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Nuclear Engineering and Design 250 (2012) 403– 411

Contents lists available at SciVerse ScienceDirect

Nuclear Engineering and Design

j ourna l ho me page: www.elsev ier .com/ locate /nucengdes

oupled neutronics thermal-hydraulics analysis using Monte Carlo andub-channel codes

iriam Vazqueza,∗, Haileyesus Tsige-Tamiratb, Luca Ammirabileb, Francisco Martin-Fuertesa

Ciemat, Avda. Complutense 40, 28040 Madrid, SpainEuropean Commission, Joint Research Centre, Institute for Energy and Transport, P.O. Box 2, NL-1755 ZG Petten, Netherlands

i g h l i g h t s

Neutronics and thermal-hydraulics coupling with MCNPX and COBRA-IV.We study the pseudo-material approach in fast reactors.We carry out a coupled calculation in a SFR fuel assembly at pin-by-pin level.SFR full core analysis with fuel assemblies grouped by radial rings.The keff difference is 200 pcm between the MCNPX stand alone calculation and the coupled solution.

r t i c l e i n f o

rticle history:eceived 15 November 2011eceived in revised form 29 May 2012ccepted 5 June 2012

a b s t r a c t

The accuracy and the degree of spatial resolution of safety studies, required for new reactor concepts,imply the use of coupled 3D neutronic and 3D thermal hydraulic codes. Tools to perform the couplingbetween neutronic codes both deterministic and stochastic with plant or sub-channel codes are beingdeveloped worldwide. With the increase of computational resources, Monte Carlo codes like MCNPX areacquiring much more relevance. They are able to obtain results without major approximations in thegeometry and with point-wise cross section representation. This paper describes the development of a

coupled neutronics/thermal-hydraulics code system based on Monte Carlo code MCNPX and the sub-channel code COBRA-IV. In the current work the temperature dependence of nuclear data is handledwith the pseudo material approach and based on JEFF 3.1 data libraries compiled with NJOY. The codehas been applied to a sodium fast reactor (SFR) concept at both fuel assembly and full core scale. This isthe first step toward a more comprehensive tool that takes into account more phenomena and feedbackeffects.

© 2012 Elsevier B.V. All rights reserved.

. Introduction

The computational accuracy required for design and safetynalysis of current and future nuclear reactors is continuouslyncreasing. The on-going tendency toward improving computa-ional accuracy is based on a coupled multi-physics approach,he final goal being to be able to perform static and dynamic

nalyses of nuclear reactors integrating all relevant phenomena:eutronics, thermal-hydraulics and structural mechanics. Severalulti-physics nuclear reactor analysis computational platforms are

∗ Corresponding author. Tel.: +34 913466778; fax: +34 913466576.E-mail addresses: miriam.vazquez@ciemat.es (M. Vazquez),

aileyesus.TSIGE-TAMIRAT@ec.europa.eu (H. Tsige-Tamirat),uca.AMMIRABILE@ec.europa.eu (L. Ammirabile),rancisco.martin-fuertes@ciemat.es (F. Martin-Fuertes).

029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.nucengdes.2012.06.007

currently under development (Meplan et al., 2009), but none seemsto have achieved maturity for general release.

As a sub-part of such a multi-physics computational platform,Monte Carlo codes for neutron transport are been coupled withthermal-hydraulics codes (Hoogenboom et al., 2011; Puente-Espelet al., 2010; Sanchez and Al-Hamry, 2009; Ivanov et al., 2011) forgeneric static analysis of nuclear reactors. Most of the innovativefast reactors that are being developed within the Generation IVInternational Forum (GIF, 2002) are planned to use heterogeneouscores, capable to load minor actinides (MA) bearing fuels in a fastspectrum. Due to the characteristics of the fast reactor spectra, inorder to calculate the neutron flux and reaction rates accurately,a continuous energy representation is required rather than the

multi-group one, usually used in thermal reactors. The calcula-tion of a heterogeneous core is easier with Monte Carlo methodsthan with deterministic codes because the latter needs approxi-mations to solve the transport equation and with the Monte Carlo

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04 M. Vazquez et al. / Nuclear Engin

ethods, the transport equation is solved by simulating the behav-or of individual particles in the real geometry.

The developed code system which is presented in this paper,nables consistent steady-state three dimensional full-core cou-led neutronics/thermal-hydraulics analysis. The coupling schemeollows the traditional approach of loose coupling where the infor-

ation exchange is performed via shared file or message passing.he implementation is sufficiently generic, such that any type ofeactor which can be handled by the involved sub-codes can beimulated.

In Section 2, the methodology of the coupling and a brief descrip-ion of the codes involved are explained. The implementation of theode is object of Section 3, describing the driver program and theross section handling. Finally, in Section 4, two applications of theoupled code are shown: a coupled calculation in an average fuelssembly of a sodium cooled fast reactor (SFR) at sub-channel scale,nd in a full SFR core at channel scale.

. Methodology

.1. Coupling scheme

The traditional approach for code coupling in nuclear engineer-ng is based on loose coupling where individual validated codeserform their respective calculation and exchange informationt specified points. In a coupled transient calculation, the timedvance of each physics component can be performed simultane-usly or firstly one part is solved and then the data are used for thether set of equations. With both options only one iteration is per-ormed per time step, leading to nonlinearly inconsistent methodsnd first-order time discretization (Ragusa and Mahadevan, 2009).ith this coupling scheme, the time approximation order of each

solated code is masked by the approximation order of the cou-ling approach. Loss of accuracy is a main drawback. However thispproach is the state-of-the-art coupling methodology to performulti-physics studies in nuclear reactor design and safety analyses

ecause it allows the use of verified and validated codes for eachart of the problem without major modifications of the source code.ystem codes RELAP5 (Relap5 Team, 2002) or TRACE (Odar et al.,004) serve as good examples.

The use of different codes to solve each field of study is equiva-ent to solve a set of partial differential equations with the operatorplitting method (Golub and Van Loan, 1989). Operator split-ing belongs to the family of iterative methods as the Jacobi orauss–Seidel methods. If x(k) is an approximate solution of the sys-

em of equations Ax = b and the matrix can be split as A = M − N thenhe new approximate solution is computed as Mx(k+1) = Nx(k) + b.his process is repeated until the converged solution is found. Fornstance, according to the Jacobi method, for the matrices of order

that have nonzero diagonal elements, the next value is calculatedsing the most current estimate of the solution as:

(k+1)i

=bi −

∑i−1j=1aijx

(k+1)j

−∑n

j=i+1aijx(k)j

aii(1)

If we express the matrix A as a combination of upper, lowernd diagonal matrices, A = L + D + U, the system can be rewrittenn a compact way: M = D and N = − (L + U). This loop is iter-ted until the convergence and the solution found is consistent.f the convergence is slow an over-relaxation method such us(k+1) = (M−1Nx(k) + M−1b)ω + (1 − ω)x(k) can be applied.

When a coupled system of partial differential equations is solved

ith the operator-splitting method, the time discretization can be

xplicit or implicit. It is implicit when the new variables values aresed to evaluate the Jacobian coefficients and explicit when the oldalues are used. If the coupling is carried out with two separated

and Design 250 (2012) 403– 411

codes, only an explicit or semi-implicit scheme can be used withoutmodifying the original codes. It leads to restrictions in the time stepused in order to avoid stability problems.

In a steady state analysis, as it is our case, the final solution isfound with an iterative approach without inconsistent problems.Since the calculation is performed at the same time point, theapproximation order of the time discretization is irrelevant. Themain critical aspect is the stability when the heterogeneity of thesystem leads to big differences between one iteration and the nextone. In this type of systems, relaxation methods should be includedin order to achieve the convergence or to speed it up.

Apart from thermal expansion effects, neutronics and thermal-hydraulics effects are linked together via the fission powergenerated and the temperature feedback produced due to coolantdensity changes and cross section Doppler broadening. Therefore,the neutron transport equation and the dynamic flow equations arecoupled. There are modern multiphysics algorithms like Jobian-Free-Newton–Krylov method (JFNK) (Knoll and Keyes, 2004) tosolve coupled systems of partial differential equations. Never-theless a new solver implies a big effort in code validation andverification. With the operator splitting technique, two validatedcodes can be coupled in a straightforward manner. The methoddescribed here is used for steady state calculations. The MonteCarlo code, MCNPX, solves the neutron transport equation witha given temperature and density field. The neutron flux obtainedin the simulation is used to calculate the system power map. Thethermal-hydraulic code, COBRA, calculates the temperature anddensity distribution in a sub-channel. The two codes are coupledvia file exchange and a driver program.

2.2. Codes

MCNPX (Hendricks et al., 2007) is a general-purpose,continuous-energy, generalized-geometry, time-independent,coupled neutron/photon/electron Monte Carlo transport code.MCNP can be used in several transport modes: neutron only,photon only, electron only, combined neutron/photon transportwhere the photons are produced by neutron interactions, neu-tron/photon/electron, photon/electron, or electron/photon. It canalso be used to compute the effective multiplication factor keff andthe fundamental mode eigenfunction in a critical system.

The COBRA-IV-I (Wheeler et al., 1976) code is an extendedversion of COBRA-IIIC being developed at the Pacific NorthwestLaboratory by Battelle-Northwest. It is a sub-channel analysis codewhich computes the flow and enthalpy distributions in nuclear fuelrod bundles or cores for both steady state and transient conditions.It can be used for thermal-hydraulic analysis of rod bundle fuelelements and cores for water, liquid metal and gas cooled reactors.

3. Implementation

3.1. Data exchange

The main temperature feedbacks in a neutron transport calcu-lation are the changes in coolant density, Doppler broadening ofabsorption cross section and thermal expansion of structural com-ponents. In Light Water Reactors (LWRs), the thermal feedback ispredominantly due to the relationship between coolant densityand temperature in the core. When the temperature increases, thecoolant density decreases causing a reduction in neutron scatter-ing rate. Another important thermal feedback effect is due to thetemperature dependence of the effective microscopic cross section.

In the fuel or structural materials as the temperature of absorbingnuclides rises, the absorption probability is higher due to the res-onance Doppler broadening of the absorption cross section. TheDoppler effect is an inherent negative reactivity coefficient with

M. Vazquez et al. / Nuclear Engineering

T claddingi , T fuel

i

T coo lanti , ρcoo lant

i

COBRAMCNPQi

giatastTfittw

tdtTcafgaiF

1

2

Fig. 1. Data exchange diagram.

reat importance in nuclear reactor safety and the dominant effectn some reactor designs. Furthermore, changes in material temper-ture affects neutron scattering probability due to the motion ofhe target nuclei in the epithermal and thermal energy range. Inddition to these two nuclear effects, the thermal expansion of thetructural components within the reactor may cause a change inhe geometry, which may affect neutron transport in the system.hermal expansion can be tailored as a key negative reactivity coef-cient in a fast reactor design. These three thermal effects implyhat the final power distribution within the core depends on theemperature of the coolant, fuel and structural materials, thoughith different time constants.

In the current study, thermo-mechanical effects are not yetaken into account, only temperatures and densities are updated toetermine the thermal-hydraulic feedback. The coupling betweenhe two mentioned codes is performed by means of data exchange.he power distribution in the fuel rod as a function of axial height isalculated by MCNP, and it is used by COBRA together with bound-ry conditions and fluid properties to obtain coolant density anduel, coolant and cladding temperature in each node. This hetero-eneous distribution of densities and temperatures is then used in

new MCNP input deck. A diagram of the data exchange is shownn Fig. 1, while the flow chart of the coupled procedure is shown inig. 2 and operates as follows:

. The coupling program starts with initial distribution of densitiesand temperatures and performs an MCNP calculation.

. When the MCNP run is finished, the output is read and the fluxtally results multiplied by fission cross section are used to calcu-late the power in each rod and axial level. The power is written

UpdateMCNPinput de ck

yes

Run MCNP

Run COBRA-IV

Calculate

START

ρcoolant

Tfuel, Tcoolant, Tclad

T fueli , T clad

i

ρcoolanti , T coolant

i

Qi

Ti−Ti−1

Ti−1

no

STOP

Fig. 2. Coupling code flow chart.

and Design 250 (2012) 403– 411 405

in an external file with the format and units system required byCOBRA.

3. COBRA is executed with the power distribution obtained in theprevious calculation. Fuel temperature is computed in each rod.Coolant properties are calculated in the sub-channels formedbetween three or four rods. Since MCNP cells are rod centered,coolant density and temperature are calculated as the averageof all the sub-channels surrounding a rod. Two files are createdby COBRA: the usual COBRA output and a second one containingdensity and temperature data for each MCNP cell.

4. The convergence criterion � is calculated in the first step as threetimes the maximum flux relative uncertainty in the MCNP fluxcalculation.

5. The convergence is achieved when the relative temperature dif-ference between the previous step and the actual is smaller thanthe established � criterion. The convergence is checked at eachaxial level and in each rod.

6. If the problem has not converged, a new MCNP input deckis created with the updated temperatures and densities. Newmaterials are also created with the suitable cross section depend-ing on the cell temperature.

Reaction probability depends on the material cross section. Thisprobability depends on the thermal motion of the nucleus in thematerial, therefore cross sections are function of temperature. Themain effort of this coupled calculation is focused on finding themost suitable solution to change the cross section.

3.2. Cross section handling

The cross section files are provided in ENDF-6 format (Hermanand Trkov, 2010) and in principle, they should be processed withNJOY (MacFarlane and Muir, 1994) to obtain point-wise cross sec-tion libraries in ACE format (suitable for MCNP) at the materialtemperature to include the movement of the atomic nuclei target.There are three different methods to use the cross section libraryat the cell temperature.

1. Generate explicit cross section libraries with temperature depen-dence (NJOY). NJOY performs the Doppler broadening of theresolved resonance data, generates the effective self-shieldedpoint-wise cross section in the unresolved energy range, whereit is difficult to measure individual resonances, and adjusts thecross section for free or bound scattering in the thermal energyrange. There are two options: run NJOY during the neutronic-thermal hydraulic iterations for each temperature needed oruse pre-generated NJOY libraries that cover the temperaturerange expected with enough accuracy. The first one is very time-consuming and is not practical for a realistic application. Thesecond one was successfully used to couple STAR-CD and MCNPwith pregenerated libraries at 5 K intervals in the fuel tempera-ture range (Seker et al., 2007).

2. On-The-Fly Doppler broadening. New methodologies to performDoppler broadening of cross section data during the MonteCarlo code execution are under development. The implementa-tion of On-The-Fly (OTF) Doppler broadening in MCNP involveshigh precision fitting of cross section data, with parameters thatdepends on the energy and temperature (Brown et al., 2012). Dif-ferent procedures have been developed for SERPENT (Viitanenand Leppanen, 2012) and TRIPOLI (Jouanne and Trama, 2010).OTF Doppler broadening feature is not yet included in the lastMCNP release.

3. Approximate approach using pseudo-materials. The pseudo-material approach consists in a weighted combination of anuclide X at lower temperature T1 and higher temperatureT2. The advantage is that it is not necessary to create new

406 M. Vazquez et al. / Nuclear Engineering and Design 250 (2012) 403– 411

Table 1Comparison of an SFR MCNPX simulation using fuel cross section libraries at different temperatures.

900 K (NJOY) 900 K (interpolated) 800 K (NJOY) 1000 K (NJOY)

keff 1.03989 (±3 pcm) 1.03999 (±3 pcm) 1.04156 (±3 pcm) 1.03854 (±3 pcm)�keff (pcm) 10 167 −135Averaged �f� (reactions) 5.09392E−03 5.09342E−03 5.10189E−03 5.08667E−03�(�f�) (%) −0.01 0.15 −0.14Capture reaction rate −5.3137E−01 −5.3134E−01 −5.3059E−01 −5.3206E−01

05

573E08

c2mcttcipsteitsulcolc

fs

˙

wh

w

w

Tss9aatwtbs

3

sts

per cycle and number of cycles being constant.The relative difference between the fuel, cladding and coolant

temperature in one iteration and in the next one is checked at everynode within the system:

0 604020 10080 1601401201.

1.02

1.04

1.06

1.08

1.1

k (trk length)

kcode data from file runtpe

Point source

Converged source

�capture (%) −0.0Fission reaction rate −3.5576E−01 −3.5�fission (%) −0.0

cross-section data sets, but the accuracy depends on the inter-polation interval.

In the present study, the NEA JEFF 3.1 cross section librariesompiled with temperature interval of 100 K were used (Cabellos,006). A different method has been adopted depending on theaterial. For the cladding and coolant the nuclear data at the

losest fixed temperature was applied because MCNP performshe automatic adjustment of the scattering cross section withhe TMP card. The Doppler broadening of the fuel absorptionross section is a key point to obtain the temperature reactiv-ty feedback and it is not performed by MCNP, therefore theseudo-material approach was used to obtain more accurate cross-ection libraries. The technique of mixing cross-section data fromwo temperatures in a fuel material was first used by Bernnatt al. (2000). Trumbull (2006) performed an analysis of differentnterpolation techniques between cross sections at two differentemperatures and calculated the relative difference in the crossection of important isotopes. Four interpolation schemes weresed (linear–linear, logarithmic–logarithmic, square root–linear,

inear–logarithmic). The first refers to the way the interpolatedross section is expressed and the second one to the calculationf the mixing coefficient. Trumbull showed that for linear interpo-ations, the linear coefficient is more accurate than the square-rootoefficient in all the studied cases.

In MCNP this interpolation was made by mixing two librariesor the same material and using a weighting factor for each dataet:

(T) = wlow˙(Tlow) + whigh˙(Thigh) (2)

here T is the actual temperature, Tlow and Thigh are the lower andigher temperatures, and the mixing coefficients are

high = T − Tlow

Thigh − Tlow(3)

low = 1 − whigh (4)

he accuracy of this approximation was studied in an MCNPXimulation of an SFR fuel assembly with MOX fuel at 900 K. Theimulation was carried out with the cross section temperature at00 K processed with NJOY, using pseudo-material at 900 K withn interpolation interval of 200 K and comparing with the resultst 800 and 1000 K. In Table 1 it is shown that the differences of usinghe library compiled with NJOY at 900 K and a pseudo-materialith an interpolation interval of 200 K is 10 times lower than using

he cross section data 100 K lower or greater. The keff differenceetween the interpolated library and the exact one is within the 3igma confidence interval.

.3. Convergence issues

The aim of a coupled calculation is to provide fairly accurateimulations in a reasonable amount of CPU time without reducinghe detail of the model. Monte Carlo criticality calculations haveome statistical limitations due to the bias in keff, reaction-rates

−0.14 0.12−01 −3.5630E−01 −3.5525E−01

0.15 −0.14

and the lower estimation of the uncertainties (Brown, 2009). Inorder to obtain correct keff and reaction rates, the cycles beforethe keff and fission source convergence should be discarded. Themultiplication factor is an integral quantity and it converges fasterthan the spacial source distribution. Fig. 3 shows the keff evolu-tion with two different initial sources: when a converged fissionsource from a prior calculation is used, the keff converges fasterthan when a point source is used. Furthermore the same calcula-tion takes 14% less CPU time with the converged source. In practice,the neutronic/thermal-hydraulic coupled calculation is carried outwith a fission source calculated previously and with more cycles.This source is used later in the next iterations with the consequentreduction of computational time.

Since the power calculated by MCNP has an uncertainty, theaccuracy of the coupled solution is fixed by the statistical uncer-tainty of the MCNP calculation and not by an arbitrary convergencecriterion. The reaction-rate uncertainty should be propagated to thethermo-hydraulic parameters calculated by COBRA and then set theconvergence criterion. In our case, the temperature convergence ischosen as three times the highest relative error. With the 3-sigmaconfidence interval, 99% of the results are within the range. This cri-terion is calculated during the first MCNP run and is used in the nextiterations to check the convergence. This is a coherent assumption,because if the criterion is not equal in all the calculations, the sys-tem would became unstable. Furthermore the MCNP uncertaintiesare nearly the same in all iterations due to the number of particles

keff Cycle Number

Fig. 3. Comparison of keff estimation using a point source and a converged fissionsource from a previous calculation.

M. Vazquez et al. / Nuclear Engineering and Design 250 (2012) 403– 411 407

wttmr

4

c

660

680

700

720

740

760

780

800

820

840

860

5 10 15 20 25 30

Co

ola

nt te

mp

era

ture

(K

)

Axial nodes

step1step2step3

(a)Coolanttemperature

0.83

0.835

0.84

0.845

0.85

0.855

0.86

nt d

en

sity (

g/c

m3

)

step1step2step3

Fig. 4. SFR fuel assembly model with MCNP. Vertical and horizontal cut.

Ti − Ti−1

Ti−1< � (5)

here Ti represents either the fuel, cladding of coolant tempera-ure in the current iteration, and Ti−1 the previous value. Althoughhe temperature convergence is checked for all the materials, the

ost restrictive convergence is the fuel temperature, being directlyelated to the pin power.

. Application to SFR technology

The MCNP–COBRA coupling tool has been applied to a SFR con-ept that is being developed within the framework of CP-ESFR

1300

1400

1500

1600

1700

1800

1900

2000

5 10 15 20 25 30

Fu

el te

mp

era

ture

(K

)

Axial nodes

step1step2step3

(a) Fuel temperature

300

350

400

450

500

550

0 5 10 15 20 25 30

-0.2

0

0.2

0.4

0.6

0.8

Pow

er

density (W

/cm

3)

Re

lative

diffe

ren

ce

(%

)

Axial nodes

step1step2step3

Difference

(b) Rod power densit y and relative difference betwee n

first and last it erati on per axial level

Fig. 5. Central rod properties along the iterations.

0.81

0.815

0.82

0.825

5 10 15 20 25 30

Co

ola

Axial nodes

(b)Coolantdensity

Fig. 6. Temperature profile in the FA central rod along the iterations.

project (Vasile et al., 2011). The first test case is a detailed analy-sis of an SFR fuel-assembly with pin-by-pin resolution and 30 axiallevels. The advantage of using a Monte Carlo code is that the reso-lution level is larger. The next step has been the application of thecode to an SFR full core to assess the feasibility of using this tool inextended simulations.

In a Monte Carlo calculation changes in temperature are relatedto changes in cell dimensions and densities and changes in thecross section temperature. In systems with high density varia-tion such as Boiling Water Reactors (BWR), Super Critical WaterReactors (SCWR) or High Pressure Light Water Reactors (HPLWR),the coolant density feedback is the responsible of the axial powerchange in the shape and in the peak value. Several proposals forfast reactor concepts use liquid metal as coolant, where the den-sity change with temperature is lower than with other coolants. Forinstance, an increment of 180 K in liquid water causes 0.185 g/cm3

density decrease, whereas the same temperature increment in liq-uid sodium changes the density in 0.045 g/cm3. In sodium fastreactors one of the main reactivity feedback is due to the radialexpansion of the core. As a first approximation, in our study thermomechanical effects are not taken into account, and the reactivityfeedback due to thermal hydraulic parameters will be explored.

4.1. SFR fuel assembly calculations

The geometry consists of a hexagonal fuel-assembly (FA) withreflecting surfaces to simulate the behavior of an assembly locatedin the core center. The fuel-assembly is composed of 271 fuel pins.Each pin is surrounded by helical wire wrap spacers. The hexagonal

408 M. Vazquez et al. / Nuclear Engineering and Design 250 (2012) 403– 411

st to l

wcdPoda

l9aat

tbaac

a

TE

Fig. 7. Power density map evolution (from fir

rapper tube is made of ferritic martensic steel (EM10). The fuel pinonsists of (U,Pu)O2 pellets in ODS steel cladding. The theoreticalensity of the Uranium Oxide (UO2) is 10.95 g/cm3 and from thelutonium Oxide (PuO2) is 11.46 g/cm3. Since the volume fractionf the fuel is 88.8% and the Pu mass content is 14.05%, the smearedensity of the MOX is 9.78 g/cm3. The dimensions of the fuel sub-ssembly are summarized in Table 2.

In the MCNP model, each pin contains fuel (100 cm activeength), upper and lower gas plena for storing fission gases (11 and1 cm respectively) and two axial segments (30 and 70 cm) withn homogenized steel and Na composition. An MCNP plot of a fuel-ssembly elevation is shown in Fig. 4, where it can be seen that onlyhe active length is simulated in detail.

Apart from the reactor geometry and materials, other impor-ant parameters to perform the study are the thermal-hydraulicoundary conditions: pressure, fluid temperatures, friction factornd heat transfer correlations. The thermal properties of the fuelnd cladding and the gap conductance are parameters needed to

alculate the temperature distribution inside the pellet.

The reactor is of pool type and the system pressure is set attmospheric pressure, i.e. 1 bar. The inlet temperature is 395 ◦C and

able 2SFR fuel sub-assembly dimensions.

Sub-assemblies pitch (mm) 210.8Sodium gap width inter assembly (mm) 4.5Wrapper tube thickness (mm) 4.5Wire wrap spacer diameter (mm) 1.0Number of fuel pins per sub-assembly 271Pin pitch (mm) 11.73Outer clad diameter (mm) 10.73Inner clad diameter (mm) 9.73Fuel pellet diameter (mm) 9.43

ast iteration) in the 5th and 20th axial levels.

the average mass flux for each sub-channel is 5126 kg/cm2 s to havea �T across the core of 150 ◦C.

The flow regime in a liquid metal reactor is turbulent and singlephase. The Ushakov correlation (Ushakov et al., 1977) is used for theheat transfer in the rod bundle and the sodium. The Nusselt factoris calculated with the simplified equation proposed by (Mikityuk,2009):

Nu = hD

k= 7.55x − 20

x13+ 0.41

x2Pe0.56+0.19x (6)

where x is the pitch–diameter ratio. It is recommended in triangu-lar lattice of rods, with x between 1.3 and 2.0 and Peclet numbers(Pe = RePr) up to 4000. Using x = 1.09319, the heat transfer coeffi-cient h then becomes

h = k

D(0.0343Re0.7677Pr0.7677 + 1.974) (7)

where k is the conductivity, D is the hydraulic diameter, Re is theReynolds number defined as Re = (�vD)/� and Pr is the Prandlt num-ber Pr = (cp�)/k being � the dynamic viscosity, cp the specific heat,� the density and v fluid velocity.

The friction factor is used to calculate the pressure drop in thesub-channel. This correlation depends on the channel type andthe roughness of the pipe and therefore must be implemented bythe user. In this simulation, the Rehme correlation has been used(Rehme, 1973):

f = 1.017Re71.55 + 0.1 (8)

The simulation was performed on a Linux cluster at the Insti-

tute for Energy, JRC Petten. The total computational time requiredfor this simulation was 129 h, using 48 processors each with 2 GBRAM memory. The MCNP statistical uncertainty is provided in one� confidence interval, meaning that the result has a 66% probability

M. Vazquez et al. / Nuclear Engineering

Table 3Initial temperatures and densities in the MCNP model.

Fuel temperature 1500 KStructural materials and coolant temperature 743 KAbsorber temperature 900 KFuel density 9.95 g/cm3

Coolant density 0.84 g/cm3

ODS steel density 7.25 g/cm3

Table 4FA keff in all the iterations of the coupled analysis.

1 iteration 2 iteration 3 iteration

oltavt

FdawfFs

control rods. The CSD (Control and Shutdown Device) is a hexago-

keff 1.02749 1.02568 1.02567� 0.00002 0.00002 0.00002

f being correct. The keff uncertainty in the simulation with 3 mil-ion particles per cycle is 2 pcm. The initial average conditions forhe MCNP simulation are shown in Table 3. The first keff guess usingveraged temperature and density values is 1.02749. The final con-erged keff result (Table 4) decreases by 200 pcm when a 3D map ofemperatures is used instead of an averaged constant temperature.

The power density distribution in all the iterations is shown inig. 5(b). In a sodium fast reactor the thermal-hydraulic feedbackoes not produce a significant change in the power distributionlong the sub-assembly because sodium properties change slowlyith temperature. The power only changes by 0.2% in average

rom one calculation to another when the relative error is 0.06%. Inig. 6 coolant properties in the central sub-channel are shown. Theodium temperature varies around 200 K in the core (from 670 K

1200

1250

1300

1350

1400

1450

1500

1550

1600

101 102 103 104 105 106 107 108 109 110

Fu

el te

mp

era

ture

(K

)

Axial nodes

step1step2step3

(a) Fuel temperature

660

680

700

720

740

760

780

800

101 102 103 104 105 106 107 108 109 110

Co

ola

nt te

mp

era

ture

(K

)

Axial nodes

step1step2step3

(b) Coolant temperature

Fig. 8. Axial temperature evolution in the first ring of the SFR core.

and Design 250 (2012) 403– 411 409

in the bottom to 850 K on top) but in this range the density onlychanges by 0.04 g/cm3. The temperature distribution in the centralrod is shown in Fig. 5(a) and the average value in the FA accordingto COBRA-IV simulation is 1706 K.

The power radial distribution in two axial levels in the first andthe last iteration are shown in Fig. 7. In spite of the statistical fluc-tuations, it is clear that, in the axial end regions of the assembly thepower is greater in the final iteration and lower in the middle plane.This change is due to the Doppler effect, since the final fuel temper-ature in the bottom is lower than the initial guess value of 1500 K(see Fig. 5(a)) and the opposite happens in the middle. The averageddifferences in power density in an axial level are shown in Fig. 5(b).As a consequence of the coolant heating inside the core, the fueltemperature is higher at the top than at the bottom. This asym-metry produces a positive difference (0.6%) at the top and smallerdifference at the bottom part (0.1%) for the power density.

4.2. SFR core calculations

The oxide core contains 225 inner fuel sub-assemblies and 228outer sub-assemblies. The difference between the inner and theouter is the Pu mass content of the MOX fuel (14.05% and 16.35%respectively). The fuel sub-assembly, studied in detail in the pre-vious section, consists of a hexagonal wrapper tube that containsa triangular arrangement of 271 fuel pins. There are two types of

nal wrap filled with 37 B4C rods (natural boron). The rod claddingas well as the wire spacer is made of ODS steel. The hexagonaltube containing the rod is made of EM10 steel and it is surrounded

220

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0

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(b) Radial power evoluti on and relative difference be-

tween the first and the last it erati on in the midd le plane

Fig. 9. Power density profile in the SFR core.

410 M. Vazquez et al. / Nuclear Engineering

650

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-0.3

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Difference

Fc

bD

dcfptbtacrF

ptaattfipito

igdp3it0

5

atcgu

p

ig. 10. Coolant radial temperature profile in 10th axial level (top) of the full SFRore and relative difference between the first and the last iteration.

y sodium. The other shutdown system DSD (Diverse Shutdownevice) is a cylindrical tube filled with the 55 B4C rods (90% B10).

The SFR MCNP model studied here consists of the fuel assembliesescribed in the previous section filling the core and both sets ofontrol rods withdrawn. Due to the radial symmetry of the model,uel assemblies have been grouped in 11 radial rings with similarower density in order to limit the number of cells and minimizehe computational time. In the COBRA core model, each fuel assem-ly is a channel and the rod temperature profile is calculated withhe average power developed in the assembly. The power, temper-ture and density are calculated in each fuel-assembly. Later theoupling code calculates the average temperature and density of aing and updates the MCNP input in the same way than in the SFRA calculation.

The coupled calculation converges in 3 cycles. Using 2 millionarticles per cycle and 150 criticality cycles, each MCNP calcula-ion takes 12 h using 64 processors from the Linux cluster placedt CIEMAT. The keff uncertainty is 3 pcm, the maximum flux rel-tive uncertainty is 0.3% and the average uncertainty among allallies is 0.17%. The results show that the keff difference betweenhe first guess with the homogeneous temperature model and thenal result with the heterogeneous temperature distribution is 188cm, the final keff value being equal to 1.01356. Note that in our case,

nitial average properties come from a good guess of the averageemperature. If the initial assumption had not been close to the realne the difference would have been higher.

As a result of the coupled calculation the fuel temperaturencreases for 25 K in the first ring, and the coolant temperature is 4 Kreater in the upper part (see Fig. 8). Fig. 9 shows the axial poweristribution in the first ring and the radial power in the middlelane. It can be seen that the power relative difference goes from% in the first ring to −2% in the last one. On the other hand, as

t is shown in Fig. 10, the maximum relative difference in coolantemperature between the first guess and the converged solution is.5%.

. Conclusions

A computational tool has been developed in order to perform theutomatic coupling between MCNPX and COBRA-IV, that improveshe calculation of neutronic reactor safety related parameters. Theode is written in C++and it allows the simulation of any kind of

eometry that can be analyzed with the individual code. It can besed for sub-channel analysis in a fuel-assembly or full core studies.

With the analysis of the SFR concept, the feasibility of the cou-ling scheme has been demonstrated. The code has been applied to

and Design 250 (2012) 403– 411

an SFR fuel assembly with a pin-by-pin description and to an SFRfull core where fuel assemblies are grouped by radial rings. Thismethod is robust for systems with low coolant density change suchus liquid metals and the coupled solution is found after three itera-tions. The reactivity change from the initial temperature estimationto the final one obtained from the thermal-hydraulic calculation is200 pcm in both cases. In the FA analysis the maximum power dif-ference between the first and the last step is 0.6% whereas in thefull core the maximum change is 3% in power.

A special effort has been made to deal with the cross sectiondependence with the temperature. The pseudo-materials approachwas used. Other options to handle the dependence or shorter inter-polation intervals have to be explored in order to find the suitablesolution in terms of accuracy and computational resources. In spiteof the parallel capabilities of MCNP, more work has to be done toreduce the computational time required by each MCNP job with-out precision loss or to optimize the number of MCNP calculations.The capabilities developed here are the first step toward morecomprehensive tool capable to perform transient calculations, andincluding additional reactivity effects important in fast reactorssuch us radial expansion.

Acknowledgments

The first author wishes to thank the European Commission, DGJoint Research Centre, Institute for Energy and Transport for provid-ing the traineeship funding and the facilities and for the stimulatingenvironment to carry out this work, as well as the Spanish Sci-ence and Innovation Ministry for the financial support through theFPI-CIEMAT grant.

Views and opinions expressed herein do not necessarily reflectthose of the European Commission.

References

Bernnat, W., Matters, M., Keinert, J., Conti, A., 2000. Application of the Monte CarloMethod for the Calculation of Safety Related Reactor Physics Parameter forPower Reactor with Consideration of Detailed Burn-up Distributions. Universityof Stuttgart, IKE-6-2000.

Brown, F.B., Martin, W.R., Yesilyurt, G., Wilderman, S., 2012. Progress with On-The-Fly Neutron Doppler Broadening in MCNP. Transactions of the American NuclearSociety, p. 106.

Brown, F.B., 2009. A review of Monte Carlo criticality calculations – convergence,bias, statistics. In: Proceedings of the International Conference on Mathematics,Computational Methods and Reactor Physics (M&C 2009), Saratoga Springs, NewYork, May 3–7, 2009.

Cabellos, O., 2006. Processing of the JEFF-3.1 Cross Section Library into a ContinuousEnergy Monte Carlo Radiation Transport and Criticality Data Library. OECD NEAData Bank, NEA/NSC/DOC.

U.S. DOE Nuclear Energy Research Advisory Committee and the Generation IV Inter-national Forum, 2002. A Technology Roadmap for Generation IV Nuclear EnergySystems. http://www.gen-4.org/PDFs/GenIVRoadmap.pdf.

Golub, G.H., Van Loan, C.F., 1989. Matrix Computations. The Johns Hopkins UniversityPress.

Hendricks, J.S., McKinney, G.W., Fensin, M.L., James, M.R., Johns, R.C., Durkee, J.W.,Finch, J.P., Pelowitz, D.B., Water, L.S., 2007. MCNPX, Version 26D. Los AlamosNational Laboratory, LA-UR-07-4137.

Herman, M., Trkov, A., 2010. ENDF-6 Formats Manual, Data Formats and Proceduresfor the Evaluated Nuclear Data Files ENDF/B-VI and ENDF/B-VII. BrookhavenNational Laboratory, BNL-90365-2009.

Hoogenboom, E., Diop, Ch., Ivanov, A., Sanchez, V., 2011. Development of couplingmethodologies for TRIPOLI and FLICA in the frame of NURISP project. In: The2011 International Conference on Mathematics and Computational MethodsApplied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil,May 8–12, 2011.

Ivanov, A., Imke, U., Sanchez, V., 2011. Development of a coupling scheme betweenMCNP5 and SUBCHANFLOW for the pin- and fuel-assembly-wise simulation ofLWR and innovative reactors. In: The 2011 International Conference on Mathe-matics and Computational Methods Applied to Nuclear Science and Engineering(M&C 2011), Rio de Janeiro, RJ, Brazil, May 8–12, 2011.

Jouanne, C., Trama, J.C., 2010. Linux is Binary of TRIPOLI-4 Capable of ProducingPoint Cross-Sections at Various Temperatures. NURISP Deliverable D.1.1.1, FP7-EURATOM-FISSION-232124.

Knoll, D.A., Keyes, D.E., 2004. Jacobian-free Newton–Krylov methods: a survey ofapproaches and applications. J. Comput. Phys. 193, 357–397.

eering

M

M

M

O

P

R

R

S

M. Vazquez et al. / Nuclear Engin

acFarlane, R.E., Muir, D.W., 1994. The NJOY Nuclear Data Processing System Ver-sion 91. Los Alamos National Laboratory, LA-12740-M.

eplan, O., Wilson, J., Bidaud, A., David, S., Capellan, N., Havluj, F., Vocka, R., 2009.MURE, MCNP Utility for Reactor Evolution – User Guide – Version 1.0. LPSC 0912,IPNO-09-01.

ikityuk, K., 2009. Heat transfer to liquid metal: review of data and correlations fortube bundles. Nucl. Eng. Des. 239, 680–687.

dar, F., Murray, C., Shumway, R., Bolander, M., Barber, D., Mahaffy, J.,2004. TRACE V4.0 User’s Manual. U.S. Nuclear Regulatory Commission,USA.

uente-Espel, F., Avramova, M., Ivanov, K., 2010. New developments of theMCNP/CTF/NEM/NJOY code system-Monte Carlo based coupled code for highaccuracy modeling. In: Proceedings of Advances in Reactor Physics to Power theNuclear Renaissance (PHYSOR-2010), Pittsburgh, Pennsylvania, USA, May 9–14,2010.

agusa, J.C., Mahadevan, V.S., 2009. Consistent and accurate schemes for coupledneutronics thermal-hydraulics reactor analysis. Nucl. Eng. Des. 239, 566–579.

ehme, K., 1973. Pressure drop correlations for fuel element spacers. Nucl. Technol.

1, 15–23.

anchez, V., Al-Hamry, A., 2009. Development of a coupling scheme between MCNPand COBRA-TF for the prediction of the pin power of a PWR fuel assembly. In:Proceedings of International Conference on Mathematics, Computational Meth-ods and Reactor Physics (M&C 2009), New York, USA.

and Design 250 (2012) 403– 411 411

Seker, V., Thomas, J.W., Downar, T.J., 2007. Reactor physics simulations with coupledMonte Carlo calculation and computational fluid dynamics. In: InternationalConference on Emerging Nuclear Energy Systems (ICENES-2007), Istanbul,Turkey, June 3–8, 2007.

The Relap5 Development Team, 2002. RELAP5/MOD3.3 Code Manual, vols. 1–8.Idaho National Engineering Laboratory, NUREG/CR-5535, Rev1.

Trumbull, T.H., 2006. Treatment of nuclear data for transport. Nucl. Technol. 156,75–86.

Ushakov, P.A., Zhukov, A.V., Matyukhin, M.M., 1977. Heat transfer to liquid metalsin regular arrays of fuel elements. High Temperature 15, 868–873 (translatedfrom Teplofizika Vysokikh Temperatur 15 (5), 1027–1033 (1977)).

Vasile, A., Fiorini, G.L., Dufour, Ph., Bonnerot, J.M., Latge, Ch., Riou, B., Stainsby, R.,Rini, M., Verwaerde, D., Struwe, D., Stieglitz, R., Badea, F., 2011. The collaborativeproject for a European sodium fast reactor CP ESFR. In: Proceedings of the Inter-national Congress on Advances in Nuclear Power Plants (ICAPP), Nice (France),May 2–6, 2011.

Viitanen, T., Leppanen, J., 2012. Explicit temperature treatment in Monte Carloneutron tracking routines-first results. In: Proceedings of Advances in Reactor

Physics Linking Research, Industry, and Education (PHYSOR-2012), Knoxville,Tennessee, USA, April 15–20, 2012.

Wheeler, C.L., Stewart, C.W., Cena, R.J., Rowe, D.S., Sutey, A.M., 1976. COBRA-IV-I: AnInterim Version of COBRA for Thermal-Hydraulic Analysis of Rod Bundle NuclearFuel Elements and Cores, BNWL-1962.