Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 Application of Quasi-Static Method to Whole Core Transient Calculation in nTRACER Junsu Kang and Han Gyu Joo Department of Nuclear Engineering, Seoul National University, 1 Gwanak-gu, Seoul, Korea 08826 * Corresponding author: joohan@snu.ac.kr 1. Introduction k . The symbol M is migration and loss operator, F and F represent static fission production and quasi- d The direct whole core calculation code nTRACER stationary delayed neutron production respectively. S d performs 3D sub-pin level transient calculation for high denotes actual delayed neutron source. fidelity multi-physics reactor simulation [1]. In order to The quasi-static method is based on a factorization of alleviate the heavy computational burden, the quasi- the neutron flux into two components, ‘amplitude’ and static method is implemented to nTRACER. This ‘shape’: method has been used to several reactor dynamics applications for efficient transient calculation [2, 3]. By ( , , ) ( ) ( , , ). (2) r E t p t r E t factorizing neutron flux into only time-dependent fast varying amplitude and slow varying shape, larger time- step size is used for expensive shape calculation while The amplitude p t ( ) represents overall amplitude maintaining solution accuracy. changes of neutron flux and it is only dependent on time. Most implementations of the quasi-static method in r The shape function ( , , ) still depends on all E t reactor dynamics come in two major variations, namely, variables but it has comparatively small time variation improved quasi-static (IQS) method and predictor- than the neutron flux. For unique factorization, the corrector quasi-static (PCQS) method. IQS solves the constraint condition is required. The normalization nonlinear system of the shape and amplitude equations can be * while PCQS linearly corrects the flux level with condition weighted with initial adjoint flux 0 amplitude [2]. By avoiding the computational cost from used as the constraint condition: nonlinear iteration, PCQS usually shows better computational efficiency and even shows better 1 * ( , ) , (3) K accuracy in several cases. Both IQS and PCQS are 0 0 v examined with nTRACER. In addition the exponential transformation (ET) method [4] that assumes an 1 exponential variation of the regional flux is investigated * where the initial value K ( , ) is constant. 0 0 0 v noting that ET resembles IQS in that it applies the Applying factorization in Eq. (2) into the Eq. (1) temporal discretization to the factorized component. yields the time-dependent shape equation: There were several applications of quasi-static methods to the diffusion solvers [2] and PCQS application to pin-resolved transport solution [3]. This work examined 1 1 1 dp ( ) . (4) M F F S the applicability of IQS to the sub-pin level transport d d v t p v dt calculation of nTRACER noting that it already uses nonlinear iteration for convergence of whole core With known p t and its time derivative term, shape ( ) transport solution. The characteristics and effectiveness component can be computed by integrating Eq. (4). On of the three methods (IQS, PCQS and ET) are also the other hand, integrating Eq. (1) with weighting compared and analyzed in the work here. yields exact point kinetics equation (PKE): * function 0 2. Quasi-Static Approaches ( ) ( ) 1 dp t t p t ( ) ( ) t The time-dependent neutron balance equation is k k ( ) dt t written in operator form as k 0 (5) d F t ( ) k ( ) t ( ) ( ), t p t k 1, K , k k k dt F 1 0 M ( F F ) S d d v t (1) with PKE parameter set defined as C k F C , k 1,..., K . dk k k t 1 * * ( ) ( , ) / ( , ), (6) t F 0 0 Where is the neutron scalar flux, is the v C k delayed neutron precursors density for precursor group
Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 * * ( ) t ( ,[ F M ] )/ ( , F ), (7) 1 1 * 0 0 ( , ) / , p K p 0 p 0 p v p p * * ( ) ( , )/ ( , ), (8) t F F 3: Generate PKE parameters in Eqs. (6)-(9) 0 0 d 4: Update corrected p by solving Eq. (5) with t c * * ( ) ( , ) / ( , ). (9) t C F 5: Get corrected p k 0 dk k 0 c c 6: t t t The amplitude can be computed by integrating Eq. (5) with knowledge of PKE parameters in Eqs. (6)-(9). T he low subscript ‘p’ and ‘c’ in above algorithm Since integration of PKE equation doesn’t impose represents predictor and corrector respectively. This heavy computational cost, temporal discretization with algorithm assumes that the shape function calculated , which is much smaller than macro micro time-step t is accurate enough. PCQS with macro time-step t used for discretization of shape derivative, time-step t doesn’t require nonlinear iteration since the solut ion of is possible. However PKE parameters are dependent on the time-dependent neutron balance equation (1) can’t the shape. Therefore evaluation of PKE parameters be improved with corrected amplitude from PKE should be accompanied by accurate computation of solution [2]. shape. 2.3 Exponential Transform Method 2.1 Improved Quasi-Static Method The ET method was designed to enhance accuracy of In advance, it was shown that the shape equation Eq. the conventional theta method [4]. This method (4) and PKE equation (5) are nonlinearly coupled. IQS anticipates exponential variation of the neutron flux. directly solve this nonlinearly coupled system with Similar with quasi-static methods, the regional flux is iterative method. Following is typical algorithm of IQS factorized into two components in ET method: involving Picard iteration: ( , ) r E t ( ,E,t) r e ( ,E,t). r (10) Algorithm 1 IQS algorithm i 1: do 1,2 until convergence, r The transformed flux ( , , ) E t will vary slowly when from Eq. (4) with 2: Evaluate t i neutron flux changes exponentially. Introducing Eq. 3: Generate PKE parameters in Eqs. (6)-(9) (10) into Eq. (1) yields equation similar with Eq. (4) : 4: Update p by solving Eq. (5) with t i with updated 5: Get corrected p m m d 1 i i m t g g m m m e g ( F F M ) S . (11) 6: end do d g d g g m m v dt v g g 7: t t t Eq. (11) is presented in discretized multi-group form Above algorithm doesn’t guarantee the constraint for simplicity where subscri pt ‘m’ denotes region index condition in Eq. (3) until it is fully converged. Some and ‘g’ denotes energy group index. Similar with IQS implementations of IQS guarantee constraint condition the transformed flux is computed directly by solving Eq. by rescaling the shape function after step 3 in above (11). The difference of this method from quasi-static 1 * algorithm with the factor / ( , ) [2]. method is that exponential function in ET has regional K 0 0 i v and energy dependency while amplitude in quasi-static method has only time dependency. The main focus of 2.2 Predictor Corrector Quasi-Static Method this method is make the transformed flux less variable over time for small truncation error while IQS focused Unlike IQS, the concept of flux factorization is to find out the unique factorization of shape and maintained but Eq. (4) is not solved in PCQS. The amplitude based on exact perturbation theory. shape is computed from the neutron flux with constraint condition in Eq. (3). Therefore the constraint condition 3. Application in nTRACER is always preserved in PCQS. The algorithm of PCQS could be presented as The transport solution in nTRACER is composed of 3 components: radial 2D method of characteristics (MOC), Algorithm 2 PCQS algorithm 3D coarse mesh finite difference (CMFD), and 1D axial by solving Eq. (1) with MOC. These three components form the nonlinear 1: Predict t p system and it is solved in Picard iteration manner. Since into p and based on Eq. (3) 2: Factorize p p there is already nonlinear iteration procedure in
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