Annals of Nuclear Energy 63 (2014) 228–232

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Annals of Nuclear Energy

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Sensitivity analysis of core neutronic parameters in accelerator drivensubcritical reactors

0306-4549/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.anucene.2013.07.046

⇑ Corresponding author. Tel.: +98 21 22431595; fax: +98 21 29902546.E-mail addresses: A_feghhi@sbu.ac.ir, A.feghhi@gmail.com (S.A.H. Feghhi).

M. Hassanzadeh a, S.A.H. Feghhi b,⇑a Nuclear Science & Technology Research Institute, AEOI, Tehran, Iranb Radiation Application Department, Shahid Beheshti University, G.C, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 December 2012Received in revised form 15 May 2013Accepted 28 July 2013Available online 26 August 2013

Keywords:Sensitivity analysisNeutronic parametersMonte Carlo calculationsADSR

In this paper, sensitivity of the ADSRs core neutronic parameters to the accelerator related parameterssuch as beam profile, source multiplication coefficient (ks) and proton beam energy (Ep) has been inves-tigated. TRIGA reactor has been considered as the case study of the problem. Monte Carlo code MCNPXhas been used to calculate neutronic parameters such as: effective multiplication coefficient (keff), netneutron multiplication (M), spallation neutron yield (Yn/p), energy constant gain (G0), energy gain (G),importance of neutron source (u*), axial and radial distributions of neutron flux and power peaking factor(Pmax/Pave) in two axial and radial directions of the reactor core for three eigen values levels (ks) including:0.91, 0.97 and 0.99. According to the results, using a parabolic spatial distribution instead of a uniformspatial distribution increases the relative differences of spallation neutron yield, net neutron multiplica-tion and energy gain by 4.74%, 4.05% and 10.26% respectively. In consequence the required acceleratorcurrent (Ip) will be reduced by 7.14% to preserve the reactivity. Although safety margin is decreased inhighest case of ks, but energy gain increases by 93.43% and the required accelerator current and impor-tance of neutrons source decrease by 48.3% and 2.64% respectively. In addition, increasing Ep from115 MeV up to1 GeV, improves spallation neutron yield and energy gain by 2798.71% and 205.12% anddecreases the required accelerator current and power by 96.83% and 72.44%, respectively. Therefore,our results are indicative of the fact that investigating sensitivity of the core neutronic parameters tothe accelerator related parameters are necessary in order to optimally design a cost-efficient ADSR.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Accelerator Driven Subcritical Reactors (ADSRs), called hybridreactor, are modern nuclear reactors producing energy and trans-mutation of radioactive wastes in a clean and safe manner. Oneof the essential issues for an ADSR is the condition that the reactorcore is designed in such a way, that it remains sub-critical undernormal operation and off-normal conditions.

Number of neutrons produced per proton-nucleus reaction inthe spallation target is an effective parameter in a subcritical reac-tor driven by a proton accelerator. On the other hand, one of theobjectives in designing an ADSR is to obtain as high power as pos-sible in the core using as low proton beam power as possible (Kadiand Revol, 2001; Nifenecker et al., 2001, 2003).

Furthermore, the system is driven by an accelerator with ahigh-energy proton beam that smashes a target atom into manyatomic fragments producing a large number of neutrons. Therefore,the optimum proton beam energy for production of neutrons by

spallation depends on heavy metal target, in terms of costs, systemefficiency and etc. (Eriksson et al., 2005; Nifenecker et al., 2001).

Since the construction of a reliable high-power proton acceler-ator is a difficult technical task and its operation is very expensive,investigating sensitivity of the core neutronic parameters to theaccelerator related parameters has a significant impact on theoverall design of a future ADSR and on the economy of itsoperation.

Therefore, in this paper we study the effects of several varia-tions in the accelerator related parameters such as beam profile,ks and Ep on the core neutronic parameters of the TRIGA reactor(Borio di Tigliole et al., 2010) as the case study of the problem usingMCNPX (version 2.4) code (Hughes et al., 2002).

The TRIGA reactor is a pool-type research reactor moderatedand cooled by light water which is utilized in TRADE project (TRIG-A Accelerator Driven Experiment). This project is based on couplingof a linear proton accelerator with TRIGA reactor. The conceptualdesign is carried out. Due to a lack of enough fund, the protonaccelerator was replaced with californium sources and a smallD–T neutron generator placed in the centre of the reactor fuel in2004 (Borio di Tigliole et al., 2010; Rubbia et al., 2002a,b, 2004).

Table 1TRIGA core characteristics.

Description Value

Initial fuel mixture UZrHInitial fuel mass (g) 235.2Initial U concentration (%) 8.5Initial fissile enrichment (%) 20Proton beam energy (MeV) 115Accelerator current (mA) 2Number of fuel elements 116Number of control rods (B4C) 4Diameter of the core (cm) 56Height of the core (cm) 72AISI-304 thickness (cm) 0.508Density of AISI-304 (g/cm3) 7.5Density of fuel mixture (g/cm3) 5.8Density of clad (Zr) (g/cm3) 6.49Clad thickness (cm) 0.25Density of graphite rod (g/cm3) 2.25Diameter of the graphite rod (cm) 8.7Height of the graphite rod (cm) 3.63Pin external diameter (cm) 3.73Pin active length (cm) 38.1Pitch (cm) 3.97External fuel radius (cm) 1.815Internal fuel radius (cm) 0.25Fuel volume in a pin (cm3) 387.407Pin exchange surface (cm2) 446.46

AISI-304

Fuel

Graphite

Zr

Air

Fig. 2. A schematic view of the fuel rod in TRIGA core.

M. Hassanzadeh, S.A.H. Feghhi / Annals of Nuclear Energy 63 (2014) 228–232 229

2. Materials and method

2.1. Subcritical TRIGA core model

Monte Carlo analyses were all performed using an identicalgeometrical model of the subcritical TRIGA core shown in Fig. 1(Hassanzadeh et al., 2013). The core is divided into seven rings.Some of the regions contain fuel pins, graphite dummy elementsand control rods as well. Central channel has been designed forloading a neutron source.

Three different cases of simulations are considered as follow-ings (Rubbia et al., 2002a,b, 2004):

I. Criticality operation mode: The core configuration consistsof 108 fuel elements of 20% enrichment, four B4C controlrods, some tubes loaded for various experiments, elevengraphite rods and a tungsten neutron source.

II. Subcritical operation mode: For ks = 0.97–0.98, the structureis similar to the critical structure, except that 102 fuel pins inaddition to three control rods have been loaded into thecore.

III. Low level subcritical operation mode: This core configura-tion has been studied with 79 fuel pins and three controlrods. One of control rods is fully extracted; the second oneis positioned at 32% fuel �68% absorber and the third oneis positioned at 46.5% fuel �53.5% absorber that results inks = 0.90–0.91.

Also, spallation target neutronic source parameters for twocases of uniform and parabolic beam spatial distribution have beeninvestigated in accelerator driven subcritical TRIGA reactor. More-over, these parameters have been calculated at ks = 0.97 for protonenergies of 115, 300, 600, and 1000 MeV using the MCNPX code.Details of the TRIGA core characteristics are presented in Table 1.A schematic view of the fuel rod is shown in Fig. 2.

2.2. Calculation of neutronic parameters

MCNPX code reads the input file, where the geometry, thematerials, the neutron source and etc are described. Results ofinterest can be scored by using tallies. A tally is a specification ofwhat should be included in the problem output, for example theneutrons flux (F2 or F4) through a certain area or the number ofneutrons in a particular energy interval. In MCNPX code it is possi-ble to calculate integrals of the following equation:

F2;4 ¼ CZ

UðEÞRðEÞdE ð1Þ

Graphite (N=11)

Rabbit (N=1)

Control Rod (B4C) (N=4)

Loop (N=1)

Source (N=1)

Fuel Rod (N=108)

Fig. 1. Schematic view of the horizontal cross section of the TRIGA core.

where C is a multiplication constant, R (E) is any combination ofsums and products of energy-dependent quantities known toMCNPX and U (E) is the neutron flux. In this way, reaction rateswith different materials can be determined (Hughes et al., 2002).

Heating and energy deposition (MeV/g) can be determined byMCNPX code using F6 and F7 cell tallies according to the Eq. (2)(Hughes et al., 2002):

F6;7 ¼ qa=qg

ZHðEÞUðEÞdE ð2Þ

where qa is atom density, qg is gram density, U (E) and H (E) arethe neutron flux and heating response respectively. Power peakingfactors within the core has been calculated using these tallies.

Another powerful property of MCNPX code is the possibility ofcriticality calculation using KCODE card to give an estimation ofkeff. Continuous energy cross section data from nuclear data evalu-ation file, version ENDF/B-VII.0, has been used. Additionally, ther-mal neutron scattering cross sections has been used from theENDF/B-VI library (Hughes et al., 2002).

Neutronic parameters such as Yn/p, ks, keff, M, G0, G, Ip, u*, Pacc, axialand radial distributions of neutron flux and Pmax/Pave ratio in two axialand radial directions of TRIGA reactor core model have been calcu-lated for three eigen values levels (ks) including: 0.91, 0.97 and 0.99.

In the present study, the parameter u*, which represents therelative efficiency of external source neutrons is defined as the ra-tio of the average importance of the external neutrons source tothe average importance of the fission neutrons (Gudowski et al.,2001). The u* describes the difference between the real externalsource multiplication and the multiplication inherent to the distri-bution of neutrons corresponding to the fundamental modeaccording to the Eq. (3) (Seltborg, 2003, 2005):

u� ¼ 1keff� 1

� ��1ks� 1

� �ð3Þ

Fig. 3. Axial distribution of the neutron flux in the fuel pin.

Fig. 4. Radial distribution of the neutron flux in the TRIGA core.

230 M. Hassanzadeh, S.A.H. Feghhi / Annals of Nuclear Energy 63 (2014) 228–232

In order to study the beam power amplification in an ADSR theparameter G should be calculated using Eqs. (4)–(6). Ratio of pro-duced energy due to fission process in an ADSR core to the energyof primary particles colliding to the spallation target is defined asenergy gain, G (Nifenecker et al., 2001, 2003; Kadi and Revol,2001):

G ¼ Yn=p�Ef :ks

�mð1� ksÞEpð4Þ

where Yn/p is the number of neutrons produced per proton in thespallation target, �Ef is average energy per fission and �m is the aver-age neutron yield per fission. According to energy constant gain, G0,the equation is modified as below.

G0 ¼Yn=p

�Ef

�mEp) G ¼ G0

ks

1� ksð5Þ

The importance of neutrons source divided by the proton en-ergy (u*/Ep) is closely related to the energy gain of a source drivensubcritical system. The parameter G, as defined in Eq. (6), repre-sents the total power produced in the core over the acceleratorpower (Seltborg, 2003, 2005).

G ¼ Ptot

Pacc¼ keff

1� keff:Yn=pu�

Ep:�Ef

�mð6Þ

This can also express the accelerator current as Eq. (7) (Seltborg,2003, 2005).

Ip ¼Pacc

Ep¼ 1� keff

keff:

�m�Ef:

Ptot

Yn=pu�ð7Þ

Assuming that the accelerator current needed to drive the sub-critical core, Ip and the reactivity of the subcritical core have beenfixed, the total core power, Ptot and accelerator power, Pacc are gi-ven in kW.

3. Results and discussion

Axial and radial distributions of neutron flux in the core haveinfluence on the other important neutronic parameters such aspower and power peaking factor. Axial and radial distributions ofthe flux have been calculated using the MCNPX code and usingEqs. (1) and (2) for 0.91, 0.97 and critical eigen values.

Figs. 3 and 4 show the radial flux distribution averaged over thechannels of a given cylindrical ring and the axial flux distributionfor the hottest fuel pin respectively at different sub-criticality lev-els. As shown in Figs. 3 and 4, neutron flux decreases both in axialand radial directions towards the core edge. Clearly, the neutronflux has a maximum in the reactor core centre, where neutronsare moving in from all directions and decreases toward the bound-aries due to neutron leakage. The value of neutron flux calculatedusing Eq. (1) has been normalized to the accelerator current andspallation neutron yield according to the following procedure:

U ¼ 2� 10�3 C=smA

� 1p

1:6� 10�19C� F2=4� Yn=p

The power distribution in a sub-critical system is sensitive tothe core intrinsic multiplication coefficient (keff). As the keff in-creases towards unity (criticality conditions), the neutron flux dis-tribution flattens on the core fundamental harmonic.

The effective multiplication coefficient (keff) of the reactor coredoes not depend on the external neutron source intensity and itis defined as an intrinsic property of the system. For a system with-out an external source only fundamental mode of the eigen func-tion determines the neutron flux distribution, but in an ADSR,there is an external source and the transport equation has an

inhomogeneous part. In such systems neutron flux distribution de-pends on the external neutron source properties such as spatiallocation, energy spectrum, etc. If the neutron flux distribution ina source driven system is not determined only by fundamentalmode but also the higher order harmonics, source multiplicationcoefficient (ks) differs from keff. The importance of higher order har-monics increases with increasing sub criticality as the system ismore source dominated. One can formally define neutron sourcemultiplication as ks = 1�(1/M).

The neutronic parameters such as Yn/p, ks, M, G0, G, Ip and Pmax/Pave in two axial and radial directions have been calculated fortwo uniform and parabolic distributions of proton beam usingEqs. (4)–(7) based on MCNPX criticality calculations.

The main parameters of the TRIGA core for two cases of uniformand parabolic spatial beam distributions are shown in Table 2. Theresults show that neutronic parameters such as Yn/p, M and G havebeen increased by 4.74%, 4.05% and 10.26% respectively. Althoughthe Pmax/Pave ratio in the core is fairly improved (0.67–0.69%) by theuse of the uniform spatial distribution, but the required Ip has beenreduced by 7.14% for parabolic spatial distribution. The statisticaluncertainty associated with criticality calculations is approxi-mately 60.0 pcm in each case. The relative error for other calcu-lated parameters by MCNPX code generally lies within 5%.

The mentioned parameters have been calculated at 0.91, 0.97and 0.99 eigen values and are shown in Table 3. Source multiplica-tion coefficient (ks) of 0.97 is for the reference configuration. This

Table 2The main parameters of the TRIGA core for two cases of uniform and parabolic spatial distributions.

Main parameters Proton beam spatial distributions Relative differences (%)

Parabolic Uniform

Yn/p 0.464 ± 0.003 0.443 ± 0.003 4.74ks 0.9740 ± 0.0006 0.9730 ± 0.0006 0.10M = 1/(1�ks) 38.5 ± 0.8 37.0 ± 0.7 4.05G 13.11 ± 0.3 11.89 ± 0.3 10.26G0 0.35 0.33 6.06Ip (mA) 0.13 ± 0.003 0.14 ± 0.003 �7.14Pmax/Pave (Radial) 1.46 1.45 0.69Pmax/Pave (Axial) 1.50 1.49 0.67

Table 3the main parameters of the TRIGA subcritical for three cases of ks at 0.91, 0.97, 0.99 eigen modes.

Main parameters Reference High ks Relative differences (%) Low ks Relative differences (%)

ks 0.9761 ± 0.0006 0.9875 ± 0.0006 1.17 0.9078 ± 0.0006 �7.00keff 0.9745 ± 0.0006 0.9870 ± 0.0006 1.28 0.9012 ± 0.0006 �7.52u* 1.067 ± 0.008 1.041 ± 0.008 �2.64 1.079 ± 0.008 1.00G 13.1 25.3 93.43 3.2 �75.89Ip (mA) 0.13 ± 0.03 0.07 ± 0.03 �48.3 0.55 ± 0.03 314.8Pacc (kW) 6.87 ± 0.15 3.55 ± 0.16 �48.3 28.49 ± 0.20 314.8Pmax/Pave (Radial) 1.464 1.458 �0.41 1.603 9.49Pmax/Pave (Axial) 1.501 1.481 �1.33 1.735 15.59

Table 4The main parameters of the TRIGA subcritical configurations for proton energies of 115, 300, 600, and 1000 MeV.

Main parameters Reference case Case 1 Relative differences (%) Case 2 Relative differences (%) Case 3 Relative differences (%)

Ep (MeV) 115 300 160.87 600 421.74 1000 769.57Yn/p 0.464 2.984 543.10 7.773 1575.22 13.45 2798.71M = 1/(1�ks) 41.5 40.3 �2.89 39.2 �5.65 38.2 �8.03ks 0.9759 ± 0.002 0.9752 ± 0.002 �0.14 0.9745 ± 0.002 �0.15 0.9738 ± 0.002 �0.22G 14.2 33.9 138.73 42.8 201.69 43.3 205.12G0 0.35 0.86 145.71 1.12 220.81 1.17 233.06Ip (mA) 0.13 0.021 �83.85 0.007 �94.37 0.004 �96.83Pacc (kW) 14.95 5.89 �60.60 4.39 �70.63 4.12 �72.44

M. Hassanzadeh, S.A.H. Feghhi / Annals of Nuclear Energy 63 (2014) 228–232 231

value has been chosen in such a way that criticality conditions areprevented, with adequate margins, under all normal conditions aswell as following transient and accident conditions. As shown inTable 3, the parameter G has been increased by 93.43% and the re-quired Ip and u* have been decreased by 48.3% and 2.64% respec-tively in highest case of ks, although safety margin is decreased.In addition, Pmax/Pave ratio in the core increases in lowest case of ks.

Therefore, optimizing the importance of neutrons source andthereby minimizing the proton beam requirements can have anessential impact on the overall design of an ADSR and on the econ-omy of its operation. However, if the reactor is operating at a reac-tivity level closer to criticality, the reactivity insertion effect on thecore power will be higher and is more important than the u*.Moreover, the neutronic parameters have been calculated atks = 0.97 for proton energies of 115, 300, 600, and 1000 MeV usingthe MCNPX code and the results are shown in Table 4. According tothe results, increasing Ep from 115 MeV up to1 GeV, increases Yn/p

and G by 2798.71% and 205.12%, but decreases the required Ip andPacc by 96.83% and 72.44%, respectively. According to the results, Gis significantly affected by Ep and Yn/p. The statistical uncertaintyassociated with criticality calculations is approximately200.0 pcm in all cases.

4. Conclusion

Sensitivity of the ADSRs core neutronic parameters to the accel-erator related parameters such as beam profile, ks and Ep was

investigated. TRIGA reactor was considered as the case study ofthe problem. Monte Carlo code MCNPX has been used to calculateneutronic parameters for three levels of ks.

Power peaking factor in the core is fairly improved by the use ofa uniform spatial distribution. But using parabolic spatial distribu-tion improves the neutronic parameters and decreases the re-quired accelerator current and power.

At deeper sub criticality the system’s response is dominated bythe source behavior and requires higher accelerator current whichcan be difficult and expensive to realize. As the system is close tothe criticality, the ‘core dominated regime’ becomes more impor-tant. It was found that u⁄ remains approximately constant in theinterval 0.9078 < keff < 0.9875.

Therefore, the results show that investigating sensitivity of thecore neutronic parameters to the accelerator related parametershave an important impact on optimizing the overall design of anADSR and on the economy of its operation.

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