and the statistical errors of the k eff lead density sensitivity coefficient in the outer fuel region of ALFRED. In Fig. 5, the estimated sodium density reactivity coefficients for the ten fuel regions of the OECD/NEA SFR core are shown for different latent generations, along with 1 sigma statistical error bars. Differently from the previous case, five generations appear to give a satisfactory estimate of the whole coolant density map, and moving to ten generations provides no statistically significant improvement. Figure 5: Effect of latent generations on k eff sodium density sensitivity coefficient in all the considered fuel region of the OECD/NEA SFR core. As a general remark, a single optimal value for the number of latent generations for different systems 3499

8 and core zones can not be suggested. For this reason, the capability of the present collision history approach to simultaneously produce results with different number of latent generations revealed to be very useful for the analysis of coolant density reactivity maps. In any case, the number of required latent generations shows to fall well within the capabilities of the available Monte Carlo codes, confirming the adequacy of the present approach for LFR and SFR Coolant density coefficient decomposition The peculiar behavior presented in the previous sections suggested that a deeper investigation of the sensitivity coefficient for the ALFRED outer fuel region could be of interest. This investigation is also presented as a further example of the capabilities of the Monte Carlo sensitivity/perturbation approach. In Fig. 6, the energy and reaction decomposition of the coolant density reactivity coefficient is shown, as sum over 204 Pb, 206 Pb, 207 Pb and 208 Pb. Results are scored on the standard ECCO 33 groups energy grid and are normalized per lethargy width of the energy bins. The reactivity effect is the sum of large opposite terms. The largest contribution is represented by the elastic scattering reactions, impacting the leakage term mainly between a few kev and a few MeV. Inelastic scattering close to the Mev region has a large opposite spectral effect, impacting the average number of fission neutron produced per neutron absorption in the fuel. Radiative capture has a smaller effect, directly related to the loss term of the neutron balance. In Fig. 6b, the effect of latent generations is highlighted, showing the difference in the sensitivity profiles between ten and zero latent generations. As discussed above, the elastic scattering component introduces a perturbation in the fission source distribution, leading to a large difference between converged results and zero latent generations estimates. Spectral and direct effect (capture and inelastic scattering) have a smaller impact on the fission source and lower differences are observed between the two cases. These plots are consistent with the unexpected result of coolant density sensitivity coefficient changing from negative to positive through the latent generations convergence. In Table III, the lead density sensitivity coefficient for the ALFRED Outer Fuel Active region is further decomposed by isotope contributions. This analysis shows that the global effect is the results of the sum of positive ( 208 Pb) and negative ( 204 Pb and 206 Pb) terms. In order to confirm this perturbation-based reactivity decomposition, direct perturbation results have been obtained by running separate Serpent calculations in which the atomic density of a single isotope was modified by 10%. These results are presented in Table III and compared to perturbation theory estimated. 3500

9 Table III: Lead density k eff sensitivity coefficient in the ALFRED Outer Fuel Active region. Decomposition by lead isotopes. Comparison between sensitivity/perturbation and direct perturbation Serpent results. k eff lead density sensitivity coefficient lead isotope reaction Direct Pert. # Pert. Theory Difference 204 Pb total 0.61 ± ± elastic scattering 0.34 ± 0.04 capture 0.75 ± 0.00 inelastic scattering 0.27 ± Pb total 2.08 ± ± elastic scattering 3.36 ± 0.15 capture 1.77 ± 0.00 inelastic scattering 4.19 ± Pb total 5.73 ± ± elastic scattering 8.23 ± 0.23 capture 0.34 ± 0.00 inelastic scattering 2.02 ± 0.02 Sum + total 2.69 ± ± elastic scattering ± 0.31 capture 3.99 ± 0.01 inelastic scattering 9.06 ± 0.04 pcm/% ± 1 sigma # Direct pert. with 10% atomic density variation + The small contribution of 207 Pb is not presented in the table, but is taken into account in the Sum section 3501

10 ALFRED - Lead density sensitivity coefficient Decomposition by energy and reaction - 0 and 10 latent generations ALFRED - Lead density sensitivity coefficient Difference between 10 latentent generations and 0 latent generations estimates Sensitivity coefficient (per lethargy unit) Total - 10 gen Total - 0 gen Inelastic - 0 gen ( ) Elastic - 0 gen ( ) Capture - 0 gen ( ) Inelastic - 10 gen ( ) Elastic - 10 gen ( ) Capture - 10 gen ( ) Total 0 gen Total 10 gen Sensitivity coefficient (per lethargy unit) Total total Total inelastic Total capture Total elastic Energy [ev] (a) Lead density sensitivity coefficient Energy [ev] (b) Zero to ten latent generations difference. Figure 6: Energy and reaction decomposition of the k eff lead density sensitivity coefficient. ALFRED, outer fuel active region. 3. PRELIMINARY QUANTIFICATION OF THE EFFECT OF 208 Pb CROSS SECTIONS UNCERTAINTIES ON ALFRED In the previous sections, results from the decomposition of the reactivity effects in ALFRED highlighted that lead density feedback are composed by very large opposed terms. This suggested that large uncertainty on these estimates might arise from nuclear data, and called for further investigation of this aspect. For this reason, two advanced approaches for nuclear data uncertainty quantification were adopted to study the impact of 208 Pb cross section: namely, the Total Monte Carlo method (TMC) and the XGPT methods. The Total Monte Carlo method (TMC) [12] is an innovative approach to nuclear data uncertainty propagation. In TMC, a large number of independent ENDF files are randomly generated starting from nuclear models parameters and their uncertainties. These files are then processed with NJOY or equivalent codes to produce a set of formatted continuous-energy cross sections (i.e., ACE files), and independent Monte Carlo neutron transport simulations are run with these different ACE files as input. Finally the distributions of the considered response functions are obtained directly from the results of the Monte Carlo runs. In the present work, 3000 independent 208 Pb ACE files were produced with NJOY adopting the reference TENLD-2013 evaluation [13] along with the MF2 MT151 (resonance parameters) and MF3 MT1,2,16,51,52,53,54,55,56,102 sections of random ENDF files. In this way, 3000 continuous energy 208 Pb nuclear data sets were produced with random elastic scattering, inelastic scattering, (n, 2n) and capture cross sections independent Serpent criticality source simulations were run for the AL- FRED LFR core geometry, each using one of those ACE files. Each run employed active 3502

11 histories in order to determine the effective multiplication factor k eff within statistical errors of 20 pcm or less. The runs required about 23 minutes (wall-clock time) each on 20 cores Intel(R) Xeon(R) CPU E GHz nodes. The XGPT method [? ] is a newly developed, perturbation-based approach, which allows reproducing the results of thousands TMC runs within a single Serpent simulation (total cpu time below 1 hour on a single node). In Fig. 7, the effective multiplication factor probability distribution is presented, as estimated by the two methods. Results are expressed with reference to the average estimate of k eff. Apart from a fairly good agreement between the two approaches, Fig. 7 shows a rather large k eff uncertainty, in the order of 350 pcm (1 sigma), only considering a single isotope ( 208 Pb). This large uncertainty is consistent with previously published works involving fast spectrum lead cooled systems (e.g., see [14]) ALFRED - keff uncertainty - XGPT vs. TMC keff distribution from 208 Pb cross sections uncertainty (from TENDL-2013) Total Monte Carlo XGPT Number of counts per bin [-] keff - keff [pcm] Figure 7: k eff distribution in ALFRED due to 208 Pb cross section uncertainties. Comparison between TMC and XGPT results. As a simple extrapolation of these results, the k eff uncertainty from 208 Pb can be employed as a first order approximation of the direct term of the uncertainty in the global coolant density reactivity coefficient in ALFRED. This provides an estimate of 3.5 pcm/% (1 sigma) in the feedback effect, which is slightly more than 10% of the reference value. 4. CONCLUSIONS In this work, the collision history approach to sensitivity/perturbation calculations implemented in an extended Serpent version has been adopted to study the coolant density reactivity feedback in two LFR and SFR cores. In the considered cases, the ALFRED LFR core showed negative coolant void reactivity feedback (positive coolant density k eff coefficients) in most of the core zones. On the other hand, the 3503

12 OECD/NEA SFR core presented mainly positive sodium void effects. Perturbation-based results were extensively verified against direct perturbation calculations. The perturbation approach proved able to efficiently generate accurate spatial sensitivity maps for full-core fast reactors analysis. The convergence of the adjoint-weighted Iterated Fission Probability estimators arose as an important parameter which might compromise the accuracy of the results. The calculation of coolant reactivity feedback resulted to be particularly sensitive to the number of latent generations, especially for regions in which leakage effects are dominant. The LFR core has been selected for a deeper investigation via the decomposition by energy, isotopes and reactions of the coolant density coefficient. In the considered case, the lead density coefficient resulted from the contributions of large and opposed terms. In particular, elastic scattering provides a large positive contribution to the lead density k eff coefficient, mainly acting on the leakage term. Inelastic scattering and capture reactions give negative contributions via spectral effects and direct terms. A preliminary analysis of the 208 Pb cross sections uncertainty suggests that the impact of nuclear data on the coolant void estimated might be rather large, and calls for a deeper investigation of the problem. REFERENCES [1] L. Buiron et al. Evaluation of large 3600mwth sodium-cooled fast reactor neutronic oecd benchmarks. In: PHYSOR 2014, Kyoto, Japan, Sep Oct. 3, 2014 (2014). [2] G. Grasso et al. The core design of alfred, a demonstrator for the european lead-cooled reactors. Nuclear Engineering and Design, 278: pp (2014). [3] S. Bortot et al. European benchmark on the astrid-like low-void-effect core characterization: neutronic parameters and safety coefficients. In: Proceedings of ICAPP 2015, Nice, France, May 03-06, [4] S. Lorenzi, A. Cammi, and L. Luzzi. Spatial neutronics modelling to evaluate the temperature reactivity feedbacks in a lead-cooled fast reactor. In: Proceedings of ICAPP 2015, Nice, France, May 03-06, 2015 (2015). [5] M. L. Williams. Perturbation theory for nuclear reactor analysis. CRC Handbook of Nuclear Reactors Calculations, 3: pp (1986). [6] M. Aufiero et al. A collision history-based approach to sensitivity/perturbation calculations in the continuous energy monte carlo code serpent. Annals of Nuclear Energy, 85: pp (2015). [7] PSG2 / Serpent Monte Carlo Reactor Physics Burnup Calculation Code. URL (2011). [8] S. Pelloni et al. Sodium void map preparation for the safety analysis of sodium-cooled fast reactors by using the monte carlo code serpent. In: Proceedings of ICAPP 2015, Nice, France, May 03-06, 2015 (2015). 3504

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