Development and Verification of an SP 3 Code Using Semi-Analytic Nodal Method for Pin-by-Pin Calculation

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1 Journal of Physical Science and Application 7 () (07) 0-7 doi: 0.765/ / D DAVID PUBLISHIN Development and Verification of an SP 3 Code Usin Semi-Analytic Chuntao Tan Shanhai Nuclear Enineerin Research and Desin Institute, Shanhai 0033, China Abstract: SP 3 (simplified P 3 ) theory is widely used in LWR (liht water reactor) analyses to partly capture the transport effect, especially for pin-by-pin core analysis with pin size homoenization. In this paper, a SP 3 code named is developed and verified at SNERDI (Shanhai Nuclear Enineerin Research and Desin Institute). For SP 3 method, neutron transport equation can be transformed into two coupled equations in the same mathematical form as diffusion equation. In this wor, SANM (semi-analytic nodal method) is used to solve diffusion-lie equation, due to its easy to handle multi-roup problem. Whole core nodal boundary net current couplin is used to improve converence stability in SANM, instead of solvin two-node problem. CMFD (coarse-mesh finite difference) acceleration method is employed for 0-th SP 3 equation, which represents the neutron balance relationship. Three benchmars are used to verify the SP 3 code,. The first one is a self-defined one dimensional problem, which demonstrates SP 3 method is extremely accurate, due to no academic approximation in one dimensional for SP 3. The second one is a two dimensional one-roup problem cited from Larsen s paper, which is usually used to verify and prove the SP 3 code correct and accurate. And the third one is modified from D C57-MOX benchmar, whose numerical results indicate that is accurate and efficient in pin size level, compared to diffusion model. Key words: SP 3 method, semi-analytic nodal method, pin-by-pin, CMFD, C57-MOX benchmar.. Introduction The current eneration LWR (liht water reactor) core physics calculation methods are based on the neutron diffusion theory framewor, two-roup enery structure and the eneralized equivalent homoenization theory. The potential deficiencies of current method are as follows. Firstly, two-roup enery structure cannot deal with the enery spectrum interference effect very well, such as UO and MOX fuel mixed loadin problem. Secondly, two-step calculation method which decouples the assembly homoenization and the actual core calculation will induce the depletion historical effect, and usually micro-depletion method is used to improve at the core level in today s industry nuclear desin code. Finally, there exists quite lare power distribution error in the reion with hih leaae or stron absorption, such as core periphery or control rod neihborin fuel Correspondin author: Chuntao Tan, Ph.D., research fields: reactor physics methods & reactor core desin. assemblies. As the fuel loadin pattern of core desin becomes more and more complex, it is necessary to improve the existin calculation methods to improve the calculation accuracy. Considerin that diffusion calculation needs many approximation and ordinary transport calculation methods such as MOC or S N method usually taes more computin resources, isotropic SP 3 (simplified P 3 ) method is encouraed to be a middle choice for enineerin practice use for whole core pin-by-pin calculation with pin size homoenization [-4]. For SP 3 method, neutron transport equation can be transformed into two coupled equations in the same mathematical form as diffusion equation. Therefore, with the advantae of the SP 3 method, all the effective methods for solvin the diffusion equation can also be used to deal with the SP 3 equation. A next eneration nuclear desin code system for whole core pin-by-pin calculation with pin homoenization is under development by SNERDI (Shanhai Nuclear

2 Development and Verification of an SP3 Code Usin Semi-Analytic Enineerin Research and Desin Institute). In this code system, core neutronics enine will intend to employ SP 3 method, and will be coupled with sub-channel thermal-hydraulics code for pin size feedbac calculation. In this paper, we will report the development and verification of the SP 3 code,, at SNERDI. There are several numerical methods for solvin diffusion-lie equation. Nodal method is one of the best choices, specifically for its better performance and superior accuracy. Many inds of nodal diffusion method have been developed, such as NEM (Nodal Expansion Method), the ANM (Analytic Nodal Method) and the NFM (Nodal reen s Function Method), etc. In this wor, SANM (Semi-Analytic Nodal Method) will be used to solve diffusion-lie equation, due to that it is easy to handle multi-roup problem [5]. We use whole core nodal boundary net current couplin to improve converence stability in SANM, instead of solvin two-node problem, which does not need nonlinear iterations. CMFD (Coarse-Mesh Finite Difference) acceleration method is employed for 0-th SP 3 equation, which represents the neutron balance relationship. Three benchmars will be used to verify the SP 3 code,, in this paper. The first one is a self-defined one dimensional problem, which will demonstrate SP 3 method is extremely accurate, due to no academic approximation in one dimensional for SP 3. The second one is a two dimensional one-roup problem cited from Larsen s paper, which is usually used to verify and prove the SP 3 code correct and accurate. And the third one is modified from D C57-MOX benchmar, whose numerical results indicate that is accurate and efficient in pin size level, compared to diffusion model. In Sec., fundamental theory and detailed formulations are described. Numerical results are shown in Sec. 3, followed by conclusions in Sec. 4.. Calculation Model. SP 3 Formulation The SP 3 transport equation with isotropic scatterin can be written as follow with the standard notation. D0, 0, () r r0, 0, () r S 0, () r r0,, () r () D,, () r r,, () r S 0, () r r0, 0, (). r 3 3 r0, t, s, r, t, r0, 0, 0, ' 0, ', ' ' ' S () r [ ()- r ()] r eff v f, ' 0, ' r, ' ' [ ( )- ( r)] D, 0, (a) (b) 9 D (c) 7. SANM Formulation In order to simplify the description, the followin one-dimensional sub-diffusion interal equation is iven directly. d ( u) D ( ) r u du ( ) ( u) L ( u) ' ' f' ' eff (3) In this paper, the -order exponential functions and -order polynomials are used for expansion of the transverse interal flux. The followin is the expansion form functions.

3 Development and Verification of an SP3 Code Usin Semi-Analytic p () t 0 p() t t u ( ) ( ) (4) u 4 u ai pi i p () t (3t ) sinh( t) m (sinh) p( t) p3 () t sinh( ) (sinh) m cosh( t) m (cosh) p ( t) m (cosh) p ( t) p4 () t cosh( ) (cosh) (cosh) tu u m r 0 0 m0 m (sinh) sinh( t) p( t) dt N m (cosh) cosh( t) p( t) dt ( i0,) i i N i N (i) ( i0,, ) i u D (5) (6) For transverse leaae, the quadratic approximation method is adopted. u L u L L p (7) ( ) 0 i i ( ) i u In this paper, we use residual weiht method to construct the moment weiht equation for solution of expansion coefficients of transverse interal flux. A. Zero-order moment equation 4D ( u ) Where a a4 (3 ) ' ' 0 ' ( B B ) L 0 (8) B B r ' ' f' eff sinh( ) 3 m (cosh) m0 m cosh( ) (cosh) (cosh) B. First-order moment equation a A B a 3 A ' ' u ' A ( B a L ) sinh( ) m (sinh) rm (sinh) C. Second -order moment equation a C C B a 4 ' ' u ' C ( B a L ) cosh( ) m (cosh) m (cosh) 0 rm(cosh) (9) (0) () () (3) D. Usin transverse neutron flux expansion coefficients to represent the boundary net current D J ( a 3 a H a a ) R 3 4 u D J ( a 3 a H a a ) L 3 4 u H cosh( ) m (sinh) m sinh( ) (sinh) (4) (5)

4 Development and Verification of an SP3 Code Usin Semi-Analytic 3 Eqs. (8), (0), () and (4) are used to obtain the relationship between nodal averae flux and expansion coefficients of transverse interal flux. ( J J ) ( B L ) R L ub B ( R L) ( ) R L 3 3( R L) 3( ) 4 4( R L) 4( L a a J J a B a L a a ( J J ) a ( B a L ) a a J J a B a L a a J J a B a ) u 4D H a, a A, HAB HAB u 4D a, a C, 3CB 3CB u 4D a3 AB, a3 A, HAB HAB u 4D 3 a4 CB, a4 C. 3CB 3CB 0 (6) (7) Based on the continuity condition of surface flux after multiplyin the discontinuous factors, the followin equation is established to represent the boundary net current couplin relationship of neihborin nodes. n n ( CC) f uj ( C C ) f u J n ( CC) fu ( C C ) f u J (8) ( S0 L0 ) B C3 ( S L ) C4 ( S L ) fu ( S0L0) B C3( SL) C4( S L ) f u C a a3, C a a4, ub (9) C3 a a3, C4 a a4. Once the boundary net current is obtained, the nodal averae flux can be calculated by usin the nodal neutron balance equation. And then the neutron source term can be updated and the iterative process can be established..3 CMFD Acceleration for SP 3 Formulation CMFD (Coarse-Mesh Finite Difference) is an effective acceleration converence technique, which is widely used in the field of three dimensional diffusion calculation and heteroeneous transport calculation. In this paper, we have developed a CMFD acceleration method for the SP 3 equation with reference to recent research proress, and obtained a ood acceleration effect. D () r () r r ' f' ' eff [ ] ( r) D [ ( r) ( r)] ( r) 0, 0,, r 0, 0, f' 0, ' eff (0a) [ ] ( r ) (0b) Eq. (0a) is the neutron diffusion equation, and Eq. (0b) is the 0-th equation of SP 3. Comparin Eqs. (0a) and (0b), the remainin items are identical except for the first item on the left side. Therefore, as lon as eepin the first term of Eq. (0a) same as Eq. (0b), the flux and -effective of Eq. (0a) can convere to the flux and -effective of 0-th equation of SP 3. Furthermore, it is found that the -th flux of SP 3 equation is usually about to orders of manitude smaller than the 0-th flux. Therefore, the converence of the 0-th flux of the SP 3 equation is dominated by the solution of the SP 3 equation. In this paper, the CMFD equation is proposed to accelerate the converence of 0-th flux of the SP 3 equation, and this practice enhances the stability of the CMFD acceleration. The formula is as follows.

5 4 Development and Verification of an SP3 Code Usin Semi-Analytic J SP, 30 th FDM, SP, 3 D ( ) D ( ) () The nodal couplin correction factor is updated accordin to Eq. (), in the iterative process of solvin the SP 3 SANM equations. The remainin process is identical to the conventional CMFD acceleration for the nodal diffusion method, which is omitted for simplicity. D FDM, SP, 30th D ( ) SP, 3 J ().4 Iteration Process The code is written in the Fortran-95 standard. The source iterative method is used to solve the SP 3 equations. Firstly, iven the initial source, 0-th SP 3 equation with CMFD acceleration is solved, the number of iterations should be fixed (usually 5 to 8). Secondly, usin the 0-th flux to update the source term of -th SP 3 equation, and then drive the SANM to solve the -th SP 3 equation. Finally, tain both 0-th and -th flux contribution into account, update the -effective and source term until the iterations are reached to converence. 3. Numerical Results 3. Self-defined D Problem cross-section parameters of materials are from the D C57-MOX benchmar [6]. The left side is the vacuum boundary condition, and the riht side is the reflective boundary condition. The width for each material is 0 cm shown in Fi.. Due to the lare differences in enery spectrum amon the various materials and the stron leaae, it s a ood case for code accuracy verification. The reference solution includin -effective and power distribution of the problem is iven by the Monte Carlo proram MCM [7]. In order to obtain a reliable reference result, we use,000 cycles (includin 50 inactive cycles) with 00,000 samplin for each cycle. The result of is iven by the SANM-SP 3 module and the mesh is divided into cm per mesh. Table shows the results comparison of the and MCM. Since there is no academic approximation in one dimensional for SP 3 theory, the difference should be very small. It can be seen from Table that is very accurate, with -effective error of only - pcm and the maximum power error of -0.5%. 3. Larsen s One-roup Problem Brantley and Larsen [] have established a two dimensional one-roup isotropic problem. Fi. shows the layout of the problem, and Table ives the cross-section parameters. Boundary condition and eometry size are also shown in Fi.. A one dimensional 7-roup benchmar problem is self-defined. Fi. shows the layout of this problem. Materials from the left side to riht are H O, UO, 4.3% MOX, 7.0% MOX and 8.7% MOX. The Fi. Layout of the self-defined D problem. Table Results comparison between and MCM. MCM Err. eff pcm a UO % Power 4.3% MOX % 7.0% MOX % 8.7% MOX % a pcm is defined as percent-milli, i.e., 0-5.

6 Development and Verification of an SP3 Code Usin Semi-Analytic 5 The benchmar reference solution is established by the two dimensional S N transport code TWODANT, with the quadrature roup S 6 and the meshes. In reference paper only provides -effective and does not provide power distribution. In order to further evaluate the benchmar problem, a two dimensional MOC transport code PEACH [8] is used to provide the detailed power distribution; PEACH uses meshes and 0.0 cm ray spacin, with 3 azimuthal anles and optimal polar anles in an octant. P and SP 3 model uses the same 0 0 meshes division. Accordin to Table 3, we can see that the correspondin deviations of -P and -SP 3 are 35.6 pcm and 5.3 pcm respectively. We also find that the -effective error is only.8 pcm between reference and PEACH-MOC, so the results from PEACH-MOC are reliable and the power distribution of PEACH-MOC can be used as the benchmar reference. The maximum power error is.38% between PEACH-MOC and SP 3 accordin to Fi. 3. Compared with those reference results, the quite small deviation can prove that the theoretical derivation and code development are correct in this paper. 3.3 D C57-MOX Problem A D C57-MOX homoenized pin problem is enerated from oriinal C57 benchmar [6]. For the homoenized pin problem, the reference solution is obtained by a two dimensional MOC code PEACH, in which the sinle pin is subdivided with 5 5 meshes, and the ray spacin is about 0.04 cm. Fi. 4 is the layout of this problem. has two modules of neutronics solution, one is diffusion calculation module with SANM (named P ), and the other is SP 3 module introduced in Sec.. In this paper, the problem of the D C57-MOX homoenized pin problem is solved by usin both P model and SP 3 model. The sinle fuel pin mesh is not subdivided, and still uses oriinal mesh. Fi. layout of the one-roup problem. Table Material parameters of the one-roup problem. Material M F t.0.5 s f f Table 3 K-effective comparison of Larsen s problem. Larsen s paper This paper Model eff Err. /pcm Model eff Err. /pcm S Reference -P a P SP b SP PEACH c a Err. = (-P Larsen P )*0 5. b Err. = (-SP 3 Larsen SP 3 )*0 5. c Err. = (PEACH Larsen S 6 )*0 5.

7 6 Development and Verification of an SP3 Code Usin Semi-Analytic Fi. 3 Comparison of power distribution between PEACH-MOC and SP 3. Fi. 4 Layout of the D C57-MOX homoenized pin problem. Table 4 K-effective difference and performance comparison. eff Err. /pcm Time /s CMFD PEACH.876 Reference -P No -P Yes -SP No -SP Yes Table 5 Power distribution difference comparison between P and SP 3. -P -SP 3 Inner UO Assembly power err. /% MOX Outer UO Max. power pin -0.9 (.368) a 0.4 (.368) a Fuel pin power err. /% Max. 3.8 (0.660) a (0.557) a RMS a Number in parentheses stands for normalized power. Fi. 5 Pin power error (%) distribution of diffusion calculation module. Fi. 6 Pin power error (%) distribution of SP 3 calculation module.

8 Development and Verification of an SP3 Code Usin Semi-Analytic 7 Table 4 ives the results of the D C57-MOX benchmars -effective calculation. The results show -effective difference and performance comparison. We can see that the P model -effective error is 90 pcm and the SP 3 model is 55 pcm. We also find that CMFD can sinificantly accelerate the converence rate and reatly reduce the computation time without affectin the calculation results. Table 5 shows the power distribution difference comparison between P and SP 3. For the assembly power error and fuel pin power error, the result of SP 3 is more accurate compared with P, with maximum error in assembly power of -0.06% and RMS error in pin power of 0.9%. Fi. 5 shows pin power error distribution of P module compared with reference. Fi. 6 ives pin power error distribution of SP 3 module. The results of these two fires show that the diffusion model has a lare error at the interface of fuel-reflector and UO -MOX fuel cell, and the SP 3 model can obtain perfect results, which shows that the SP 3 model is better than the diffusion model in accuracy. 4. Conclusions In this paper, we report the development and verification of the SP 3 code,, at SNERDI. We use SANM with CMFD acceleration to solve the diffusion-lied SP 3 equations. Numerical results of several benchmars demonstrate that SP 3 module is accurate at pin-by-pin level, compared to diffusion model. In the future, we will extend the code for whole core pin-by-pin calculation. Acnowledments The author would lie to express his sincere ratitude to Dr. Y. A. Chao for invaluable discussions. References [] Brantley, P. S., and Larsen, E. W The Simplified Approximation. Nuclear Science and Enineerin 34: -. [] TATSUMI, M., and YAMAMOTO, A Advanced PWR Core Calculation Based on Multi-roup Nodal-transport Method in Three-dimensional Pin-by-Pin eometry. Journal of Nuclear Science and Technoloy 40 (6): [3] Lee, C. H., and Downar, T. J A Hybrid Nodal Diffusion/SP 3 Method Usin One-Node Coarse-Mesh finite Difference Formulation. Nuclear Science and Enineerin 46: [4] Becert, C., and rundmann, U Development and Verification of a Nodal Approach for Solvin the Multiroup SP 3 Equations. Annals of Nuclear Enery 35: [5] Zimin, V.., Ninoata, H., and Poosbeyan, L. R Polynomial and Semi-Analytic Nodal Methods for Nonlinear Iteration Procedure. Proceedins of the International Conference on the Physics of Nuclear Science and Technoloy : [6] Lewis, E. E., Smith, M. A., Tsoulfanidis, N., et al. 00. Benchmar Specification for Deterministic -D/3-D MOX Fuel Assembly Transport Calculations without Spatial Homoenisation (C57 MOX). OECD/NEA report, NEA/NSC/DOC(00)4, [7] Den, L., Xie, Z. S., and Zhan, J. M A 3-D Multi-roup P-3 Monte Carlo Code and Its Benchmars. J. Nucl. Sci. Technol. 37: [8] Tan, C. T., and Zhan, S. H Development and Verification of an MOC Code Employin Assembly Modular Ray Tracin and Efficient Acceleration Techniques. Annals of Nuclear Enery 36: 03-0.