Science and Technology of Nuclear Installations

Volume 2016 (2016), Article ID 7328131, 30 pages

http://dx.doi.org/10.1155/2016/7328131

## PWR Containment Shielding Calculations with SCALE6.1 Using Hybrid Deterministic-Stochastic Methodology

Department of Applied Physics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia

Received 25 July 2016; Accepted 2 November 2016

Academic Editor: Arkady Serikov

Copyright © 2016 Mario Matijević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The capabilities of the SCALE6.1/MAVRIC hybrid shielding methodology (CADIS and FW-CADIS) were demonstrated when applied to a realistic deep penetration Monte Carlo (MC) shielding problem of a full-scale PWR containment model. Automatic preparation of variance reduction (VR) parameters is based on deterministic transport theory ( method) providing the space-energy importance function. The aim of this paper was to determine the neutron-gamma dose rate distributions over large portions of PWR containment with uniformly small MC uncertainties. The sources of ionizing radiation included fission neutrons and photons from the reactor and photons from the activated primary coolant. We investigated benefits and differences of FW-CADIS over CADIS methodology for the objective of the uniform MC particle density in the desired tally regions. Memory intense deterministic module was used with broad group library “v7_27n19g” opposed to the fine group library “v7_200n47g” used for final MC simulation. Compared with CADIS and with the analog MC, FW-CADIS drastically improved MC dose rate distributions. Modern shielding problems with large spatial domains require not only extensive computational resources but also understanding of the underlying physics and numerical interdependence between -MC modules. The results of the dose rates throughout the containment are presented and discussed for different volumetric adjoint sources.

#### 1. Introduction

The Monte Carlo (MC) simulation of deep penetration shielding problems is a very challenging task. Shortcomings of the classical MC variance reduction (VR) techniques, which are absolutely necessary to get any answer at all, applied to such complex shielding problems are a much known issue. A hybrid deterministic-stochastic shielding methodology is frequently used today when facing such problems [1]. This relatively novel approach in automatic VR preparation has an ultimate goal of achieving acceptable precision of the MC results in a reasonable time. Deterministic transport theory methods [2], typically discrete ordinates , are used for numerical calculation of particle transport over space-energy domain of the problem. The solution is a mesh-based adjoint function in phase-space and it is used for source and transport biasing. Such formalism is known as CADIS [3] and it is based on the concept of the adjoint function [4] (i.e., solution of the adjoint Boltzmann equation) being the importance function toward the specified objective of user’s interest, with zero-variance solution in the limit. The typical use of CADIS is when optimization of localized results is needed, such as point or region detectors, comprising small entities (units) of much larger global unit. Generalization of this adjoint methodology for obtaining global MC distributions is far more difficult task. For the MC simulation resulting in uniformly small statistical uncertainties over large portions of phase-space overlaid with Cartesian mesh grid, additional forward calculation is needed for proper adjoint source weighting. In this case, the adjoint source is discretized over mesh cells, where every voxel has a biased strength (in space and energy) to result in global MC distribution with fairly uniform uncertainties. FW-CADIS (Forward-Weighted Consistent Adjoint Driven Importance Sampling) method [5, 6] in MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) shielding sequence of SCALE6.1 code package [7] was used with aforementioned hybrid methodology. The objective of this paper was to determine the fairly uniform dose rate distributions throughout typical PWR facility including containment building. This is representative deep penetration shielding problem, addressing modern engineering problems for mapping doses everywhere inside PWR [8–10]. Regarding the size and the model’s complexity, the utilization of the manually fine-tuned VR parameters is an impossible task. Therefore, we address the problem with the described modern hybrid shielding methodology.

The SCALE6.1 general geometry package (SGGP) was used for modeling of the typical PWR facility based on the H. B. Robinson-2 Pressure Vessel Benchmark (HBR-2) [11] critical core. Validation of the model (via detector reaction rates) was conducted in previous paper [12], where good agreement with referenced TORT results was obtained in accordance with US NRC regulations [13]. With previous HBR-2 benchmark calculations and later geometry expansion to primary loop elements [14], we now make the final geometry generalization to a full-sized, typical PWR facility with containment structure. Typical industrial and text-book data were used for dimensions and materials required: reactor internals, upper and lower reactor pressure vessel (RPV) head, biological shield, steam generators, primary pumps and pipes, concrete structures such as floors and walls, and finally containment building.

The preliminary shielding results of the PWR containment were obtained using the SCALE6.0 code but computational meshes were rather coarse and MC statistics was not satisfactory [15]. The shielding calculations in this paper are focused on hybrid capabilities in SCALE6.1/MAVRIC to produce well-converged particle fluxes and dose rates throughout containment structure originating from critical reactor core and activated coolant, which becomes additional gamma source in operating reactor. The adjoint source was examined as external air of the model and containment air to enable uniform particle attraction in all phase-space by using hybrid methodology, but the selection between CADIS and FW-CADIS resulted in drastic differences in final MC dose rates. This was notably evident for neutrons and photons originating from reactor core (compact volume) and was less evident for photons coming from activated coolant (distributed volume). The advantage of using FW-CADIS methodology with adjoint source weighting over CADIS was clearly quantified.

The paper is organized as follows. Section 2 gives the description of the SCALE6.1 code package with accent on the MAVRIC shielding sequence. Section 3 describes the PWR facility model preparation: ionizing sources, geometry, and calculational parameters. Section 4 gives MAVRIC dose rates inside PWR containment for analog MC, CADIS, and FW-CADIS methods. Mitigation of observed ray effects is considered in Section 5. Section 6 gives discussion and conclusions, while the referenced literature is given at the end of the paper.

#### 2. The SCALE6.1 Code Package

The SCALE6.1 code package is a comprehensive modeling and simulation suite for nuclear safety analysis and design. It was developed for the US NRC for the purpose of the evaluation of nuclear facilities and radioactive package designs. The SCALE6.1 modular code is devised in analytical (control) sequences with their functional modules for performing criticality, shielding, radiation source term, spent fuel depletion/decay, reactor physics, and sensitivity analyses. For the purpose of paper clarification, only used sequences will be mentioned with the focus on the MAVRIC hybrid shielding sequence. Additional mathematical details are available elsewhere [7].

The criticality sequence (CSAS6) uses a 3D multigroup MC transport code KENO-VI to provide problem-dependent, cross section processing followed by calculation of the neutron multiplication factor . The MAVRIC hybrid shielding sequence is based on CADIS methodology and its generalization with forward flux weighting is FW-CADIS. Excellent recent papers on methodologies were done by scientists at Oak Ridge National Laboratory [16, 17]. CADIS and FW-CADIS are based on the concept of the importance function which is the solution of the adjoint Boltzmann transport equation [2, 4]. These hybrid shielding methods are used for the calculation of space-energy-dependent VR parameters in the form of the weight windows (i.e., importance map) and biased source, which work in tandem. The VR parameters are automatically transferred to functional module Monaco which is multigroup fixed-source 3D MC transport code. The integrated transport code Denovo [18] is used for the calculation of VR parameters over orthogonal meshes using Koch-Baker-Alcouffe parallel sweep algorithm and nonstationary Krylov methods to solve within-group equations. The results from even intermediate quality Denovo calculation will provide superior VR parameters compared to user’s ability to manually tune the same ones for Monaco calculations. For shielding calculations involving localized results such as point detectors or small region responses, CADIS methodology is preferred, and it is based on single Denovo adjoint transport solution. For shielding problems with multiple tallies (point and/or region detectors) or mesh tally over large portions of phase-space, an extension of CADIS method called FW-CADIS can be used to obtain uniformly small relative uncertainties [19, 20]. FW-CADIS requires additional Denovo forward transport solution to approximate multigroup fluxes and responses used for inverse weighting of the adjoint source. In case of optimizing global MC results over large meshes, every adjoint source cell becomes a discrete source element with weighted strength to ensure the same particle population for distant and close portions of phase-space. In a hybrid shielding methodology, we are interested to find a final MC solution of the steady-state transport equation:where is the linear transport operator, is the forward flux, and is the total source. A solution of (1) is based on deterministic approximation of the steady-state, multigroup adjoint transport equation:where is the linear adjoint transport operator, is the adjoint flux, and is the adjoint source. The adequate numerical solution (i.e., approximation) will provide the means to accelerate the final MC simulation via VR parameters. For global MC answers, additional forward deterministic approximation is needed to perform forward-weighting of the adjoint source. In FW-CADIS hybrid shielding methodology, the adjoint source is typically described as a product of the geometric function and the energy spectrum corresponding to the user’s function of interest (i.e., response function), which is often cross section for reaction rate calculations or dose function [7]. In that case, adjoint source weighting is done with the integral of product of the response function and the Denovo forward flux :A particle target (average) weight is used in Monaco MC simulation for particle splitting/roulette in the form of the space-energy-dependent mesh importance map and it is inversely related to Denovo adjoint flux bywhere is the normalization constant. This ensures consistency between source biasing and particle biasing, where birth weight of the particles matches target weight of importance map. Since the objective of our PWR containment calculation was to determine global dose rates inside containment structures with uniformly small uncertainties, the aforementioned FW-CADIS methodology was a highly desirable choice. To achieve such objective, one has to construct such importance function which will represent the importance of achieving uniform MC particle distribution throughout the model [9]. It was shown that this corresponds to weighting the adjoint source with the inverse of forward Denovo response. Once the adjoint source is determined, the hybrid methodology is used for the calculation of source biasing parameters and weight windows for optimized Monaco simulation. There are several multigroup cross section libraries distributed within SCALE6.1 code package. For criticality eigenvalue calculations, the “v7-238” library was used, and, for shielding calculations, we used “v7-27n19g” library for Denovo and “v7-200n47g” for Monaco. Primary data for both libraries originate from the ENDF/B-VII.0 nuclear data library [21].

#### 3. PWR Facility Model Preparation

The simplified model of a typical PWR facility with two-loop reactor, primary loop elements, and containment building was developed using SCALE6.1/MAVRIC sequence. The reactor is the standard Westinghouse PWR type with thermal power of 2300 (710 ). The reactor core consists of 157 fuel assemblies (15 × 15 matrix, pitch 21.504 cm) and it is radially surrounded by baffle plates, core barrel, thermal shield, RPV, and biological shield. The heterogeneous fuel elements have rather complicated design which is irrelevant for the purpose of containment shielding calculations on such large scale. Therefore, the fuel elements are approximated as homogenized axial regions using referenced data from the HBR-2 benchmark [11]. The SCALE6.1/MAVRIC model of the HBR-2 nuclear reactor was validated against referenced benchmark data and was later extended to include primary loop elements [12]. The final geometry generalization now includes complete containment structure with simplified interior. Typical industrial and text-book data were used for PWR components such as reactor internals, upper and lower reactor pressure vessel (RPV) head, biological shield, steam generator, primary pumps and pipes, and massive concrete structures such as floors and walls.

##### 3.1. Definition of Ionizing Sources

The critical reactor core was uniformly sampled in space (“flat” spatial profile) and had Watt spectrum distribution for thermal fission of ^{235}U with* a* = 1.028 MeV and* b* = 2.249/MeV ( is the normalization constant) [7]. The total neutron intensity was n/s which corresponds to one half of total thermal power (2300 ) to better simulate neutron spatial gradient from the center to the core periphery (i.e., core self-shielding). The fission photons were also included with mean value of 7.04 per ^{235}U fission and the secondary gamma emission in neutron transport was explicitly included in MC simulation. To include the fission photons, a mesh-based version of the neutron fission source was required and obtained using KENO-VI eigenvalue calculations. This option in SCALE6.1 is known as the CAAS (Criticality Accident Alarm System) [7] for saving space-energy fission distribution over user-defined mesh for active neutron cycles. To correctly account for how many source photons are released per source neutron, the system -parameter calculated by KENO-VI was used (2.46 neutrons/fission) with = . The activated primary coolant becomes additional gamma emitter in operating reactor, since the product of reaction with ^{16}O in water is beta-active having a half-life of 7.13 s and additionally emitting gamma ray of high energy, 6.123 MeV, with the emission probability of 92%. The energies of the emitted photons from the coolant of a typical PWR were taken from ANSI/ANS-18.1-1999 [22], where activities for more than 50 isotopes are listed, together with plant specific scaling parameters. The details of the source term derivation can be found in the previous paper [14], together with discrete energy spectrum of the activated nitrogen which accounts for 94–97% of the total coolant activity. Using the total coolant volume of cm^{3}, its total mass of g, and its working density of 0.79 g/cm^{3}, we obtained total gamma source strength of photons/s. Together with photon emission spectra of , prepared for the multigroup library “v7_27n19g,” we obtained complete information for distributed coolant gamma source.

##### 3.2. Geometry Extension

The* hole* option inside MAVRIC was used extensively for placing smaller units inside larger units utilizing SGGP combinatorial geometry. The homogenization process was conducted in the same way as with reactor internals, which is mass conservation based on best available input data. The MAVRIC model of the full-sized PWR reactor (coolant not shown) is shown in Figure 1 and the primary loop components are depicted without air for the purpose of clarity. Symbolically added arrows show direction of primary coolant motion in reality. Figure 1 is also showing MAVRIC model of PWR containment with front quarter removed. One can notice the reactor, the cylindrical biological shield, the primary loop elements, and the concrete shielding structures. The walls and the floors have thickness of 100 cm and 50 cm, respectively. All concrete structures were made of 02-B type concrete with density of 2.275 g/cm^{3}. The reactor midplane is in = 0 cm plane and the primary pipes are in = 342 cm plane. The radius of containment inner steel protective shell (3.8 cm thickness) is 1600 cm and the radius of outer concrete layer was 1826 cm (76 cm thickness). The global unit, corresponding to cylindrical containment building, has diameter of about 36 m and height of about 78 m, which results in formidable MC shielding model. Although many small auxiliary components inside real containment were discarded in this phase of research, they are not relevant for the paper objective, since their inclusion would dramatically slow down MAVRIC calculations. Only gross containment structures are important because of their immense attenuating power, such as the concrete floors and the walls.