Transient Capability of a Multi-group Pin Homogenized SP 3 Code - PDF document

Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 Transient Capability of a Multi-group Pin Homogenized SP 3 Code SPHINCS Hyun Ho Cho 1) , Junsu Kang 1) , Joo Il Yoon 2) and Han Gyu Joo 1)* Seoul National University, 1 Gwanak-ro, Gwanak-Gu, Seoul, 08826, Korea 1) KEPCO Nuclear Fuel Co. Ltd., 242, Daedoek-daero 989beon-gil, Yuseong-gu, Daejeon, 34057, Korea 2) * Corresponding author: joohan@snu.ac.kr 1. Introduction The time dependent SP 3 equation can be derived from time dependent Boltzmann transport equation as same The conventional two-step method is still a powerful as the derivation of steady state formulation. The and practical tool for nuclear design and analyses which difference is that precursor balance equation also should involve repeated steady-state and transient calculations be considered so that coupled kinetics equations would with reasonable accuracy. In this regard, be solved. For simplicity, derivation is done in 1-D. nTRACER/SPHINCS advanced two-step calculation Time dependent 1-D Boltzmann transport equation and system employing the simplified P 3 (SP 3 ) method precursor balance equation are coupled so that it can be utilizing pin-homogenized group constants (GCs) and written as follows. ( ) ( ) ∂ ϕ µ ∂ ϕ µ pin-sized finite difference method (FDM) is developed x E , , , t x E , , , t 1 + µ at Seoul National University [1]. ( ) ∂ ∂ v E t x The C5G7-TD benchmark (Deterministic Time ( ) ( ) +Σ ϕ µ x E t , , x E , , , t Dependent Neutron Transport Benchmark without t Spatial Homogenization) [2] is proposed to ensure   np 1 ( ) ( ) ( ) ( ) ( ) = − β χ ψ + ∑ χ λ reliable modeling of reactor physics based on neutron  1 E x t , E C x t ,  π p dk k k   4 = (1) k 1 kinetics equations without the use of diffusion ( ) + Σ ′ → Ω → Ω ˆ ′ ˆ Ω ˆ ′ ′ approximation and spatial homogenization. It contains ∫ ∫ x E , E , , t d dE ′ Ω ′ s E six series of space-time neutron kinetics test problems with a heterogeneous domain description for solving the ( ) ∂ time-dependent multi-group neutron transport equation C x t , ( ) ( ) = β ψ − λ k x t , C x t , without feedbacks. ∂ k k k t Recently, the transient capability of nTRACER [3], Following the well-known derivation of P n equation direct whole core transport code, has been examined from Boltzmann transport equation, Legendre expansion with C5G7-TD [4]. In accordance with the purpose of angular flux and scattering XSs is introduced. aiming for analyzing space-time neutron transport Throughout the application of addition theorem, equation with heterogeneous domain description, the orthogonal property of Legendre polynomial and C5G7-TD benchmark problems were solved by recursive relation, 1-D multi-group time-dependent P n nTRACER employing faithful models of the core equation. configuration and transient control parameters. In addition to applying well-known assumptions for However because of its large computational burden SP 3 steady state derivation, assume odd moment time as well as computing time, in order to establish an derivative terms as zero. With those assumptions, multi- efficient core analyses system, conventional two-step group time-dependent SP n equation can be obtained. method is still required. In this regard, for the complete  ∂  2 ( ) ( ) ( )   − − φ pin-by-pin core analyses, a transient calculation module D x t , 2 D x t ,  x t ,  ∂  0, g 0, g   2 0, g x  has been recently implemented in the SPHINCS code.  2 ( ) 4 ( ) 3 ( )  − − −  ∂  D x t , D x t , D x t , 2 ( )   φ  0, g 0, g 2, g   x t ,  5 5 5 The transient capability of SPHINCS involves the  ∂ 2, g  2 x ( ) ( ) solution of the time-dependent SP 3 equation that is  Σ   φ  x t , 0 x t , +  r g , 0, g    ( ) ( ) Σ φ properly reformulated to be applicable to the FDM 0 x t , x t ,     t g , 2, g solver. In the nTRACER/SPHINCS pin-wise two-step ( )    ∂ φ  np G x t , ( ) ∑ ( ) ∑ ( ) ( ) 1 − β χ ψ + χ λ + Σ φ − 0, g     1 C x t , x t , x t , calculation system, nTRACER provides pin- ′ ∂ p g , d g k , , k k 0, g g 0, g   v t   = ′ = = k 1 g 1 g   homogenized GCs by single assembly level calculations. ( ) ∂ φ x t , 1   − 2, g SPHINCS then performs core calculation based on pin-  ∂  v t   g wise GCs with super-homogenization (SPH) factors. In (2) this work, the implementation of the transient In order to properly apply finite difference method, calculation features of SPHINCS is provided and the introduce summed flux as Eq. (3) so as to make 0 th assessment of that is done solving C5G7-TD. moment and 2 nd moment linearly dependent. 2. Derivation of Time Dependent SP 3 Equation φ ˆ = φ + φ (3) 2 g 0, g 2, g

Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 With time discretization, multi-group time-dependent SP 3 equation re-formulated for finite difference method is as follows.   ( ) 2 ( ) ( ) d 1 2 − + + Σ + + − Σ + − n 1 n 1 n 1  D x t , x t , 2 x t ,   ( )  φ ˆ + 0, g r g , ∆ r g , ∆ n 1 2 x t ,  dx v t v t   g  g g    ( )  φ + ( ) ( ) 2 ( ) ( ) n 1 2 2 d 4 5 3 x t ,  − Σ + − − + + Σ + + Σ + +    n 1 n 1 n 1 n 1 x t , D x t , x t , x t , 2, g r g , ∆ 2, g r g , t g , ∆  2   3 3 v t dx 3 3 v t  g g  ( ) ( )  1 + + φ n 1 n  q x t , x t ,  ∆ 0, g 0, g v t   g =     ( ) ( ) ( ) 2 1 5   − + + φ + φ  n 1 n  n q x t , x t , x t ,    0, g ∆ 0, g ∆ 2, g  3 v t 3 v t     g g (4) In this formulation, main difference compared to time-independent SP 3 equation is the augmentation of removal cross section. Fig.1. Necessity of generation of SPH factors with core power 3. Core Modeling and Group Constants & SPH behavior for TD1-5 problem Factors Generation Radial geometry of the C5G7-TD core is exactly same as that of the C5G7-MOX benchmark core. In nTRACER modeling, sufficient number of flat source regions were used and the ray tracing parameters as well as sub-pin modeling were applied identically for both group constants generation and the reference solution generation. Simple Crank-Nicolson method was used as temporal discretization scheme in nTRACER and 2.5ms time step size was used and finite time discretization whose time step size is same as that of nTRACER was used in SPHINCS. In the C5G7-TD problem sets, postulated transient Fig. 2. Necessity of generation of SPH factors with reactivity event is approximated as a step change or ramp change change for TD1-5 problem of material composition as well as change of moderator density. When utilizing pin-homogenized GCs 4. Analyses of 2-D Problems generated in single assembly unit into two-step calculation, SPH method is introduced in SPHINCS in The 2-D problems consist of TD0 through TD3 order to reduce pin-homogenization error. problems that contain their own sub-problems. Unless proper SPH factors are incorporated into pin- Simulation of a postulated control rod insertion and homogenized GCs, the effect of rod insertion would be withdrawal event is modeled by time-dependent change distorted as shown in Fig. 1 and Fig. 2. Each figure in the cross sections which is step change for TD0 and shows the necessity of generating SPH factors ramp change for TD1 and TD2. The TD3 problem throughout the core power behavior and reactivity involves the ramp changes in the moderator density. change respectively using TD1-5 problem. Fractional total core fission rate shown in Fig. 1 shows about - 4.1. TD0-Set with 5 sub-problems 7.5% discrepancy and reactivity change also shows large difference. At that time step of maximum The TD0 problems postulate control rod insertion and difference of total core fission rate, pin power withdrawal as a step change of the material composition. distribution has even much more discrepancy. Largest It is assumed that all the control rods are fully removed pin power difference is about -17% so that it can be said from the core initially and the transient is initiated by an that the postulated system is completely wrong. abrupt control rod insertion. Detailed control rod However when properly generated SPH factors are movement is depicted in benchmark specifications [2]. incorporated into pin-homogenized GCs, large Pin-wise GCs and incorporated SPH factors are discrepancy disappears and the comparison of results generated for every type of fuel assemblies at each show excellent agreement. fractional control rod insertion points. In other words, 10% and 5% of rod insertion cases respectively for initial 1s and next 1s are considered. For all the sub-problems for TD0, results of SPHINCS show excellent agreement compared with those of nTRACER in spite of abrupt step-wise rod