high-fidelity light water reactor analysis with the...

High-Fidelity Light Water Reactor Analysiswith the Numerical Nuclear Reactor

David P. Weber, Tanju Sofu, and Won Sik Yang

Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

Thomas J. Downar, Justin W. Thomas, and Zhaopeng Zhong

Purdue University, 1290 Nuclear Engineering, West Lafayette, Indiana 47907-1290

Jin Young Cho, Kang Seog Kim, and Tae Hyun Chun

Korea Atomic Energy Research Institute150 Dukjin-dong, Daejeon 305-600, Korea

and

Han Gyu Joo* and Chang Hyo Kim

Seoul National University, San 56-1, Sillim-dong, Seoul 151-742, Korea

Received January 31, 2006Accepted May 18, 2006

Abstract – The Numerical Nuclear Reactor (NNR) was developed to provide a high-fidelity tool for lightwater reactor analysis based on first-principles models. High fidelity is accomplished by integrating fullphysics, highly refined solution modules for the coupled neutronic and thermal-hydraulic phenomena.Each solution module employs methods and models that are formulated faithfully to the first principlesgoverning the physics, real geometry, and constituents. Specifically, the critical analysis elements that areincorporated in the coupled code capability are a direct whole-core neutron transport solution and anultra-fine-mesh computational fluid dynamics / heat transfer solution, each obtained with explicit (sub-fuel-pin-cell level) heterogeneous representations of the components of the core. The considerable com-putational resources required for such highly refined modeling are addressed by using massively parallelcomputers, which together with the coupled codes constitute the NNR. To establish confidence in the NNRmethodology, verification and validation of the solution modules have been performed and are continuingfor both the neutronic module and the thermal-hydraulic module for single-phase and two-phase boilingconditions under prototypical pressurized water reactor and boiling water reactor conditions. This paperdescribes the features of the NNR and validation of each module and provides the results of severalcoupled code calculations.

I. INTRODUCTION

Design and analysis of reactor cores and nuclearpower plants have relied heavily on modeling and sim-ulation of key phenomenology, component and systembehavior, and the availability of advanced computing sys-tems. Sophisticated models of individual and coupledphenomena, such as neutron transport, thermal hydrau-

lics and heat transfer, structural and thermomechanics,and fuel behavior, have been developed to understandand predict the behavior of the reactor core and its com-ponents under steady-state and transient conditions,representing normal, off-normal, and accident condi-tions. The level of modeling sophistication in routinedesign and analysis has increased substantially becauseof improved understanding of the governing physical prin-ciples and the availability of advanced computing sys-tems to perform the analyses. Recent developments in*E-mail: [email protected]

NUCLEAR SCIENCE AND ENGINEERING: 155, 395–408 ~2007!

395

computing capability, including the development of mas-sively parallel computing systems, now enable not onlysophisticated phenomenological modeling but also theapplication of such high-fidelity models in an integratedanalytical environment.

In recognition of these developments, a project wasinitiated under sponsorship of the International NuclearEnergy Research Initiative ~I-NERI! to develop a high-fidelity, integrated core design and analysis tool basedon first-principles models, which takes advantage of ad-vanced parallel computing systems. Developed by col-leagues at Argonne National Laboratory ~ANL!, KoreaAtomic Energy Research Institute ~KAERI!, Purdue Uni-versity, and Seoul National University, this analysis sys-tem is referred to as the Numerical Nuclear Reactor1– 4

~NNR!. Originally developed for the analysis of pressur-ized water reactors ~PWRs!, it has recently been ex-tended to treat boiling water reactors ~BWRs!. Althoughthe NNR is a computationally intensive analysis system,requiring the use of parallel and massively parallel com-puting systems, it provides a robust complement to ex-isting design and analysis methodologies. Because it isbased on first-principles models being validated againstexperimental and analytical data, it can serve as an ef-fective benchmark for existing methods and can providea basis for investigation of phenomena and behavior re-quiring detailed understanding of coupled phenomenathat may not be possible with existing methods.

The traditional approach for analyzing reactor coresinvolves the separate evaluation of the different phenom-enological effects in the system. In this approach, whichis still generally widespread at nuclear vendors and util-ities, different technical groups perform the neutronic,thermal-hydraulic, and thermomechanical calculations,using information supplied by the other groups. This ap-proach often requires iterations of data between thegroups, which can make the process tedious and time-consuming. In this traditional approach, it is generallyattempted to loosely couple the different disciplines bydefining conservative safety parameter limits that haveto be met by the various groups. Obviously, this ap-proach could be very costly if the limits are either tooconservative or not conservative enough.

In the neutronic and thermal-hydraulic analysis per-formed by each group, simplifications and approxima-tions of the system are generally introduced because thedirect solution of the basic governing equations can beprohibitive because of computing resource limitations.For instance, approximations such as homogenization,energy group condensation, and neutron diffusion aregenerally used in the neutronic calculations, whereas one-dimensional flow formulations with numerous experi-mental correlations are used in the thermal-hydraulicanalysis. The separate evaluation of the coupled phenom-ena with approximate models not only introduces inevi-table errors but also lacks the details of the solution. TheNNR is designed to provide high-fidelity solutions in

reactor core analysis by integrating first-principles–based simulation models in each phenomenological area.

The critical analysis elements that are incorporatedin the coupled code capability are a direct whole-coreneutron transport solution and an ultra-fine-mesh com-putational fluid-dynamics ~CFD! 0 heat transfer solu-tion, each obtained with explicit ~sub-fuel-pin-cell-level! heterogeneous representations of the componentsof the core. The considerable computational resourcesrequired for such highly refined modeling are addressedby using massively parallel computers, which togetherwith the coupled codes constitute the NNR. Modelingparameters can also be relaxed in the NNR to make prac-tical applications tractable on clusters of workstationsand personal computers but with the assurance that themore rigorous model is available to evaluate consis-tently the errors introduced by the reduced-order model.

The development of the NNR involved a significantamount of developmental and modeling effort. First, adirect whole-core neutronic calculation code, Determin-istic Core Analysis based on Ray Tracing ~DeCART!code, was developed, which is capable of direct three-dimensional ~3-D! transport calculation for PWRs atpower-generating conditions. A coupling methodologywas then developed and implemented to couple the CFDcalculation with the whole-core neutronic calculation.The project also included extensive verification and val-idation of the individual modules of the NNR. Resultsfrom the reactor physics area showed DeCART to be veryaccurate in comparison to a range of high-fidelity MonteCarlo simulations and experimental measurements. CFDresults for single-phase flow were found adequate in com-parison to experimental results, and best-practice guide-lines for turbulence modeling were identified. The NNRwas also extended to handle two-phase-flow systems be-cause of interest in detailed understanding of integratedneutronic and thermal-hydraulic conditions in BWRs.Extension of the DeCART methodology to treat the morecomplex geometry of the BWR fuel assembly was per-formed, and a series of verification and validation activ-ities was initiated. In order to treat two-phase boilingconditions in PWRs and BWRs, new models, which arealso undergoing extensive verification and validation,were developed and implemented in the CFD code usedin the NNR. In Secs. II and III, the development and val-idation of the neutronic and CFD-based thermal-hydraulicmodules are described. The code coupling methodologyand demonstrative coupled calculation results are thenprovided in Sec. IV. Section V concludes the paper withcontinuing work and future applications of the NNR.

II. THE DECART DIRECT WHOLE-CORETRANSPORT CALCULATION CODE

The current practice in light water reactor ~LWR! coreanalysis is to use pregenerated homogenized few-group

396 WEBER et al.

NUCLEAR SCIENCE AND ENGINEERING VOL. 155 MAR. 2007

constants to obtain the whole-core power distributionand reactivity. One objective of the research here was toremove this limitation and consequently the errors asso-ciated with a priori spatial homogenization and groupcondensation. The DeCART code5 was developed forthe direct whole-core neutronic calculation and is capa-ble of generating the 3-D sub-pin-level power distribu-tion with the thermal feedback effect directly incorporated.The term “direct” is used loosely here to reflect thefact that the core calculation can be performed directly,once a multigroup ~e.g., 45 or more groups! cross-section library for the thermal reactors is available. Adirect whole-core transport calculation for practical 3-Dproblems can be computationally prohibitive, and there-fore, various innovative methods are used in the De-CART solution to reduce the computational burden suchas the planar method of characteristics ~MOC! solution-based coarse mesh finite difference ~CMFD! scheme andmultiprocessing.

The planar MOC solution–based scheme takes ad-vantage of the fact that the primary heterogeneity in theLWR occurs in the radial plane. The two-dimensional~2-D! MOC calculation6 is performed in each plane ofthe 3-D domain to generate pin-cell homogenized, equiv-alent multigroup cross sections that are then used in a3-D CMFD calculation.7 Prior to the transport calcula-tion, however, the self-shielded multigroup cross sec-tions are determined, which are composition, geometry,and temperature dependent. This is carried out by usingthe subgroup method that was formulated suitably for thedynamic generation of multigroup cross sections in theresonance region at nonuniform temperature conditions.The planar MOC solution–based CMFD method and thetemperature-dependent subgroup method are the twoprimary calculation methods of DeCART for the directwhole-core solution. These two methods and some ofthe validation of the methods are described in Secs. II.Aand II.B.

II.A. Direct Three-Dimensional Whole-CoreCalculation Methodology

For the direct whole-core calculation, local hetero-geneity at the sub-pin level should be explicitly repre-sented. MOC is an effective transport solution methodfor such highly refined heterogeneous problems, and nu-merous MOC-based whole cores such as CASMO-4E~Ref. 8! have been developed and successfully appliedto LWR problems. The 3-D MOC calculation for thewhole reactor problem is, however, computationally un-realistic even with today’s high-performance computerssince it would require a prohibitive number of neutronray tracings. Therefore, an approximate 3-D transportsolution method involving 2-D planar MOC solutionswas devised for practical calculations based on the fol-lowing rationale.9 Since most heterogeneity in a reactorproblem appears in the radial direction rather than in the

axial direction, it is sufficient to apply the MOC to re-solve only the radial dependency and to use a lower-order method for the axial direction.

In this method, the 2-D planar MOC solutions areiteratively combined with pin-cell–based 3-D CMFD so-lutions. The planar MOC solution provides the CMFDproblem with cell-homogenized group constants and alsothe radial coupling coefficients that relate the interfacecurrent with two node average fluxes. The cell homog-enization is performed during the calculation so that no apriori homogenization is necessary.10 Pin-cell–sized nodesare employed radially in the CMFD formulation whilemuch larger nodes with sizes up to 20 cm are used axi-ally. The axial dependence of the flux within the largenode can be described by a lower-order method, namely,the nodal expansion method11 with the diffusion approx-imation. It should be noted that this lower-order axialtreatment can potentially deteriorate the solution accu-racy when there is a severe axial heterogeneity. The pla-nar MOC problems are developed consistently with theglobal 3-D CMFD problem in order to preserve the neu-tron balance. This is accomplished by introducing anaxial leakage source into the planar MOC calculation ateach pin cell that is determined from the CMFD solu-tion. Another important function of the CMFD solutionis to accelerate the source convergence of the MOC cal-culation.12 In the following section, the essential ideas ofthis method are presented, but a more detailed descrip-tion can be found in the references.13

For a 3-D neutron transport problem in which sev-eral radial planes are used, a transverse integration canbe used to treat the axial neutron leakage as a sourceterm in the planar neutron balance equation. As is nor-mally done in the transverse-integrated nodal methods,the transport equation for a discretized angle m can beintegrated over the axial direction on a plane ~designatedby plane index k! to yield

�«m

]

]x� hm

]

]y� Twmk ~x, y!� St

k~x, y! Twmk ~x, y!

� OQmk ~x, y!� Lz, k

m ~x, y! , ~1!

where

Twmk ~x, y! � axially averaged angular flux

OQmk ~x, y! � axially averaged neutron source consist-

ing of fission and scattering sources

Lz, km ~x, y! � axial leakage source representing the

axial gradient in the angular fluxdistribution.

Since the axial leakage term is generally much smallerthan the neutron fission and scattering sources, it is plau-sible to introduce an approximation to the angular andspatial dependence of this term. Specifically, the approx-imate axial leakage term can be constructed iteratively

NUMERICAL NUCLEAR REACTOR 397


by using the surface average net current and flux deter-mined at the axial interface from the 3-D CMFDcalculation.

Once the axial leakage source term is determined,Eq. ~1! can be solved by a 2-D ray-tracing scheme. InDeCART, a pin-cell–based modular ray-tracing ~CMRT!scheme was originally implemented. The pin-cell–basedmethod considerably reduces the memory requirementsby storing the ray segment information only for typicalcells, which is possible by suitably choosing the azi-muthal angle and ray spacing such that the same raypattern can be repeated in all the pin cells. The CMRThas, however, limited applicability. For example, thecomplex geometry of a BWR cannot be described as arepeating array of pin cells. Therefore, the modularray-tracing scheme in DeCART was generalized by add-ing an assembly-based modular ray-tracing @AssemblyModular Ray Tracing ~AMRT!# method. With AMRT,the association of ray segments with flat source regionsis determined in two stages. The assembly module isdivided into rectangular cells aligned in a regular array;these rectangular cells provide the basis for the CMFDmesh. Each cell is further divided into flat source re-gions with a more complex geometry, which is the meshunit for the MOC calculation.

The solution of the 2-D MOC problem determinesthe scalar flux at each region within a pin cell and alsothe net current at the cell surfaces. These intracell scalarflux distributions can be used to define the cell averagecross sections while the cell surface current is used todefine the radial coupling coefficients that express theinterface current as a function of the two adjacent nodeaverage fluxes. Unlike the radial coupling relations, theaxial coupling relations are determined within the 3-DCMFD formulation. They are established by using thepartial current scheme of the nodal expansion method,which expresses the outgoing current at a surface in termsof the two incoming currents, the node average flux, andtwo source moments. The source moments here includethe contributions from the quadratic radial transverse leak-age terms. The radial and axial coupling relations yieldthe following CMFD nodal balance equation expressedin terms of the node average fluxes, the incoming partialcurrents, and the axial flux moments as

�1

h (s�1

Nradi

~ ED i, s � ZD i, s ! Nf i, s

� �Sri �

2T3i

hz

�1

h (s�1

Nradi

~ ED i, s � ZD i, s !� Nf i

� NS i �1

hz

~T1i � T2

i � 1!~Jz�i, T � Jz

�i, B!�2T5

i

hz

Ef2i ,

~2!

where

Nradi � number of radial neighboring nodes

of node i

EDi, s, ZDi, s � radial coupling coefficients determinedfrom the MOC solution

Tmi, s � coefficients needed for the axial out-

going current relation

Jz�i, T , Jz

�i, B � incoming currents at the top and bot-tom surfaces, respectively

Ef2i � second-order moment

NS i � source including the fission and thescattering.

The incoming currents and the second moments on theright side are assumed to be available from the previousiteration so that Eq. ~2! becomes essentially a 2-D prob-lem. The axial incoming currents are updated by the planesweep to solve the 3-D problem.

The MOC and CMFD problems represented byEqs. ~1! and ~2! are coupled with each other. The MOCsolution provides the CMFD problem with the cell ho-mogenized cross sections and the radial coupling coeffi-cients whereas the CMFD solution provides the MOCproblem with the cell average fluxes and axial leakages.The cell average flux is used to adjust the regionwisefission and scattering source in the MOC calculation.These two calculations are thus performed alternatelyuntil the coupled solution converges.

The transport solution process outlined above is car-ried out using the self-shielded multigroup cross sec-tions determined for each region by the subgroup method,in which the effective cross section of a resonance iso-tope ~represented by R below! for an energy group ~de-noted by g! is expressed with the following quadratureformula:

TsagR �(

i

wifisaiR

(i

wifi

, ~3!

where saiR , wi , and fi are the subgroup level, weight, and

flux, respectively, corresponding to the i ’th subgroup. Inorder to apply the above formula, the flux of each sub-group level needs to be determined for each resonanceregion. The method for determining the fluxes involvesthe solution of the so-called subgroup fixed source prob-lem, which can be written as

V{¹w~r,V!� ~NR~r!sm � lSp~r!!w~r,V!

�1

4plSp~r! , ~4!

398 WEBER et al.


where

NR � number density of the resonance absorber

sm � prespecified m’th subgroup level

l � hydrogen equivalence factor.

Equation ~4! is solved for the angular flux distribution ofa subgroup using the isotropic slowing-down source de-termined under the assumption that the scalar flux abovethe resonance group of interest is unity throughout thecore. In this problem, the same subgroup level sm is usedeverywhere, signifying that the neutrons belonging tothe subgroup would experience the same absorption crosssection regardless of their energy or position. This isreasonably valid under a uniform temperature conditionsince a neutron that survived from collision in one re-gion would leave the region with the same energy, andthus, its subgroup will not be changed. The cross sectionit would experience in the next region will be the sameas in the region it left. In the case of nonuniform temper-ature distributions, an approximate method was devel-oped in which the subgroup levels are adjusted accordingto temperature-dependent subgroup weights.14

Equation ~4! can be solved by employing the sametransport solver used for the regular transport solutionof the whole-core problem. In the case of DeCART, theMOC solver is used to determine the angular and sub-sequent scalar flux distribution. Once the scalar flux isobtained for each region, equivalence between the het-erogeneous and the homogeneous systems is imposedto represent the solution flux with the formula for thehomogeneous cases. In order to consider resonance in-teractions between different isotopes, the resulting back-ground equivalence cross section is functionalized overthe subgroup level similar to the method used in theHELIOS code.15

II.B. Solution Verification and Validation

The DeCART code was validated using a series ofbenchmark problems to assess the accuracy of variousapproximations in the calculation, as well as the accu-racy of the overall solution methodology. The first bench-mark problem involved heterogeneous core configurationswith given cross sections. As reported in previous pa-pers,9,13 the DeCART solution accuracy is excellent forboth the original and rodded C5G7MOX benchmark prob-lems. The eigenvalue error is ,200 pcm, and the pin-power error is ,2% even for the rodded cases involvingsevere axial heterogeneity. This demonstrates that thedegradation in the accuracy attributable to the low-ordermethod used for the axial treatment is not very significant.

More practical and extensive verifications were thenperformed primarily by comparing DeCART andcontinuous-energy Monte Carlo solutions. This was doneby solving a series of test problems by both codes thatspans a wide range of geometrical configurations and

temperature conditions. The DeCART and Monte Carlomodels used the same modeling parameters such as iso-topic number densities and temperatures. Specifically,the regionwise temperature distribution determined bythe DeCART calculation was used in the correspondingMonte Carlo calculation so that the identical tempera-ture distribution was used in both cases.

The first set of test problems was used to assess theaccuracy for PWR applications and was based on a typ-ical 17 �17 PWR fuel design. For the PWR cases, com-parisons were made between DeCART and the MCCARDMonte Carlo code.16 The fuel rod consists of 4.95 wt%UO2 pellets and Zr cladding whose diameters are 0.805and 0.952 cm, respectively. The isotopic number densi-ties of the fuel, cladding, and coolant are specified sothat both codes use the same composition. The activecore height is 2 m. The geometrical configurations ofthe test problems consist of pin cell, assembly, and twominicores in both two and three dimensions. The basicminicore ~case A! consists of 21 identical fuel assem-blies ~four corner assemblies removed from the 5 � 5fuel assembly lattice!, and the other ~case B! has acheckerboard-type arrangement of 4.95 and 2 wt% fuelassemblies. Both uniform temperature conditions repre-senting the hot-zero-power condition and nonuniformtemperature conditions representing the hot-full-power~HFP! and the hot-double-power ~HDP! conditions arespecified. The HDP condition is added to compensatefor the low power density of the HFP case. A 47-groupcross-section library was used in the DeCART runs whilethe continuous-energy cross sections generated with a10 K interval from 300 to 2500 K were used in MCCARD.

The reactivity obtained by DeCART and MCCARDfor the pin cell ~PC!, fuel assembly ~FA!, and minicore~MC! are compared in Table I. The full-power ~F! andthe double-power ~D! cases involve nonuniform fuel tem-peratures while the hot 600 K ~H! and the zero powercases are the uniform temperature cases. In the MC-CARD runs, sufficient neutron histories were used toyield standard deviations of the reactivity ,10 pcm. Forthe PC and FA cases, the reactivity difference betweenthe two codes is ,70 pcm for the four temperature con-ditions examined. There are various factors affecting theaccuracy of a deterministic transport code such as thetransport solver, basic cross-section data, and the reso-nance self-shielding treatment method. The good agree-ment between the DeCART and MCCARD reactivityimplies that the DeCART method and data, particularlythe resonance treatment method and data, are suffi-ciently accurate. The error in the reactivity becomes largerfor the minicore case where there is large radial leakage.However, even in these cases the difference is,200 pcm,which still represents a reasonable accuracy.

The power-generating minicore cases involve highlynonuniform fuel temperature distributions since the powervanishes at both radial and axial boundaries. Therefore,it should be a good measure of the effectiveness of the



subgroup method implementation for nonuniform tem-perature cases. The radial power distribution error ofDeCART for the checkerboard-type loading minicore case~case B! is shown in Fig. 1. As can be seen in Fig. 1, thepin-power error is ,0.7%, and no particular trend isobserved. This extent of the agreement is noted in otherminicore cases as well, as can be identified in the powerdistribution comparison summary given in Table II.

The second set of cases was used to verify the AMRTmethod in DeCART. It consists of three 2-D test prob-lems involving two types of BWR assemblies. The firstassembly type is modeled after a UO2 Atrium10 assem-bly mentioned in an Organization for Economic Coop-eration and Development0Nuclear Energy Agency~OECD0NEA! benchmark,17 and the other is similar to aGE10 assembly ~an artificial design!. The first two prob-lems are single-assembly problems for each fuel typewhile the third problem is a 2 � 2 checkerboard-typeproblem consisting of these two assembly types. Thefuel rod array of the Atrium10 assembly is 10 �10 whilethat of the GE10 assembly is 8 � 8.

The reference solutions for these problems were ob-tained by MCU Monte Carlo simulations.18 The standard

deviation in the MCU simulations was 20 pcm in allcases. The error in the eigenvalue and pin-power distri-bution of the DeCART solutions is summarized in

TABLE I

Comparison of keff for Different Temperature Profiles

Configuration CodeH

~600 K!Z

~543 K!F

~100%!D

~200%!

PC2D DeCART 1.40342 1.43914 1.42052 1.41385

MCCARD 1.40220 1.43880 1.41975 1.41347Difference, pcm 62 16 38 19

3D DeCART 1.38212 1.42110 1.39847 1.39003MCCARD 1.38147 1.42134 1.39867 1.39024Difference, pcm 34 �12 �10 �11

FA2D DeCART 1.43057 1.46181 1.44521 1.43897

MCCARD 1.43012 1.46201 1.44540 1.43914Difference, pcm 22 �9 �9 �8

3D DeCART 1.40868 1.44349 1.42299 1.41507MCCARD 1.40913 1.44458 1.42396 1.41623Difference, pcm �23 �52 �48 �58

MC2D-A DeCART 1.36700 1.34212 1.33513

MCCARD 1.36455 1.33945 1.33266Difference, pcm 131 149 139

2D-B DeCART 1.28388 1.26135 1.25470MCCARD 1.28082 1.25820 1.25155Difference, pcm 186 198 201

3D-A DeCART 1.35027 1.31977 1.31106MCCARD 1.34891 1.31813 1.30920Difference, pcm 75 94 108

3D-B DeCART 1.23995MCCARD 1.23757Difference, pcm 155

Fig. 1. Radial pin-power error of DeCART @DeCART-MCCARD ~%!# .

400 WEBER et al.


Table III, and the pin-power error of the fine-meshcheckerboard case is shown in Fig. 2. Since the eigen-value error is ;110 pcm and the pin-power error is 2%in the fine-mesh cases, these results provide confidencein the accuracy of the ARMT module in DeCART.

For the Atrium10 model, additional comparisons weremade for the intra-pin-power distribution in a few se-lected pins. The results compare 12 regions within thefuel pellet, with 3 radial divisions and 4 azimuthal divi-sions, as shown in Fig. 3. These results are comparedwith those of MCU in Table IV for a sample pin. Thissample pin is adjacent to a water channel on one side.Table IV gives the relative power in each division ob-tained from the MCU simulation and the relative error~in parentheses! in the DeCART prediction. The maxi-mum error is ,1.25%. The standard deviation in theMonte Carlo calculation was between 0.4 and 0.6%.

TABLE II

Summary of Power Distribution Errors for the Minicore Cases

Axially Averaged Three-Dimensional-Wise

Maximum Error ~%! Fr Maximum Error ~%!

Case

Root-Mean-SquareError~%! Absolute Relative Value

Error~%!

Root-Mean-SquareError~%! Absolute Relative

MC3Z-A 0.15 0.63 0.50 1.825 0.04 0.45 2.11 3.42MC3F-A 0.23 0.82 0.69 1.671 0.10 1.77 5.27 6.97MC3D-A 0.33 1.26 0.84 1.650 0.15 1.60 4.89 6.36MC3F-B 0.26 0.92 0.67 1.922 0.12 1.43 4.58 6.08

TABLE III

Comparison Summary for the 2-D BWR Assembly Cases

Model MCU keff

Error in keff

~pcm!

Root-Mean-SquarePin Power Error

~%!

Maximum PinPower Error~%!

Atrium10 1.1033 105 0.77 1.99GE10 1.0979 �96 0.98 2.07Checkerboard 1.1010 1 0.85 2.01

Fig. 2. DeCART pin-power error ~%! for the 2�2 checker-board problem ~fine angular discretization!.

Fig. 3. Pin edits regions for DeCART and Monte Carlopower distribution.



III. COMPUTATIONAL FLUID DYNAMICSFOR CORE THERMAL-HYDRAULIC

ANALYSIS

With the advance of parallel computers and theirwider availability, the analysis of thermal-hydraulic phe-nomena in a reactor core based on solutions of the Navier-Stokes equations is gaining acceptance by the reactordesign community.19,20 A highly refined CFD model withthe ability to resolve the effects of spacer grid on thermalmixing and cross flow has the potential to dramaticallyimprove the accuracy of core design. As part of the NNRdevelopment effort, the CFD methods were used to cal-culate the flow and heat transfer using a rigorous pin-by-pin representation of fuel assemblies and surroundingcoolant channels in the core.1 Using fine-mesh CFD so-lutions in each coolant channel offers an integrated whole-core analysis capability to capture important feedbackeffects between the first-principles–based models.

Many widely used CFD software codes were con-sidered, and their specific capabilities relevant to the in-tegrated simulation of a reactor core were reviewed. Thesecapabilities include turbulent flow and heat transfer sim-ulations, conjugate heat transfer between the fluid andsolid regions, steady-state and transient analyses, single-phase and multiphase flow, moving boundary and de-forming mesh structures, fluid-structure interactions, andefficient parallel computing schemes. More detailed eval-uations were performed with STAR-CD and the CFD-ACE codes, primarily because of their availability at ANLand KAERI, respectively.

The commonly used turbulence models were evalu-ated for the prediction of flow and heat transfer in variousrod-bundle configurations. Solving the Navier-Stokes mo-mentum equations in conjunction with the continuity andenergy conservation equations to simulate flow and heattransfer in a reactor core ~including the fluid and solid

domains! offers a mechanistic approach based on firstprinciples. Because of the scale of the problem, how-ever, the field variables and Reynolds stresses are oftenexpressed in terms of their ensemble averages that arelinked to the mean flow field via turbulence closure mod-els. These closure models consist of a set of additionaldifferential or algebraic equations. The most commonlyused turbulence models fall under the category of Reyn-olds Averaged Navier-Stokes ~RANS! models. The stan-dard k-« model is based on the linear eddy viscosityhypothesis for the Reynolds stresses.21–23 The aniso-tropic eddy viscosity relationship removes the assump-tion of turbulence isotropy by using quadratic24–26 andcubic27 constitutive formulations for the stress-strainrelations. Other variations of RANS models includethe renormalization group version28 and Chen’s vari-ant.29 The more complex second-order closure modelssuch as the differential Reynolds Stress Model ~RSM!are based on exact transport equations for the individualReynolds stresses as derived from the Navier-Stokesequations.30

The RANS-based turbulence models were first as-sessed for heat transfer in a pipe ~with an equivalenthydraulic diameter of a typical PWR flow channel! andcompared to the values predicted with the well-knownDittus-Boelter31 correlation. For Re numbers ranging from25 000 to 500 000 corresponding to PWR operating con-ditions from 5% to full power, the heat transfer coeffi-cients computed at the wall by most RANS models weregenerally within the experimental uncertainty of the cor-relation ~610%!. The predictions with the two-layermodel23 for Re � 254 809 differed from the correlationonly 3.5%, suggesting that the fine-mesh resolution ofthe boundary layer may result in improvement in heattransfer predictions.

The assessments of turbulence models for single-phase flow were performed for both the bare-rod-bundleconfigurations and fuel assemblies with spacer grids.32

The numerical simulation of turbulent flow structurewas performed for six-pin square-pitch33 and a 37-pintriangular-pitch34 bare-rod-bundle configurations. In bothcases, a symmetric subsection of the test subchannel wasmodeled for a set of pitch-to-diameter ratios. Theturbulence-driven secondary flows were captured withall nonlinear RANS and RSM models, exhibiting a lat-eral recirculation pattern in the subchannel caused byanisotropy in turbulence. In both studies, the CFD pre-dictions were found to be generally in good agreementwith the time-averaged experimental velocity and tem-perature profiles. However, the axial velocity distribu-tion is consistently underpredicted in the narrow gapbetween the fuel pins and overpredicted in the subchan-nel center by all turbulence models. The nonlinear k-«models provide more accurate predictions over the stan-dard k-« model; however, the best agreement with theexperimental results is obtained with the higher-order~and more computation-intensive! RSM.

TABLE IV

Comparison of Monte Carlo and DeCARTIntra-Pin-Power Distribution

Radial DivisionAzimuthalDivision 1 2 3

1 0.99 1.07 1.14~�0.06%! ~�0.59%! ~�0.95%!

2 0.96 1.00 1.06~0.60%! ~1.23%! ~0.21%!

3 0.93 0.95 0.98~0.55%! ~�0.05%! ~�0.62%!

4 0.95 0.97 1.01~�0.32%! ~�0.13%! ~0.06%!

402 WEBER et al.


The radial variations of turbulence intensities showa similar trend with the RSM predictions being in betteragreement with the measured values. While the experi-mental data suggest a significant increase in the azi-muthal turbulence in the narrow gap between the fuelpins, the predicted values were considerably lower thanthe measured ones.35 The CFD calculations also pre-dicted larger peripheral variations in wall temperatureand shear stress compared to;3% variation observed inexperiments. The discrepancy was larger for smaller pitch-to-diameter ratios, revealing the limitation of RANS meth-ods in capturing the large eddy fluctuation between twoadjacent rods.

Originally developed for PWRs, the NNR is cur-rently being extended for BWR applications under a pro-gram cofunded by the U.S. Department of Energy ~DOE!and the Electric Power Research Institute. Implementa-tion and validation of a new Eulerian two-phase-boilingheat transfer model in the fluid dynamics moduleSTAR-CD and its application for prototypic BWR fuelassembly configurations as part of the integrated analy-sis capabilities of NNR are an ongoing effort.36

IV. COUPLED CALCULATIONS

After demonstrating the validity of the individualcomponents of the NNR, the focus of the project wasthen to demonstrate an integrated analysis capability usingthe multiphysics models. Several general-purpose cou-pling schemes were developed, and the solutions werecompared as part of the quality-assurance plan. Variousiteration strategies for exchange of relevant informationamong the modules were also investigated in order toexamine such modeling options as the mapping betweenthe very different neutronic and thermal-hydraulic grids.The coupling of CFD and MOC in the NNR was verydifferent from conventional LWR coupled codes in whichmultiple neutronic nodes were generally mapped to asingle thermal-hydraulic mesh. In the NNR the meshsize in CFD is much smaller than the neutronic MOCmesh, and there are considerably larger and more com-plex data transfer requirements.

IV.A. Coupling Methodology

The coupling of DeCART and STAR-CD wasachieved using an external interface program.37,38 Thefirst task that the interface performs is to map the CFDand DeCART meshes. As shown in the pin-cell problemin Fig. 4, the CFD mesh is significantly finer than theDeCART spatial mesh. This difference is even more sig-nificant in the axial direction, where a DeCART planemay be divided into 20 to 50 layers of CFD cells. In theNNR, the modeler is required to generate the CFD meshsuch that it is a simple refinement of the coarser De-CART mesh. Thus, each CFD cell is assigned to a single

DeCART region, and no partial mapping is permitted.The temperature of a particular DeCART region is com-puted as a simple mass-weighted average of the temper-atures of the CFD cells assigned to that region. The powerdensity distribution within a particular DeCART regionis approximated to be flat, and thus, all CFD cells as-signed to the same DeCART region are given the samevalue for the power density.

The second task performed by the interface is tomanage the communication between the CFD and neu-tronic code modules. The interface serves as the masterprocess with the CFD and neutronic processes as its cli-ents. Each data exchange cycle, STAR-CD transfers thetemperature and density of each CFD cell to the inter-face. The interface then averages these distributions inorder to map them to the corresponding DeCART uni-form cross-section regions. DeCART then updates itscross sections with the new CFD data. Upon completionof several iterations of the neutron transport calculation,DeCART sends the power density of each flat sourceregion to the interface. This distribution is then reversemapped to the CFD cells and transferred to STAR-CD.These data exchange cycles continue until calculationsare converged, as illustrated in the schematic in Fig. 5.

The amount of data transferred each data exchangecycle can be significant for practical problems. Each dataexchange cycle requires the transfer of three CFD cell-wise distributions ~temperature, density, and power den-sity!. Because the CFD mesh is much finer than theDeCART spatial mesh, the data transfer requirement isgoverned by the number of CFD cells. For a representa-tive small-core model,39 there were 64 million cells, and1.5 gigabytes of data were transferred each cycle. Theinterface implements a set of customized TCP0IP socketcommunication functions, which was developed to ac-complish communication between the STARCD and De-CART processes, each of which employs its own MPIcommunicator. The socket communicator has been shownto achieve a transfer speed of 89 megabits0s across com-pute nodes on the Jazz LINUX cluster of ANL on whichthe coupled calculations were performed. The Jazz clus-ter consists of 350 computing nodes, each of which has a2.4-GHz Pentium IV CPU with 2 gigabytes of memory.

Fig. 4. Schematic showing typical mesh within a fuel pincell for ~a! STAR-CD and ~b! DeCART.



The final task for the interface is to monitor andcontrol the convergence of the coupled field solution.The interface monitors the CFD convergence by track-ing the enthalpy residual calculated by STAR-CD aftereach iteration. The enthalpy residual is an indication ofhow well the temperature distribution has converged basedon the most recent DeCART power distribution. Experi-ence has shown that for practical reactor core problems,the enthalpy residual in the fuel is the last to converge.DeCART also computes several residuals after each trans-port sweep. Typically, these residuals are reduced veryquickly, such that it is not necessary to monitor themclosely except to ensure final convergence. The inter-face computes changes in the DeCART regionwise tem-perature and power distributions from consecutive dataexchange cycles. One important convergence criterion isthe error in power density distribution:

maxn� 6qn

k � qnk�1 6

qnk � � «q , ~5!

where qnk is the most recently received power density in

DeCART flat source region n, and qnk�1 is the previously

received power density in the same region. The STAR-CDfuel enthalpy and the power density error criterion aregenerally the most important indications of coupled cal-culation convergence.

The procedure for executing a coupled DeCART0STAR-CD calculation can be summarized as follows:

1. The interface computes the mapping of STAR-CDcells onto DeCART regions based on geometry input

supplied by each code. The interface then transfers aninitial power distribution to STAR-CD.

2. STAR-CD begins iterating using the power dis-tribution supplied by the interface. Once the fuel en-thalpy residual is sufficiently small, the CFD cellwisetemperature and density distributions are transferred tothe interface. STAR-CD waits until the power distri-bution is updated again. The interface converts thecellwise distribution into a cross-section regionwisedistribution and then transfers the data to DeCART.

3. DeCART begins performing transport sweepsusing the temperature and density distributions suppliedby the interface. Once a prescribed number of transportsweeps have been performed, the power distribution istransferred to the interface. DeCART waits until the tem-perature and density distributions are updated again. Theinterface performs the reverse mapping of the powerdistribution onto the CFD cells and transfers the data toSTAR-CD.

4. Evaluate the STAR-CD fuel enthalpy residual andthe criterion in Eq. ~5!. If coupled calculations are notconverged, repeat steps 2 and 3.

The completion of steps 2 and 3 comprises the com-pletion of one data exchange cycle. Typically, between 8and 12 data exchanges are required before the coupledcalculations are converged.

IV.B. Coupled Code Convergence Analysis

A convergence study was then performed using themultiassembly checkerboard model to investigate the

Fig. 5. Schematic of the DeCART0STAR-CD coupling scheme.

404 WEBER et al.


effect of data exchange frequency on the coupled calcu-lation convergence. The power density distribution inthe CFD calculation is updated after performing N trans-port sweeps in DeCART, and the density and tempera-ture distributions in DeCART are updated after reducingthe fuel enthalpy residual by a factor of a. The purposeof the study was to determine the impact of the values ofN and a on convergence of the coupled calculations. Theproblem was executed on 13 Jazz compute nodes, 12 ofwhich were shared by DeCART and STAR-CD, and theinterface was executed on the remaining compute node.

As expected, the solution accuracy did not dependon the selection of a and N since the eigenvalues werethe same for all cases. It was also observed that the com-munication time was not a significant contribution to thetotal run time and was ,2% of the total time. The opti-mum selection of a and N occurred when the enthalpyand power distributions converged at the same rate. Theminimum execution time ~a � 0.05 and N � 3! occurswhen the two criteria are satisfied simultaneously. Itshould be noted that in the cases that showed the bestoverall performance, the time required for STAR-CDwas less than three times the time required for DeCART.The total elapsed time for each case is shown schemati-cally in Fig. 6. The results of this study suggest that thetotal computation time can be reduced by decreasing thenumber of CFD iterations and increasing the number ofray tracing sweeps in each data exchange cycle. The re-sults also suggest that the optimum combination of CFDiterations and the number of ray tracings for each dataexchange cycle depends on the enthalpy tolerance. Thisimplies that the product of inner and outer iterations ~i.e.,the total number of enthalpy sweeps! per data exchangecycle is perhaps a more appropriate parameter to track theSTAR-CD performance during the coupled calculations.

IV.C. Coupled Code Application

The practical applications of the NNR focus involvethe interaction among the various physical models. Al-though each of these modules has been developed, veri-fied, and validated for the solution of the individualphenomena, it is the integration and feedback among thefields that provides the ultimate application of interest.Because the original objective of the project was LWRanalysis, a testing sequence was developed that includeda single-pin model, a multipin model, a fuel assemblymodel, a multiassembly model, and a small-core modelfor a typical LWR. Because of space limitations, onlythe results of the largest problem will be presented here.

A model was developed for a core that is about fourtimes smaller than a typical PWR core. The DeCARTmodel for this problem is a quarter-core with four differ-ent types of 17 �17 assemblies. The core consists of fuelpins, gadolinium pins, guide tubes, and four types ofabsorber rods. There are 30 assemblies in the model, 11of which are radial reflectors. The core configuration isshown in Fig. 7; note that reflector assemblies along theperiphery are omitted in Fig. 7. Fuel pins are discretizedinto three annular rings and eight azimuthal slices. Thecladding and moderator each constitute an additional ringper fuel pin, and there are 12 planes in the model includ-ing lower and upper plena that function as reflectors.There are approximately 3 1

2_ million flat flux zones in the

core, and 45 energy groups were used. The angular dis-cretization is such that rays are positioned 0.02 cm apart,and there are eight azimuthal and four polar angles in a90-deg domain.

The CFD model consists of a one-eighth core seg-ment. There are four axial regions: lower plenum, active

Fig. 6. Total elapsed time as a function of enthalpy resid-ual reduction factor for different numbers of transport sweeps. Fig. 7. Pin-by-pin configuration.



fuel, fission gas plenum, and upper plenum. The radialdiscretization of the fuel pin is similar to DeCART, ex-cept that there are two rings in the moderator as shown inFig. 4. There are 25 azimuthal slices and 200 axial layersin the active part of the fuel pin. This discretization yieldsmore than 64 million cells.

The coupled code calculations were performed on 70processors of the ANL Jazz Linux cluster: 57 for STAR-CD, 12 for DeCART, and 1 for the interface. Steady-state calculations required ;11 h, but no effort has yetbeen made to optimize the performance. The eigenvaluefor this problem was 1.02017, and the pin average fueltemperature distribution is shown in Fig. 8 for the planewith the highest power ~plane 5!. It should be noted thatthe results for this core will differ from that of a typicalPWR core because all fuel assemblies contain fresh fueland the moderator is not borated. This exaggerates the

effect of moderation, especially at the core periphery.Nonetheless, it is interesting to note in Fig. 8 that there isa significant increase in the fuel temperature at the fuelpins adjacent to thermal neutron sources such as near theguide tubes and near the reflector region at the core pe-riphery. In the complete temperature information thatincludes the sub-pin-level fuel temperature distribu-tions, the azimuthal dependence of the clad temperaturecan also be examined so that local hot spots can be iden-tified. This detailed intra-pin-level information wouldnot be available in the current generation of coupled codes.

V. CONCLUSIONS AND FUTURE WORK

A methodology for integrating full-physics, high-fidelity reactor analysis was established and demon-strated in the work reported in this paper. It was shownthat very detailed sub-pin-level power and temperatureinformation can be obtained from the first-principles–based approach utilizing the high computing perfor-mance of a reasonably sized LINUX cluster. The half-day execution time is reasonable considering the accurateand detailed information provided by the NNR. Futureapplications of the NNR are anticipated for both LWRsand advanced reactors, though near-term emphasis willcontinue to be placed on LWRs. In addition to the capa-bilities described in this paper, transient analysis anddepletion analysis capabilities have already been devel-oped and will be validated using both numerical andexperimental data.

For many applications to both PWRs and BWRs, theability to predict thermal-hydraulic behavior under two-phase-boiling conditions is very important. A new boil-ing model has been implemented in the NNR CFDmodule, and an extensive testing and validation programhas been initiated, with particular emphasis on the OECD0NRC benchmark exercise40 for boiling behavior in BWRassemblies at prototypical temperature, pressure, and heatgeneration conditions. The extension of the boiling modelto treat other expected flow regimes is in progress alongwith comparisons to the aforementioned rod bundle ex-periments.36 An initial application of the newly devel-oped BWR version of the NNR is currently in progress.In this program the NNR will be used to identify localthermal-hydraulic conditions under BWR operating con-ditions that will help explain the development of so-called tenacious crud, which may be responsible for someobserved fuel failures.

The coupled code algorithms and methodology de-veloped initially for LWRs appear easily extendible toother reactor types. For some such applications, furthermethods development and code validation will be nec-essary, as has been done for BWR applications. Forapplications such as gas-cooled reactors, analytical tech-niques are currently available, and programs are under

Fig. 8. Average fuel pin temperature for the plane withthe highest power along selected rows of pins. Pin row 1 is forthe bottom row in Fig. 7.

406 WEBER et al.


way for higher-fidelity representation. Using the samemodular approach of the NNR, one can easily envisioncoupling neutronics, thermal hydraulics, and fuel perfor-mance in a robust analysis environment. With the impor-tance of demonstrating economic operations and meetingsafety requirements and the limitation of extensive ex-perimental facilities, it is very likely that the nuclearcommunity will move in the direction of simulation-based design, and high-fidelity, full-physics simulationcapabilities such as the NNR will play an increasinglyimportant role.

ACKNOWLEDGMENTS

This work was supported by the I-NERI program jointlyfunded by the Ministry of Science and Technology of Koreaand the DOE. Additional activities related to the BWR versionof the NNR were supported by the DOE.

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