Use of Monte Carlo and Deterministic Codes for Calculation of Plutonium Radial Distribution in a Fuel Cell

Transcription

1 Use of Monte Carlo and Deterministic Codes for Calculation of Plutonium Radial Distribution in a Fuel Cell Dušan Ćalić, Marjan Kromar, Andrej Trkov Jožef Stefan Institute Jamova 39, SI-1000 Ljubljana, Slovenia dusan.calic.@ijs.si, marjan.kromar@ijs.si, andrej.trkov@ijs.si ABSTRACT To predict accurately the formation of plutonium within a reactor lattice fuel cell it is necessary to divide the fuel into rings, because the build-up of plutonium is affected by the spatial resonance self shielding in 238 U, which results in the enhanced formation of plutonium near the surface of the fuel cell. The so called rim effect has already been studied in the past. Our goal was to check the prediction of the same effect with the WIMS-D5 deterministic code employed in the CORD-2 system, developed at the Reactor Physics Department of the Jožef Stefan Institute, and used for core design calculations of pressurized water reactors. The calculations were performed on an array of 3 3 lattice cells. The results show an overestimated prediction of the quantity of plutonium ( %) with burnup, compared to the reference case obtained with the Serpent code, while the rim effect in the outer layer of the fuel cell is not reproduced at all. Since the WIMS-D5 code failed in predicting the correct radial plutonium concentration in the fuel we performed the same calculations with Monte Carlo codes MCNP and Serpent and the deterministic code DRAGON. This paper presents the results from the listed codes and compares their results with regard to the computational efficiency and the accuracy for future consideration. 1 INTRODUCTION The reactor is such a complex object in terms of geometry, composition, and nuclear data, that it is not simple to include all details of a problem into a calculation code and obtain the results for a complete and exact simulation. We wish to select the best model or code to achieve the best results in terms of precision and reasonable computation times. Usually the precision of the results is proportional to the time of the calculations. However it is possible to combine different methods to achieve a calculation scheme which is both fast and accurate. The purpose of this paper is to perform such lattice calculation with the use of the deterministic lattice code DRAGON and WIMS-D5 and to benchmark the results obtained with the Monte Carlo codes MCNP and Serpent. For realistic systems with complex geometry and detailed energy dependency, the transport equation can only be solved by using numerical methods implemented in the lattice codes. Those calculations go through several levels before the final solution is obtained. Each level of the calculation has its own characteristics, and any simplification at any step can lead to a poor final result

2 310.2 The paper focuses on the self shielding step where we considered many resonant energy groups in the self shielding calculations. We divide the fuel cell into 10 rings, so that we can obtain the correct distribution of 239 Pu within the fuel cell. The so called rim effect can be obtained by calculating the capture rate distribution in 238 U with the use of different self shielding calculations [1], [2]. 2 SELF SHIELDING MODELS Self shielding treatment is a very important aspect of the lattice codes. It is an algorithm that produces average or self shielded cross sections defined over energy groups that cover the complete energy domain of the neutrons in a nuclear reactor. The microscopic self shielded cross section for any reaction x in group g is defined as: where σ xg = ΔEg σ x ΔEg ( E) Φ ( ) Φ E de E de, (1) ( ) Φ(E) = flux spectrum, σ x (E) = microscopic cross section for nuclear reaction x. In order to calculate the self shielded cross section σ xg in group g for the reaction x, the flux Φ(E) must be known. Since the flux is the quantity we are searching for, additional techniques are required. In general two different models are usually applied [3]: model based on equivalence in dilution model based on a subgroup approach. 2.1 Model based on equivalence and dilution This approach is based on the rational expansion of fuel to fuel collision probabilities, either in closed or open cell (or assembly). For infinite and homogeneous problems each self shielded cross section of each resonant isotope is tabulated against the dilution parameter. For heterogeneous problems a heterogeneous resonant situation is replaced with a linear combination of homogeneous resonant problems. In its simplest form this technique reduces to the use of Bell and Dancoff factors. This kind of model is implemented in the deterministic code WIMS-D5. The extension of this model has been proposed by Stamm ler and Abbate (PHOENIX code) and later by Hébert and Marleu (DRAGON code, SHI module) [3]. It is known as a generalized Stamm ler model (GSM). To achieve better accuracy with the GSM model, two additional improvements are implemented in the Dragon code [4]: 1. use of the Nordheim distributed self shielding effects in a fuel rod (GSM+1), and 2. the Nordheim distributed self shielding model and use of the Riemann integration method (GSM+2). 2.2 Model based on a subgroup approach The second class of resonant self shielding models relies on the subgroup approach where the detailed energy dependent cross section behavior in each energy group is replaced by its probability density representation which leads to so called probability tables. Due to its nature of approximation, additional corrections must be applied for the thermal lattices. Those corrections are implemented in the DRAGON code using, the USS module. Two options can be used to calculate the probability tables [4]:

3 SUBG approach (USS SUBG): where the physical probability tables are computed using the RMS approach. The slowing down operator of each resonant isotope is represented as a pure statistical resonance (ST), statistical-intermediate (ST/IR) or statistical-wide (ST/WR) resonance model, or 2. PTSL approach (USS PTSL): where mathematical probability tables and slowing down correlated weight matrices can be computed in selected energy groups using the Ribbon extended approach. With the use of the DRAGON code we tested the performance of the above mentioned self shielding models with and without the Livolant and Jeanpierre normalization scheme (LJ and NOLJ) that modifies the self shielded averaged neutron fluxes in heterogeneous geometries. We calculated k inf and reaction rates of fresh fuel and benchmarked them against the results obtained using Monte Carlo code MCNP5. We also studied the burnup calculations and benchmarked the results against those obtained using the Monte Carlo codes Serpent. 3 PHYSICAL MODEL OF THE BENCHMARK The model used in the benchmark is an example of a PWR lattice shown in Figure 1. Model characteristics, material and geometry data are given in Table 1. Figure 1 shows a typical 3 3 pin-array configuration. The extra region of Zr, Ni and water on the outside is added to preserve fuel to the moderator ratio of the overall assembly (including water holes, inter-assembly gaps and structural materials) as it is in the Krško NPP core. Full cells are divided into ten rings with equal volumes and equal atomic fractions of fresh fuel. Reaction rates are monitored only in the central cell. Figure 1: Representation of PWR lattice model used in calculations. Table 1: Material composition by natural elements is directly applicable to WIMS-D5 and Dragon codes (WIMS-D lib.); MCNP, Serpent and Dragon (Draglib lib.) use equivalent isotopic composition calculated with the MATSSF program ( ). Material Isotopic structure Temp. [K] Fuel [4.75%] U-234, U-235, U-238, O-nat 300 Clad1 Si-nat, Cr-nat, Zr-nat, Sn-nat, Hf-nat, W-nat, Pb-nat, Fe-nat, C-nat 300 Clad2 Al-27, Ti-nat, Cr-nat, Ni-nat, Nb-nat, Mo-nat, Fe-nat 300 Water H-1, O-nat and boron concentration 400 ppm 300 Gap He-4 300

4 NUCLEAR CODES USED FOR ANALYSIS In terms of solving the transport equation different methods can be used. Two different classes of codes are available in this respect: stochastic and deterministic models. In our study we used stochastic models in MCNP and Serpent, which are the most accurate but also the most time consuming, as well as deterministic codes WIMS-D5 and DRAGON Version 4. MCNP5 [5] is a well known and widely used Monte Carlo code for neutron, photon and electron transport simulation. It was tested on several criticality benchmarks, so it is verified to be a reliable and accurate code. Therefore we evaluated all of the results of k inf of fresh fuel against the results obtained with MCNP5. Serpent is a continuous-energy Monte Carlo reactor physics burnup calculation code [6] developed at the VTT Technical Research Centre in Finland and publicly available from the OECD/NEA Data Bank. Serpent code uses a single unionized energy grid for all reaction cross sections, which minimizes the number of timeconsuming grid search iterations, so the transport simulation can run significantly faster, which is important when burnup calculations are performed. The WIMS-D5 code belongs to the family of lattice codes called WIMS. The original WIMS code developed by AEE Winfrith [7] has been modified and adjusted to the special types of problems over the years. The WIMS-D5 version is available from the OECD/NEA Data Bank and was extensively used in many laboratories. The code is employed also in the CORD-2 system [8], developed at the Reactor Physics Department of the Jožef Stefan Institute, which is used for the core design calculations of PWRs. DRAGON is supported and developed at École Polytechnique de Montréal and provides 1D and 2D solutions with different modules [4]. In our calculations we used the collision probability method, using the EXCELL module. DRAGON is available from RSICC, NEA Data Bank or directly from École Polytechnique. One of the important features of the DRAGON code is the use of different cross section libraries: WIMS-AECL, WIMS-D, APOLLO and Draglib formats. 5 NUCLEAR DATA LIBRARIES In order to obtain the final solution, isotopic cross section libraries are required. In our calculations we used nuclear data originally described in the Evaluated Nuclear Data File (ENDF) format. These data are then processed using the NJOY code with different modules to obtain specific library formats. We used the following libraries: a. 172 group WIMS-D library format based on ENDF/B-VII.0 evaluated nuclear data library (WIMS-D5 and DRAGON 4.04) b. 69 group WIMS-D library format based on ENDF/B-VII.0 evaluated nuclear data library (WIMS-D5) c. 172 group Draglib library format based on ENDF/B-VII.0 evaluated nuclear data library (DRAGON 4.04) d. a compact ENDF ACE format in MCNP5 based on ENDF/B-VII.0 (MCNP5) e. Serpent cross section library based on ENDF/B-VII. This is a continuous energy ACE format data library generated using NJOY with 0.01 fractional reconstruction tolerance (Serpent v )

5 RESULTS FOR FRESH FUEL The results of k inf for fresh fuel composition for various codes are given in Table 2. From the table it can be seen that there is broad range of values, especially in the case of the DRAGON code, where we could test various self shielding models and libraries. Table 2: K inf obtained using various computer codes with various self shielding models. Computer Self shielding Library k k[pcm] code model (see section 5) inf (k inf (MCNP) - k inf ) MCNP - d ±9 pcm Reference Serpent - e ±9 pcm -44 WIMS-D5 default a WIMS-D5 default b DRAGON GSM+NOLJ+0 c DRAGON GSM +NOLJ+1 c DRAGON GSM +NOLJ+2 c DRAGON GSM +LJ+0 c DRAGON GSM +LJ+1 c DRAGON GSM +LJ+2 c DRAGON USS-SUBG c DRAGON USS-PTSL c DRAGON GSM +NOLJ+0 a DRAGON GSM +NOLJ+1 a DRAGON GSM +NOLJ+2 a DRAGON GSM +LJ+0 a DRAGON GSM +LJ+1 a DRAGON GSM +LJ+2 a Table 2 shows that results obtained with Serpent and WIMS-D5 in the case of the 172 group library are good, since k is < 100 pcm compared to the results obtained with MCNP5. In the case of the 69 group library the difference in k inf is 190 pcm. On the other hand the discrepancies in the DRAGON results are large. The best result is obtained with the use of the Draglib library and the use of GSM+LJ+2 option. There is a substantial increase in k inf of around 1000 pcm when the NOLJ option is used. Among the self shielding models based on the subgroup approach, USS-PTSL gives better result than USS-SUBG, however k inf is still under-predicted by around 100 pcm. Since the differences from the reference result for the DRAGON code obtained with the WIMS-D formatted libraries are in the range of pcm in the case of fresh fuel, we only used the Draglib library format based on ENDF/B-VII.0 in the burnup calculations. However, we will first present the effect of the self shielding model on the radial distribution of reaction rates. The fuel cell is subdivided into ten rings with equal volume, and the capture rates in 238 U are computed in each ring. Each ring has it own mixture index and identical initial isotopic composition. The distribution of the capture rates in 238 U is plotted in Figure 2. It can be seen that the three DRAGON self shielding models are close to the MCNP results. In those models the capture rate is highest for the outermost ring and reduces towards the inner rings. All models show the same behavior to that of MCNP: capture rates are lower in the outermost ring, while in the inner rings the capture rates are higher. The model based on the subgroup approach (USS PTSL) gives the best results. With the use of the GSM+LJ+0 model the rim effect is not observed. This linear increase of capture rates is a typical curve observed in the early self shielding models like the one in the WIMS-D5 code.

6 310.6 Figure 2: Rim effect in the central fuel cell calculated with various codes. 7 BURNUP CALCULATIONS Our reference case scenario for the burnup calculations was made with the Serpent code because MCNP5 has no burnup capability and Serpent results were in good agreement with MCNP5 for fresh fuel. We used an average power density of MW/tU for source normalization. The burnup calculations were performed by solving the set of Bateman equations with the CRAM method, with the predictor-corrector calculation option for more accurate results. For the isotopic one group transmutation cross section calculations we used a more precise method with direct calculation of the cross sections during the transport cycle. Serpent uses a special method by combining adjacent grid points (the default value was chosen for a fractional reconstruction tolerance of ). This reduces the amount of computer memory, which is especially important in the burnup calculations. The fuel cell was divided into 10 rings In the DRAGON burnup calculations the solution of the Bateman equations were obtained using a 4 th order Kaps-Rentrop algorithm with a linear approximation of the microscopic reaction rates at the beginning of the burnup step. The depletion calculations were performed up to MWd//tU with the burnup steps from 250 to 500 MWd/tU. Since the burnup calculations are quite time consuming on a typical PC, we divided the fuel cell into four rings with the volume ratio: 50, 35, 15 and 5%, while in WIMS-D5 burnup calculations we used 10 rings. The results of k inf calculations with the burnup are presented in Figure 3 and Table 3, and the effect of self shielding models in the various codes on the production of 239 Pu in the central fuel is presented in Figure 4 and Table 4. Table 3: Deviation [pcm] of k inf at various stages of burnup with reference to the results obtained with the Monte Carlo code Serpent. Burnup [MWd/tU] WIMS- D5 DRAGON shi+lj+0 DRAGON shi+lj+1 DRAGON shi+lj+2 DRAGON uss+ptsl

7 310.7 Figure 3: K inf for 4.75% enriched uranium as a function of burnup with different codes and various self shielding models. Figure 4: Average 239 Pu concentration in the central fuel cell calculated with different codes and various self shielding models. Table 4: Deviation [%] of 239 Pu average concentration in the fuel cell at various stages of burnup from the results obtained with the Monte Carlo code Serpent. Burnup DRAGON DRAGON DRAGON DRAGON WIMS-D5 [MWd/tU] shi+lj+0 shi+lj+1 shi+lj+2 uss+ptsl

8 CONCLUSION In this paper the analysis of different types of codes with various self shielding models was performed by comparing k inf for fresh and burned fuel and evaluating the concentrations of 239 Pu with the burnup. WIMS-D5 lattice code estimates k inf for fresh fuel with great accuracy since the difference is less than 100 pcm, however, due to the approximations in the burnup calculations, the difference in k inf is increasing with burnup to around 800 ± 80 pcm at the end of the burnup stage. From Table 4 we can see that the results of 239 Pu concentration for the most primitive self shielding models (WIMS-D5 and DRAGON GSM+LJ+0) are overestimated by around 3 %, while the results obtained with DRAGON models GSM +LJ+2 and USS+PTSL show very good agreement with the Serpent code. However, we could only used the Draglib library with the DRAGON burnup calculations, since the results using the WIMSD libraries in the case of fresh fuel show a large discrepancy. Regarding the efficiency of CPU time calculations on a typical dual core PC processor, the WIMS-D5 calculations only take a few minutes, while DRAGON calculations times are in the range from 2:30 hours for GSM +LJ+0 to 87 hours in the case of the GSM +LJ+2 model. On a Linux cluster system for only 1 million histories the Serpent calculation CPU time was around 28 hours, however with special approximations the CPU time could be decreased by a factor of 3 to 4, although some accuracy was lost. We concluded that the use of the DRAGON code with WIMSD libraries is not recommended, however additional calculation options will be tested with the Draglib libraries to decrease the computation time and maintain the accuracy of the results. Although the WIMS-D5 code fails to predict the correct radial distribution of 239 Pu due to the use of the primitive self shielding model, it is still an efficient lattice code because it predicts reasonably well the reactivity drift with burnup and the discrepancy in the average 239 Pu is in the range of a few percent. REFERENCES [1] K. Lassmann, C. O Caroll, J. Van de Laar and C.T. Walker, "The radial distribution of plutonium in high burnup UO 2 fuels", Journal of Nucl. Mat., 208, 1994, pp [2] C.C. Stoker, Z. J. Weiss, "Spatially dependent resonance cross sections in a fuel rod", Ann. Nucl. Ener., 23, 1996, pp [3] A. Hébert, Applied Reactor Physics, Presses internationals Polytechnique, Québec, 2009, pp [4] G. Marleau, A. Hébert, R. Roy, A user guide for dragon version 4, Technical report IGE- 294, École Polytechnique de Montréal, [5] X-5 Monte Carlo Team, MCNP-A general Monte Carlo N-Particle Transport Code, Version 5, April 24, [6] J. Leppänen, PSGS / Serpent a Continuous-energy Monte Carlo Reactor Physics Burnup Calculation Code, Methodology User s Manual Validation Report, [7] J. R. Askew, F. J. Fayers, F. B. Kemshell, A General Description of the Lattice Code WIMS, Journ. Of the Brit. Nucl. Energy Soc., 5, 1966, pp [8] M. Kromar, A. Trkov, "Nuclear Design Calculations of the NPP Krško core", Journal of Energ. Techn., 2, 2009, pp