Reducing Irradiation Damage in a Long-Life Fast Reactor with Spectral Softening

energies

Article Reducing Irradiation Damage in a Long-Life Fast Reactor with Spectral Softening

Andrew G. Osborne 1,2 and Mark R. Deinert 1,2,*

1 Department of Mechanical Engineering, The Colorado School of Mines, Golden, CO 80401, USA; [email protected] 2 Nuclear Science and Engineering, The Colorado School of Mines, Golden, CO 80401, USA * Correspondence: [email protected]; Tel.: +1-303-273-3602

Received: 4 May 2018; Accepted: 6 June 2018; Published: 9 June 2018

Abstract: Long-life fast reactors receive considerable attention for their potential of using uranium efficiently, and because they can operate for extended periods without refueling. However, the main obstacle to achieving maximum operating times and fuel burnup is the neutron radiation damage that accumulates in the cladding and structural materials. Simulations of metal-fueled high-burnup fast reactors showed that the damage in these reactors’ cladding material reached 200 displacements per atom (dpa) long before the maximum burnup was achieved. One possibility for overcoming this problem is spectral softening, which would reduce the kinetic energy imparted to reactor materials when neutrons collide with them. In this work, we compared the peak irradiation damage in metal- and oxide-fueled fast reactors with that in equivalent reactors containing beryllium in the fuel and reflectors. We showed that the peak damage to the cladding in a metal-fueled reactor was reduced from 273 dpa to 230 dpa when beryllium was included in the core. In an oxide-fueled reactor, the peak damage to the cladding was reduced from 225 dpa to 203 dpa. All four reactors were operated with a core-average burnup of 112 MWd/kg of initial heavy metal (IHM), without reshuffling or refueling, and contained the same initial actinide mass profiles.

Keywords: fast reactor; long-life; irradiation damage; displacements per atom (dpa)

1. Introduction The sustainability of nuclear power depends in large part on its economic competitiveness and its long-term impact on the environment. Fast-neutron spectrum reactors can achieve high levels of fuel burnup, which increases both the amount of electricity that a unit mass of fuel can produce, and the burnout of transuranic nuclides which give nuclear waste its long-term radioactive signature. Long-life fast reactors have been of interest for decades, in part because of their potential to run for extended periods without refueling, and to operate as a “nuclear battery” which could reduce both the cost and the proliferation risks associated with nuclear energy [1]. Advanced fast-reactor designs can achieve peak burnup values of over 500 MWd/kg of initial heavy metal (IHM) before the accumulation of ﬁssion products stops the chain reaction [2]. However, the main limiting factor in achieving maximum fuel burnup in fast reactors is the neutron damage to core structural and cladding materials. High-energy neutrons collide with atoms in these materials, and displace them from their equilibrium lattice positions. Over time, this leads to macroscopic changes in the materials, such as swelling and embrittlement, which can cause the core to deform, and the cladding to rupture. Studies of proposed Generation-IV fast-reactor systems showed that operating these cores produces damage levels ranging from 200 displacements per atom (dpa) to 250 dpa [3] in the cladding. At present, the best candidate material for these applications, HT-9 steel, was demonstrated to have damage levels of only ~200 dpa at temperatures below 873 K [4,5].

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Advanced materials that could withstand higher dpa levels, and maintain their mechanical properties are in development [6]; however, they are unlikely to be available for practical use anywhere in the near term. Past work showed that spectral softening could be used to reduce irradiation damage in a fission wave reactor [7]. The softer spectrum reduces the kinetic energy imparted to reactor materials when neutrons collide with them. Here, we compared the dpa levels in long-life fast-reactor cores with and without beryllium in the fuel and reflectors. Beryllium was proposed as an additive to uranium dioxide fuel for light-water reactors in order to improve its thermal conductivity [8], and beryllium reflectors were used in studies of liquid-metal-cooled fast reactors to improve their safety [9,10]. We showed that the addition of beryllium reduced the peak dpa in a mixed-oxide-fueled core from 225 dpa to 203 dpa, and in a metal-fueled core from 273 dpa to 230 dpa. In all cases, the average burnup level was 112 MWd/kgIHM, and was achieved without the refueling or reshuffling of fuel. All cores had the same initial mass of actinides, and fuel densities were adjusted where appropriate to accommodate the beryllium. A high degree of core-average burnup was achieved by axially and radially grading the initial plutonium content of the fuel so as to flatten the power profile. The cores included diluent regions of sodium to reduce neutron flux in the regions that otherwise experienced the greatest irradiation damage.

2. Results The biggest obstacle to the practical implementation of a long-life fast reactor is the high degree of irradiation damage to in-core materials. However, spectral softening, and the use of diluent regions to reduce the peak neutron fluence to materials can be used in combination to overcome both problems. In the core designs presented in this work, a reduction in the relative population of high-energy neutrons was achieved through the use of beryllium-doped oxide fuel and beryllium reflectors. The axial and radial grading of the fresh fuel, along with the beryllium reflectors and a diluent region, flattened the power profile and allowed for a high core-average burnup before maximum allowable dpa levels were reached. Figure1 shows the dpa levels in the metal-fueled core, with sodium in the reflector regions and no beryllium in the fuel. The maximum cladding dpa occurred close to the inner region, and reached a level of 273 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due to lower overall fluence away from the center of the core. Energies 2018, 11, x 3 of 17

Figure 1. Radial and axial displacements per atom (dpa) in the unsoftened metal core. The depression Figure 1. Radialin the and center axial was displacementsdue to the diluent region. per atom The peak (dpa) damage in the at 8.4 unsoftened years was 273 metal dpa in core. assembly The depression in the center wasregion due 5 between to the 180 diluent cm and region.220 cm alongThe the peak axial damage direction. atThe 8.4 dpa years was significantly was 273 dpa lower in near assembly region 5 between 180the cm sodium and 220reflector cm regions along (corre the axialsponding direction. to regions The 1, 2, dpa and was17–20 significantlyin Figure 11) due lower to a lower near the sodium overall fluence. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 232 reflector regions (corresponding to regions 1, 2, and 17–20 in Figure 11) due to a lower overall fluence. and 99, respectively. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 232 and 99, respectively.

Figure 2. Radial and axial dpa in the softened metal core. The depression in the center was due to the diluent region. The peak dpa at 8.4 years was 230 in assembly region 7 between 80 cm and 100 cm along the radial direction, and between 300 cm and 320 cm along the axial direction. The dpa was significantly lower near the beryllium reflector regions (corresponding to regions 1, 2, and 17–20 in Figure 11) due to both a softer neutron spectrum, and a lower overall fluence. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 161 and 117, respectively.

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Figure2 shows the dpa levels in the metal-fueled core, with beryllium rods in the reflector regions, and 10-atom-percent beryllium doping in the fuel. The maximum cladding damage occurred close to the inner region, and reached a level of 230 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due not only to lower fluence values, but also to a softer neutron spectrum. Figure3 shows the difference in dpa between the metal-fueled core, with beryllium reflectors and beryllium in the fuel, and the metal-fueled core with sodium in the reflector region and no beryllium Figure 1. Radial and axial displacements per atom (dpa) in the unsoftened metal core. The depression in the fuel. Thein the greatest center was reduction due to the dilu in dpaent region. was The near peak the damage axial at and8.4 years radial was 273 centerline dpa in assembly where ∆dpa was −75.8. The dparegion increased 5 between by 180a cm maximum and 220 cm along of 19.9 the axial dpa direction. in the The outer dpa radialwas significantly regions lower due near to the reflectors causing an increasethe sodium in fissionreflector regions rate; however,(corresponding in to the regions outer 1, 2, radial and 17–20 regions, in Figure the 11) dpadue to value a lower remained well below 200, as shownoverall fluence. in Figure The peak2. dpa in the fuel regions adjacent to the inner and outer reflectors were 232 and 99, respectively.

Figure 2. Radial and axial dpa in the softened metal core. The depression in the center was due to the Figure 2. Radialdiluent andregion. axial The dpapeak indpa the at 8.4 softened years was metal 230 in core. assembly The region depression 7 between in the80 cm center and 100 was cmdue to the diluent region.along the The radial peak direction, dpa at and 8.4 between years was300 cm 230 and in 320 assembly cm along regionthe axial 7direction. between The 80 dpa cm was and 100 cm along the radialsignificantly direction, lower near and the between beryllium 300reflector cm regi andons 320 (corresponding cm along theto re axialgions 1, direction. 2, and 17–20 The in dpa was significantlyFigure lower 11) due near to theboth berylliuma softer neutron reflector spectrum, regions and a (correspondinglower overall fluence. to The regions peak dpa 1, 2,in andthe 17–20 in fuel regions adjacent to the inner and outer reflectors were 161 and 117, respectively. Figure 11) due to both a softer neutron spectrum, and a lower overall fluence. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 161 and 117, respectively. Energies 2018, 11, x 4 of 17

Figure 3. ChangesFigure 3. inChanges dpa in in due dpa to in spectral due to spectral softening soften ining the in metal the metal core. Thecore. coreThe core centerline centerline experienced experienced the greatest reduction in dpa, with a corresponding axial position of 180 cm to 220 cm the greatest(assembly reduction region in dpa,3 in Figure with 11). a corresponding The rise in dpa at axial the outer position edge ofof 180the core cm was to 220 due cm to neutrons (assembly region 3 in Figurebeing 11). reflected The rise back in dpainto the at core the when outer a edgeberyllium of the reflector core was was present. due to neutrons being reﬂected back into the core when a beryllium reﬂector was present. Figure 4 shows the dpa levels in the mixed-oxide-fueled core, with sodium in the reflector regions and no beryllium doping in the fuel. The maximum cladding damage occurred close to the inner region, and reached a level of 225 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due to lower overall fluence away from the center of the core. The overall damage levels were lower relative to either of the metal-fueled cores due to the presence of oxygen atoms which produced a softer neutron spectrum. Figure 5 shows the dpa levels in the mixed-oxide-fueled core, with beryllium rods in the reflector regions and a beryllium doping of 10 atom percent in the fuel. The maximum cladding dpa occurred in the vicinity of the diluent, and reached a level of 203 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due not only to lower flux values, but also a softer neutron spectrum. Here, the presence of oxygen and beryllium in combination lowered the average neutron energy even further. Figure 6 shows the difference in dpa between the oxide-fueled core with beryllium reflectors and beryllium in the fuel, and the oxide-fueled core with sodium in the reflector region and no beryllium in the fuel. The greatest reduction in dpa was near the axial and radial centerline, where ∆dpa was −61.6. The dpa increased by a maximum of 19.0 dpa in the outer radial regions due to the reflectors causing an increase in fission rate; however, in the outer radial regions, the dpa value remained well below 200, as shown in Figure 5.

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Figure4 shows the dpa levels in the mixed-oxide-fueled core, with sodium in the reflector regions and no beryllium doping in the fuel. The maximum cladding damage occurred close to the inner region, and reached a level of 225 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due to lower overall fluence away from the center of the core. The overall damage levels were lower relative to either of the metal-fueled cores due to the presence of oxygen atoms which produced a softer neutron spectrum. Energies 2018, 11, x 5 of 17

Figure 4. Radial and axial dpa in the unsoftened mixed-oxide core. The depression in the center was Figure 4. Radial and axial dpa in the unsoftened mixed-oxide core. The depression in the center was due to the diluent region. The peak dpa at 8.4 years was 225 in assembly region 6 between 80 cm and due to the100 diluent cm along region. the radial The direction, peak dpa and at between 8.4 years 300 cm was and 225 320 incm assembly along the axial region direction. 6 between The dpa 80 cm and 100 cm alongwas thesignificantly radial direction,lower near the and beryllium between reflector 300 cmregions and (corresponding 320 cm along to regions the axial 1, 2, direction.and 17–20 The dpa was significantlyin Figure lower 11) due near to a lower the beryllium overall fluence. reflector The peak regions dpa in (correspondingthe fuel regions adjacent to regions to the inner 1, 2, and and 17–20 in outer reflectors were 181 and 73, respectively. Figure 11) due to a lower overall fluence. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 181 and 73, respectively.

Figure5 shows the dpa levels in the mixed-oxide-fueled core, with beryllium rods in the reflector regions and a beryllium doping of 10 atom percent in the fuel. The maximum cladding dpa occurred in the vicinity of the diluent, and reached a level of 203 dpa at 8.4 years. The maximum damage levels were lower close to the reflectors due not only to lower flux values, but also a softer neutron spectrum. Here, the presence of oxygen and beryllium in combination lowered the average neutron energy even further. Figure6 shows the difference in dpa between the oxide-fueled core with beryllium reflectors and beryllium in the fuel, and the oxide-fueled core with sodium in the reflector region and no beryllium in the fuel. The greatest reduction in dpa was near the axial and radial centerline, where ∆dpa was −61.6. The dpa increased by a maximum of 19.0 dpa in the outer radial regions due to the reflectors causing an increase in fission rate; however, in the outer radial regions, the dpa value remained well below 200, as shown in Figure5. Figure7 shows the neutron flux in the mixed-oxide-fueled core, with beryllium rods in the reflector regionsFigure and 5. Radial a beryllium and axial dpa doping in the softened of 10 mixed-ox atomide percent core. The in depression the fuel. in the The center profiles was due are shown at to the diluent region. The peak dpa at 8.4 years was 203 in assembly region 7 between 80 cm and 100 t = 0 years, and t = 8.4 years, the time at which peak cladding damage hit 203 dpa. The increase in cm along the radial direction, and between 300 cm and 320 cm along the axial direction. The dpa was flux in the centralsignificantly region lower of thenear reactorthe beryllium was reflector clearly regi evidentons (corresponding at 8.4 years, to regions and 1, was2, and due 17–20 to in an increase in transuranic contentFigure 11) in thatdue to region both a softer of the neutron reactor. spectrum, The and axial a lower and overall radial fluence. plutonium The peak grading, dpa in the along with the diluent and reflectors,fuel regions produced adjacent to the a inner shifting and oute fluxr reflectors distribution were 130 that and 91, produced respectively. a more even burnup profile. This allowed for a high core-average burnup before maximum dpa levels were reached, without the need for reshuffling or reforming the fuel. Plots of neutron flux and dpa levels, including error bars reflecting Monte Carlo statistical uncertainty, can be found in Supplementary Note 1. Energies 2018, 11, x 5 of 17

Figure 4. Radial and axial dpa in the unsoftened mixed-oxide core. The depression in the center was due to the diluent region. The peak dpa at 8.4 years was 225 in assembly region 6 between 80 cm and 100 cm along the radial direction, and between 300 cm and 320 cm along the axial direction. The dpa was significantly lower near the beryllium reflector regions (corresponding to regions 1, 2, and 17–20 Energies 2018in, 11Figure, 1507 11) due to a lower overall fluence. The peak dpa in the fuel regions adjacent to the inner and 5 of 16 outer reflectors were 181 and 73, respectively.

FigureFigure 5. Radial 5. Radial and and axial axial dpa dpa inin the softened softened mixed-ox mixed-oxideide core. core. The depression The depression in the center in the was center due was to the diluent region. The peak dpa at 8.4 years was 203 in assembly region 7 between 80 cm and 100 due to the diluent region. The peak dpa at 8.4 years was 203 in assembly region 7 between 80 cm and cm along the radial direction, and between 300 cm and 320 cm along the axial direction. The dpa was 100 cm along the radial direction, and between 300 cm and 320 cm along the axial direction. The dpa significantly lower near the beryllium reflector regions (corresponding to regions 1, 2, and 17–20 in was significantlyFigure 11) due lower to both near a softer the beryllium neutron spectrum, reflector regionsand a lower (corresponding overall fluence. to regions The peak 1, 2,dpa and in 17–20the in Figurefuel 11) regions due to adjacent both a softerto the inner neutron andspectrum, outer reflectors and awere lower 130 overall and 91, fluence.respectively. The peak dpa in the fuel regions adjacent to the inner and outer reflectors were 130 and 91, respectively. Energies 2018, 11, x 6 of 17

FigureFigure 6. Changes 6. Changes in dpa in dpa in duein due to to spectral spectral softening softening in the the oxide oxide core. core. The The core core centerline centerline experienced experienced the greatest reduction in dpa, with a corresponding axial position of 180 cm to 220 cm (assembly the greatest reduction in dpa, with a corresponding axial position of 180 cm to 220 cm (assembly region region 3 in Figure 11). Again, the rise in dpa at the outer edge of the core was due to neutrons being 3 in Figure 11). Again, the rise in dpa at the outer edge of the core was due to neutrons being reﬂected reflected back into the core when a beryllium reflector was present. back into the core when a beryllium reﬂector was present. Figure 7 shows the neutron flux in the mixed-oxide-fueled core, with beryllium rods in the reflector regions and a beryllium doping of 10 atom percent in the fuel. The profiles are shown at t = 0 years, and t = 8.4 years, the time at which peak cladding damage hit 203 dpa. The increase in flux in the central region of the reactor was clearly evident at 8.4 years, and was due to an increase in transuranic content in that region of the reactor. The axial and radial plutonium grading, along with the diluent and reflectors, produced a shifting flux distribution that produced a more even burnup profile. This allowed for a high core-average burnup before maximum dpa levels were reached, without the need for reshuffling or reforming the fuel. Plots of neutron flux and dpa levels, including error bars reflecting Monte Carlo statistical uncertainty, can be found in Supplementary Note 1. Figure 8 shows the position-dependent burnup profile, as well as the uranium utilization at 8.4 years in the mixed-oxide-fueled core, with beryllium rods in the reflector regions and a beryllium doping of 10 atom percent in the fuel. The maximum fuel burnup at this point was 163.4 MWd per kg of initial heavy metal, which occurred in the vicinity of the diluent. The core-average burnup at this time was 112 MWd per kg of initial heavy metal, with a minimum of 22.3 MWd per kg of initial heavy metal in the upper and lower axial locations, and outermost radial locations. The center region also experienced the greatest uranium burnup—21.6 atom percent of the initial depleted uranium. The core-average burnup was 14.7 atom percent of the initial uranium, fixed by the power, burn duration, and initial uranium mass of the core. This diminished to 2.7 atom percent in the upper and lower axial regions.

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Figure 7. Three dimensionaldimensional (3D) (3D) neutron neutron flux flux profiles profiles in in the the softened softened mixed-oxide mixed-oxide core. core. (a)Neutron (a) Neutron flux fluxat the at beginning the beginning of life of (BOL). life (BOL). Peak Peak flux wasflux 2.76was× 2.7610 15× 10n/cm15 n/cm2-s in2-s assemblyin assembly region region 7 between 7 between 80 cm80 cmand and 100 100 cm cm along along the the radial radial direction, direction, and and between between 300 300 cm cm and and 320320 cmcm alongalong the axial direction. direction. (b) Neutron flux flux at at the the end end of of life life (EOL (EOL),), 8.4 8.4 years. years. Peak Peak flux flux was was 2.792.79 × 1015× n/cm1015 2n/cm-s in assembly2-s in assembly region 7region between 7 between 180 cm 180and cm 220 and cm 220along cm the along axial the directio axial direction.n. The flux The gradually flux gradually changed changed shape shapedue to duethe increasingto the increasing concentration concentration of Pu in of the Pu ininner the region inner region of the ofcore. the core.

Figure8 shows the position-dependent burnup profile, as well as the uranium utilization at 8.4 years in the mixed-oxide-fueled core, with beryllium rods in the reflector regions and a beryllium doping of 10 atom percent in the fuel. The maximum fuel burnup at this point was 163.4 MWd per kg of initial heavy metal, which occurred in the vicinity of the diluent. The core-average burnup at this time was 112 MWd per kg of initial heavy metal, with a minimum of 22.3 MWd per kg of initial heavy metal in the upper and lower axial locations, and outermost radial locations. The center region also experienced the greatest uranium burnup—21.6 atom percent of the initial depleted uranium. The core-average burnup was 14.7 atom percent of the initial uranium, fixed by the power,

Energies 2018, 11, 1507 7 of 16 burn duration, and initial uranium mass of the core. This diminished to 2.7 atom percent in the upper andEnergies lower 2018, axial 11, x regions. 8 of 17

Figure 8.8. Position-dependentPosition-dependent fuel fuel burnup burnup at 8.4at years8.4 years in the in softened the softened mixed-oxide mixed-oxide core. (a core.) Overall (a) Overall burnup. Peakburnup. burnup Peak wasburnup 163.4 was MWd/kg 163.4 MWd/kg of initial of heavy initial metalheavy (IHM) metal in(IHM) assembly in assembly region region 7 between 7 between 80 cm and80 cm 100 and cm, 100 and cm, between and between 300 cm 300 and cm 320 and cm 320 along cm al theong axial the axial direction. direction. Core-averaged Core-averaged burnup burnup was 112was MWd/kgIHM. 112 MWd/kgIHM. (b) Uranium (b) Uranium burnup. burnup.The peak The uranium peak uranium burnup fractionburnup wasfraction 21.6 a/owas in21.6 assembly a/o in regionassembly 8 between region 8 100 between cm and 100 120 cm cm and along 120 the cm radial alon direction,g the radial and direction, between and 280 cmbetween and 300 280 cm cm along and the300 axialcm along direction. the axial The direction. core-averaged The core-averaged uranium burnup uranium fraction burnup was 14.7fraction a/o. was The 14.7 diluent a/o. The is shown diluent in theis shown empty in region the empty in both region ﬁgures in both where figures no burnup where was no burnup experienced. was experienced.

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Energies 2018, 11, x 9 of 17 Figure9 shows the neutron multiplication factor as a function of time in the mixed-oxide-fueled core, withFigure beryllium 9 shows rods the in neutron the reflector multiplication regions factor and as a a beryllium function of doping time in the of 10mixed-oxide-fueled atom percent in the fuel.core, Axial with and beryllium radial grading rods in the of thereflector fresh regions core, alongand a withberyllium the usedoping of reflectors,of 10 atom percent had a significant in the fuel. Axial and radial grading of the fresh core, along with the use of reflectors, had a significant effect effect on the reactivity of the reactor. The reactor experienced a reactivity swing because of breeding on the reactivity of the reactor. The reactor experienced a reactivity swing because of breeding and and burning, but it stayed above critical levels throughout the 8.4 years. The core started its life with burning, but it stayed above critical levels throughout the 8.4 years. The core started its life with keff k = 1.015, peaked at 1.028, and was 1.001 at 8.4 years. The beginning of life k was smaller than that eff = 1.015, peaked at 1.028, and was 1.001 at 8.4 years. The beginning of life keff waseff smaller than that for for somesome conventional conventional fast-reactorfast-reactor de designssigns which which run run oxide oxide fuel, fuel, as aswas was the the reactivity reactivity swing swing from from the the beginningbeginning to the to endthe end of life of life [11 [11].].

1.03

1.025

1.02

eff 1.015 k

1.01

1.005

1 02468 Time [years]

Figure 9. keff as a function of time in the softened mixed-oxide core. The neutron multiplication factor Figure 9. keff as a function of time in the softened mixed-oxide core. The neutron multiplication factor increased for the first 2.8 years as plutonium was bred in. The peak keff was 1.028 at t = 2.8 years. At t increased for the first 2.8 years as plutonium was bred in. The peak k was 1.028 at t = 2.8 years. = 8.4 years, corresponding to 203 dpa, keff = 1.001. The keff shown is theeff average of five burnup At t =simulations 8.4 years, correspondingof the softened mixed-oxide to 203 dpa, core keff =usin 1.001.g different The k effrandomshown number is the seeds. average The of error five bars burnup simulationsare shown of thefor one softened standard mixed-oxide deviation, corresponding core using different to the 68% random confidence number level. More seeds. details The on error the bars are showncalculation for one of uncertainties standard deviation, are given corresponding in Supplementary to theNote 68% 8. confidence level. More details on the calculation of uncertainties are given in Supplementary Note 8. Figure 10 shows the axial temperature profiles of the fuel, the cladding, and the coolant in the hottest radial position at the beginning of life in the mixed-oxide-fueled core, with beryllium rods in Figure 10 shows the axial temperature profiles of the fuel, the cladding, and the coolant in the the reflector regions and a beryllium doping of 10 atom percent in the fuel. The temperatures in each hottestregion radial remained position well at thebelow beginning material oflimits. life inThese the temperature mixed-oxide-fueled profiles were core, perturbed with beryllium to estimate rods in the reflectorthe reactivity regions coefficients and a beryllium at the beginning, doping ofmiddle 10 atom, and percent end of inlife the of the fuel. core, The corresponding temperatures to in 0 each regionyears, remained 4.4 years, well and below 8.4 years material of operation. limits. These temperature profiles were perturbed to estimate the reactivity coefficients at the beginning, middle, and end of life of the core, corresponding to 0 years, 4.4 years, and 8.4 years of operation. The reactivity coefficients are listed in Table1. All reactivity coefficients quoted were isothermal, corresponding to spatially uniform changes in fuel or coolant temperatures. The large positive void and the coolant reactivity coefficients were in part mitigated by a large negative Doppler reactivity feedback. The restructuring of oxide fuel with burnup introduced large uncertainties in the estimates of the behavior of the fuel as a function of temperature [12]. Because of this, thermal expansion of the fuel was an unreliable mechanism for additional negative reactivity feedback, and was not included in this analysis.

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1500 Fuel Cladding Inner 1400 Coolant

1300

1200

1100

1000 Temperature [K] Temperature 900

800

700

01234 Axial position [m] Figure 10. Temperature profiles of the fuel, cladding, and coolant. Blue, green, and red lines represent Figure 10. Temperature profiles of the fuel, cladding, and coolant. Blue, green, and red lines represent the temperatures of the fuel, inner cladding, and coolant, respectively. Data represent the axial the temperaturestemperature in ofassembly the fuel, region inner 8 of cladding, the reactor andwhich coolant, reached respectively.the highest fuel Datatemperature—1423 represent the K. axial temperatureThe profiles in assembly were computed region at 8 the of thebeginning reactor of which life. The reached central dip the feature highest was fuel due temperature—1423 to the diluent at K. The profilesthe z = 2were m axial computed position. at the beginning of life. The central dip feature was due to the diluent at the z = 2 m axial position. The reactivity coefficients are listed in Table 1. All reactivity coefficients quoted were isothermal, Tablecorresponding 1. Reactivity to spatially coefficients. uniform The changes coefficients in fuel were or computedcoolant temperatures. by perturbing The the large fuel positive temperature void only,and the the coolant coolant reactivity temperature, coefficients and the coolantwere in densitypart mitigated only. by a large negative Doppler reactivity feedback. The restructuring of oxide fuel with burnup introduced large uncertainties in the estimates of the behavior of the fuel as a function of temperature [12]. BecauseMixed of Oxide,this, thermal expansionMixed Oxide, of the Core Metal, Unsoftened Metal, Softened fuel was an unreliable mechanism for additional negative reactivityUnsoftened feedback, and wasSoftened not included in Beginningthis analysis. of life −1 −6 −6 −6 −6 αDoppler (K ) −2.7 ± 0.2 × 10 −5.0 ± 0.2 × 10 −8.2 ± 0.2 × 10 −9.5 ± 0.2 × 10 −1 −6 −5 −6 −6 αTablecoolant 1.(K Reactivity) 9.7coefficients.± 0.2 × 10 The coefficients1.1 ± 0.02 were× 10computed5.3 by± perturbing0.2 × 10 the fuel7.1 temperature± 0.2 × 10 −4 −4 −4 −4 αvoidonly,(per the % void)coolant temperature,3.3 ± 0.006 × and10 the coolant3.5 ± 0.006density× only.10 1.8 ± 0.007 × 10 2.1 ± 0.007 × 10

Middle of life Metal, Mixed Oxide, Mixed Oxide, α Core(K −1) −6 Metal, Softened− 6 −6 −6 Doppler −2.0Unsoftened± 0.2 × 10 −4.0 ± 0.2 × 10 −6.0Unsoftened± 0.2 × 10 −6.9Softened± 0.2 × 10 −1 −5 −5 −6 −6 αcoolant (K ) 1.2 ± 0.06 × 10 1.3 ± 0.02 × 10 7.7 ± 0.2 × 10 7.7 ± 0.2 × 10 Beginning of life −4 −4 −4 −4 αvoid (per % void) 4.1 ± 0.006 × 10 3.8 ± 0.006 × 10 2.7 ± 0.007 × 10 2.6 ± 0.007 × 10 αDoppler (K−1) −2.7 ± 0.2 × 10−6 −5.0 ± 0.2 × 10−6 −8.2 ± 0.2 × 10−6 −9.5 ± 0.2 × 10−6 αEndcoolant of (K life−1) 9.7 ± 0.2 × 10−6 1.1 ± 0.02 × 10−5 5.3 ± 0.2 × 10−6 7.1 ± 0.2 × 10−6 α (K−1) −6 −6 −6 −6 αvoidDoppler (per % void) 3.3−2.3 ± 0.006± 0.2 ×× 1010−4 3.5−2.7 ± 0.006± 0.2 ×× 1010−4 −1.84.7 ±± 0.0070.2 × × 10−4 −2.15.5 ±± 0.0070.2 × × 1010−4 −1 ± × −5 ± × −5 ± × −6 ± × −6 Middleαcoolant of(K life) 1.3 0.02 10 1.3 0.02 10 8.5 0.2 10 8.9 0.2 10 α (per % void) 4.4 ± 0.007 × 10−4 4.0 ± 0.007 × 10−4 3.0 ± 0.007 × 10−4 2.9 ± 0.007 × 10−4 voidαDoppler (K−1) −2.0 ± 0.2 × 10−6 −4.0 ± 0.2 × 10−6 −6.0 ± 0.2 × 10−6 −6.9 ± 0.2 × 10−6 αcoolant (K−1) 1.2 ± 0.06 × 10−5 1.3 ± 0.02 × 10−5 7.7 ± 0.2 × 10−6 7.7 ± 0.2 × 10−6 3. Discussionαvoid (per % void) 4.1 ± 0.006 × 10−4 3.8 ± 0.006 × 10−4 2.7 ± 0.007 × 10−4 2.6 ± 0.007 × 10−4 End of life SimulationsαDoppler (K−1 of) long-life−2.3 ± fast-reactor 0.2 × 10−6 cores−2.7 ± were 0.2 × 10 done−6 using−4.7 Serpent ± 0.2 × 10− 2.1.28,6 and−5.5 ± confirmed 0.2 × 10−6 with MCNPXα 2.7.0.coolant (K The−1) metal-fueled1.3 ± 0.02 × 10 core−5 without1.3 ± 0.02 beryllium × 10−5 experienced8.5 ± 0.2 × 10−6 the greatest8.9 ± 0.2 peak × 10−6 damage −4 −4 −4 −4 to claddingαvoid (per (273 % void) dpa), 4.4 which ± 0.007 was × 10 brought 4.0 ± down0.007 × 10 to 2303.0 dpa ± 0.007 when × 10 the fuel2.9 ± was 0.007 doped × 10 with beryllium, and when beryllium reflectors were added. The mixed-oxide-fueled core without beryllium experienced a peak cladding damage level of 225 dpa, which was brought down to 203 dpa when the fuel was doped with beryllium, and when beryllium reflectors were added. This showed that, by configuring a fast-reactor core with light elements, the dpa could be reduced by 25%. Energies 2018, 11, 1507 10 of 16

The reactor that experienced the least cladding damage was configured with an oxide fuel doped with 10 atom percent beryllium, and with plutonium content, graded axially and radially. When coupled with beryllium reflectors and a sodium diluent, this enabled the simulated reactor to operate for 8.4 years at a thermal output of 4740 MW, with a peak cladding damage of 203 dpa. The reactor developed an average uranium burnup of 14.7 atom percent of the initial uranium after 8.4 years. The average overall burnup was 112 MWd per kg of initial heavy metal. This was achieved without reshuffling or recladding the fuel, and without any additional input of fresh fuel. Crucially, this performance was for an unoptimized core, and was achievable with materials for which there was operational history. By using an optimized fuel shuffling strategy, the overall and uranium burnup values would be increased significantly. The results assumed that all fission products remained in the fuel. In reality, a significant fraction of the fission product gas would either escape into a plenum, or leave the core altogether in vented-fuel-pin designs. The partial removal of fission products would produce a small increase in criticality. While venting fission products to the sodium coolant with subsequent removal was proposed for high-burnup fast reactors [13], the implications for safety and licensing should be addressed for any design using this operating strategy. The simulations which are shown here indicated that Doppler broadening in the fuel could compensate for the positive coolant temperature reactivity effect at the beginning of life in spectrally softened cores. However, for the simulations shown here, this was no longer the case after the middle of life. Additionally, the results in Table1 show that the core suffered from a large void reactivity. One approach to mitigating these issues would be to use a gas expansion module system, which injects neutron poison into the core when the coolant heats up. These types of control devices were successfully demonstrated at the Fast Flux Test Facility [12]. A similar approach using a lithium injection device was proposed for reactivity control in breed-burn fast reactors [14], and transient analyses showed that this type of device could enable fast reactors with large positive coolant temperature reactivity coefficients to operate safely [15]. Additionally, mechanisms such as grid-plate expansion and axial fuel expansion could help reduce the large coolant temperature reactivity coefficient in both metal- and oxide-fueled cores. However, these mechanisms were not included in this work as their effect on the reactivity coefficients was found to be small relative to Doppler broadening, and coolant thermal expansion or voiding, for metal-fueled sodium-cooled fast reactors with low leakage [14]. Axial fuel-rod expansion was also understood to be an unreliable mechanism for reactivity control in oxide-fueled cores, due to the lack of structural integrity at high levels of burnup [12]. However, these effects, as well as other secondary effects such as control-rod driveline expansion and assembly bowing, should be investigated, as they could produce a strong negative reactivity feedback when combined. Importantly, the insertion of moderating materials into a fast-reactor core introduced side effects, such as disturbing the power profile of the core, as well as significantly reducing the neutron multiplication factor. These effects were consistent with past studies on fast reactors operating with additional moderating materials (e.g., References [16] and [17]). However, the usual limiting factor in achieving high burnup in sodium-cooled fast reactors is not reactivity, but rather, damage to cladding [12,18]. Figure9 shows that k eff remained above 1 when the cladding damage reached 200 dpa. More details on these specific effects are provided in Supplementary Note 2. Additionally, while the densities of fuel in this work were adjusted to accommodate mixing in beryllium, the effect of this on other thermo-physical properties of the fuel, as well as fuel performance under irradiation and burnup, was not addressed. Although experimental data on the thermo-physical properties of fresh UO2-BeO pellets exist [8], the optimal moderator composition and geometry need to be explored. Moderator materials, such as boron carbide (enriched to 99% 11B to limit neutron capture) and zirconium hydride, are options with significant operational experience, and are known to improve the safety characteristics of sodium-cooled fast reactors [16,17]. Energies 2018, 11, 1507 11 of 16

4. Materials and Methods Four core designs were simulated: (i) a metal-fueled (U-Pu-Zr) core with sodium reflectors (an unsoftened metal core); (ii) a metal-fueled, beryllium-doped (U-Pu-Zr-Be) core with beryllium reflectors (a softened metal core); (iii) a mixed-oxide-fueled core (an unsoftened mixed oxide core); and (iv) a mixed-oxide-fueled, beryllium-doped core with beryllium reflectors (a softened mixed oxide core). Each core had an identical geometry and mass of actinides in order to compare the effect of spectral softening on an even basis. The main design parameters of the simulated cores are given in Table2.

Table 2. Design parameters for all four cores. The fuel densities shown were the actual densities used in the simulation. The fuel density of the softened mixed oxide core was computed by adding mixed-oxide fuel at a density of 11 g/cm3 with beryllium at 1.85 g/cm3 in a density-weighted linear fashion, given 9.675 a/o Be and assuming a 95% smear density. The fuel densities of the softened metal core and the unsoftened cores were computed using the softened mixed-oxide core fuel as a basis, and by requiring that the overall mass of actinides in each core was the same.

Metal, Metal, Mixed Oxide, Mixed Oxide, Core Unsoftened Softened Unsoftened Softened Power (MWth) 4740 Number of assemblies 1039 Fuel form U-Pu-Zr U-Pu-Zr-Be Mixed oxide (MOX) MOX-Be Beryllium a/o, relative to actinides 0% 9.675% 0% 9.675% Zirconium w/o, overall 10% 10% 0% 0% Fuel density (g/cm3) 9.95 9.98 10.16 10.26 Fuel/coolant/void/structural volume ratios 42.18/25.24/7.44/25.13% Inlet coolant temperature (K) 659 Core flat-to-flat diameter (cm) 401.66 Assembly flat-to-flat diameter (cm) 11.5917 Duct thickness (cm) 0.1 Pins per assembly 127 Cladding thickness (cm) 0.038 Fuel pellet diameter (cm) 0.754 Helium gap thickness (cm) 0.01 Helium void diameter (cm) 0.2435 Assembly pitch (cm) 11.7917 Pin pitch (cm) 1.0175 Pin diameter (cm) 0.85 238Pu 0.8 atom % 239Pu 63.18 atom % Relative abundance of plutonium nuclides 240Pu 32.71 atom % 241Pu 1.24 atom % 242Pu 2.07 atom %

The simulated reactors were 400 cm long, with a flat-to-flat diameter of 401.67 cm. The thermal power (4740 MW) was chosen to keep the maximum inner cladding temperature of the softened mixed-oxide core below 823 K [19]. In the simulations, the cores were divided into ten axial regions of 20 cm in height, and 20 radial regions ranging from 36 to 66 assemblies each. The cores were radially and axially heterogeneous. The plutonium content of the fuel was radially and axially graded to flatten the power and flux profiles during the lifetime of the cores. The innermost 109 assemblies and the outermost 204 assemblies made up the reflectors, which were pins filled with either beryllium (in the case of the softened cores) or sodium (in the case of the unsoftened cores). A sodium diluent was incorporated into the regions situated ±100 cm from the axial center of the cores, in order to further flatten the flux and reduce the dpa in regions that would have otherwise experienced the most damage. Figure 11 shows a schematic of the core to scale, with individual material regions numbered. The specific positions and compositions of the fuel, reflector, and diluent regions, corresponding to the numbered schematic in Figure 11, are listed in Table3. Top-down diagrams of the core composition at every axial location are given in Supplementary Note 3, Figures S10–S19. In all four core designs, Energies 2018, 11, x 13 of 17

further flatten the flux and reduce the dpa in regions that would have otherwise experienced the most damage. Figure 11 shows a schematic of the core to scale, with individual material regions numbered. EnergiesThe specific2018, 11 ,positions 1507 and compositions of the fuel, reflector, and diluent regions, corresponding12 of to 16 the numbered schematic in Figure 11, are listed in Table 3. Top-down diagrams of the core composition at every axial location are given in Supplementary Note 3, Figures S10–S19. In all four thecore hexagonal designs, the assemblies hexagonal contained assemblies 127 contained annular fuel,127 annular reﬂector, fuel, and reflector, diluent and rods diluent clad in rods HT-9 clad steel, in withHT-9 sodium steel, coolantwith sodium ﬂowing coolant between flowing the pins. between More the details pins. on More the pin-level details coreon the geometry pin-level and core the materialsgeometry can and be the found materials in Supplementary can be found Notein Supplementary 4. Note 4.

20 20 20 20 20 19 19 18 19 19 20 20 19 18 17 17 18 19 20 20 18 17 16 16 16 17 18 20 19 18 17 16 15 15 16 17 18 19 19 18 16 15 14 14 14 15 16 18 19 20 18 16 15 14 13 13 14 15 16 18 20 20 18 16 14 13 13 12 13 13 14 16 18 20 20 18 16 14 13 12 11 11 12 13 14 16 18 20 18 16 14 12 11 11 11 11 11 12 14 16 18 19 16 14 12 11 10 10 10 10 11 12 14 16 19 19 17 14 12 11 10 9 9 9 10 11 12 14 17 19 20 17 15 12 11 10 9 8 8 9 10 11 12 15 17 20 18 15 13 11 9 8 8 8 8 8 9 11 13 15 18 19 16 13 11 9 8 7 7 7 7 8 9 11 13 16 19 20 16 14 11 10 8 7 6 6 6 7 8 10 11 14 16 20 17 14 12 10 8 7 6 6 6 6 7 8 10 12 14 17 18 15 13 10 8 7 6 5 5 5 6 7 8 10 13 15 18 20 16 13 11 9 7 6 5 4 4 5 6 7 9 11 13 16 20 17 14 11 9 7 6 5 4 4 4 5 6 7 9 11 14 17 19 15 12 10 8 6 5 4 4 4 4 5 6 8 10 12 15 19 20 16 13 11 8 6 5 4 3 3 3 4 5 6 8 11 13 16 20 18 14 11 9 7 5 4 3 3 3 3 4 5 7 9 11 14 18 19 16 13 10 8 6 4 3 2 2 2 3 4 6 8 10 13 16 19 17 14 11 8 6 4 3 2 2 2 2 3 4 6 8 11 14 17 19 15 12 9 7 5 4 2 2 2 2 2 4 5 7 9 12 15 19 20 17 13 10 8 6 4 3 2 1 1 2 3 4 6 8 10 13 17 20 18 15 11 9 6 4 3 2 1 1 1 2 3 4 6 9 11 15 18 20 16 13 10 7 5 4 2 1 1 1 1 2 4 5 7 10 13 16 20 18 14 11 8 6 4 3 2 1 1 1 2 3 4 6 8 11 14 18 20 16 12 10 7 5 3 2 1 1 1 1 2 3 5 7 10 12 16 20 18 14 11 8 6 4 2 1 1 1 1 1 2 4 6 8 11 14 18 19 16 12 9 7 5 3 2 1 1 1 1 2 3 5 7 9 12 16 19 18 14 11 8 6 4 2 1 1 1 1 1 2 4 6 8 11 14 18 19 16 12 9 7 5 3 2 1 1 1 1 2 3 5 7 9 12 16 19 18 14 11 8 6 4 2 1 1 1 1 1 2 4 6 8 11 14 18 20 16 12 10 7 5 3 2 1 1 1 1 2 3 5 7 10 12 16 20 18 14 11 8 6 4 3 2 1 1 1 2 3 4 6 8 11 14 18 20 16 13 10 7 5 4 2 1 1 1 1 2 4 5 7 10 13 16 20 18 15 11 9 6 4 3 2 1 1 1 2 3 4 6 9 11 15 18 20 17 13 10 8 6 4 3 2 1 1 2 3 4 6 8 10 13 17 20 19 15 12 9 7 5 4 2 2 2 2 2 4 5 7 9 12 15 19 17 14 11 8 6 4 3 2 2 2 2 3 4 6 8 11 14 17 19 16 13 10 8 6 4 3 2 2 2 3 4 6 8 10 13 16 19 18 14 11 9 7 5 4 3 3 3 3 4 5 7 9 11 14 18 20 16 13 11 8 6 5 4 3 3 3 4 5 6 8 11 13 16 20 19 15 12 10 8 6 5 4 4 4 4 5 6 8 10 12 15 19 17 14 11 9 7 6 5 4 4 4 5 6 7 9 11 14 17 20 16 13 11 9 7 6 5 4 4 5 6 7 9 11 13 16 20 18 15 13 10 8 7 6 5 5 5 6 7 8 10 13 15 18 17 14 12 10 8 7 6 6 6 6 7 8 10 12 14 17 20 16 14 11 10 8 7 6 6 6 7 8 10 11 14 16 20 19 16 13 11 9 8 7 7 7 7 8 9 11 13 16 19 18 15 13 11 9 8 8 8 8 8 9 11 13 15 18 20 17 15 12 11 10 9 8 8 9 10 11 12 15 17 20 19 17 14 12 11 10 9 9 9 10 11 12 14 17 19 19 16 14 12 11 10 10 10 10 11 12 14 16 19 18 16 14 12 11 11 11 11 11 12 14 16 18 20 18 16 14 13 12 11 11 12 13 14 16 18 20 20 18 16 14 13 13 12 13 13 14 16 18 20 20 18 16 15 14 13 13 14 15 16 18 20 19 18 16 15 14 14 14 15 16 18 19 19 18 17 16 15 15 16 17 18 19 20 18 17 16 16 16 17 18 20 20 19 18 17 17 18 19 20 20 19 19 18 19 19 20 20 20 20 20

401.67 cm

FigureFigure 11. 11.Reactor Reactorcore core structure.structure. NumberedNumbered cells belong to a single material material region. region. Material Material content content variedvaried with with axial axial andand radialradial position.position. Material regions 1, 2, 17, 18, 19, and 20 contained beryllium beryllium assembliesassemblies (softened(softened cores),cores), oror sodium-ﬁlledsodium-filled assemblies (unsoftened (unsoftened cores) cores) in in all all axial axial locations. locations. Material regions 3 to 16 contained fuel, with each numbered cell belonging to a single burnup region, Material regions 3 to 16 contained fuel, with each numbered cell belonging to a single burnup region, or sodium diluent, which varied axially. Specific contents of each radial and axial material region are or sodium diluent, which varied axially. Speciﬁc contents of each radial and axial material region are listed in Table 3. listed in Table3.

Serpent 2.1.28 [20] was used for the simulations, and the results were corroborated using MCNPXSerpent 2.70 2.1.28 [21]. [Additional20] was used details for the can simulations, be found in and Supplementary the results were Note corroborated 5. The spatial using mesh MCNPX used 2.70in the [21 simulations]. Additional had details an axial can grid be foundof 20 segments in Supplementary with Δz = 20 Note cm. 5.The The materials spatial were mesh burned used in at the a simulationsthermal power had anof 4740 axial megawatts grid of 20 segments with a time with step∆z =of 20114 cm. days, The until materials a peak were irradiation burned atdamage a thermal of power of 4740 megawatts with a time step of 114 days, until a peak irradiation damage of 203 dpa was reached. The simulations tracked a total of 19.2 million neutron histories per burnup step. A thermal resistor model was used to determine the approximate fuel and coolant temperatures in all radial and axial regions, and these were fed back into the Monte Carlo simulations at the beginning, middle and Energies 2018, 11, 1507 13 of 16 end of life, in an effort to compute the isothermal reactivity coefﬁcients. More information on the heat transfer model can be found in Supplementary Note 6.

Table 3. Radial and axial plutonium concentration. Column headings refer to axial positions. The axial concentration from 0 cm to −200 cm is the reﬂection of the concentration from 0 cm to 200 cm, and the core was reﬂected about z = 0 cm. Values are plutonium concentrations as a proportion of total heavy metal in each region, in atomic percent.

0–20 20–40 40–60 60–80 80–100 100–120 120–140 140–160 160–180 180–200 Axial Position cm cm cm cm cm cm cm cm cm cm Material Region 3 12.00 12.44 12.89 13.33 13.78 14.22 14.67 15.11 15.56 16.00 4 12.31 12.72 13.13 13.54 13.95 14.36 14.77 15.18 15.59 16.00 5 12.62 12.99 13.37 13.74 14.12 14.50 14.87 15.25 15.62 16.00 6 12.92 13.26 13.61 13.95 14.29 14.63 14.97 15.32 15.66 16.00 7 13.23 Na Na Na Na 14.77 15.08 15.38 15.69 16.00 8 Na 13.81 14.09 14.36 14.63 14.91 15.18 15.45 15.73 16.00 9 13.85 14.09 Na Na Na 15.04 15.28 15.52 15.76 16.00 10 14.15 14.36 14.56 14.77 14.97 15.18 15.38 15.59 15.79 16.00 11 14.46 14.63 14.80 14.97 15.15 15.32 15.49 15.66 15.83 16.00 12 14.77 14.91 15.04 15.18 15.32 15.45 15.59 15.73 15.86 16.00 13 15.08 15.18 15.28 15.38 15.49 15.59 15.69 15.79 15.90 16.00 14 15.38 15.45 15.52 15.59 15.66 15.73 15.79 15.86 15.93 16.00 15 15.69 15.73 15.76 15.79 15.83 15.86 15.90 15.93 15.97 16.00 16 16.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00

4.1. Irradiation Damage The displacements per atom of cladding and structural materials were computed using ZZ dpa = σD(Ei)φ(Ei, t)dE dt. (1)

Here, σD is the energy-dependent microscopic displacement cross section (barns), ϕ is the neutron ﬂux (1/b-s), and Ei is the incident neutron energy (eV). The integrals are over both energy and time. The displacement cross section took into consideration the different types of neutron–material interactions that were sufﬁcient to dislodge an atom. For steels, the elastic scattering on 56Fe was the dominant mechanism through which atoms were displaced, but neutron absorption, and inelastic scattering, (n, p), (n, α), (n, n), (n, 2n), etc., all contributed. The total microscopic displacement cross section was then the sum of the displacement cross sections for the individual mechanisms:

σD = σD,s(elastic) + σD,s(inelastic) + σD(n,γ) + σD(n,n) + σD(n,2n) + . . . . (2)

The output of the Serpent simulations was used to compute a ﬂux in 494 energy bins ranging −4 from 10 to 20 MeV for each location and time step. We used the NJOY [22] code to compute σD in the same 494-group structure as the ﬂux. More details can be found in Supplementary Note 7.

4.2. Fuel and Cladding Temperature Temperatures for the fuel, coolant, and cladding were computed as a function of axial and radial position in the reactor using a one dimensional (1D) thermal resistor model [12,23,24], and thermo-physical quantities were drawn from References [25–27]:

∆T q0 = , (3) R0 where q0 is the linear power density of the fuel (W/m), and R0 and ∆T are the linear thermal resistance (m-K/W) and the temperature difference (K), respectively, between the fuel and the coolant. Energies 2018, 11, 1507 14 of 16

In order to reduce peak temperatures, the fuel was comprised of annular pins, which were clad in HT-9 steel with a helium bond. While sodium bonds are normally used in combination with metal fuel, we chose a helium bond for all fuel types in order to keep as many design parameters constant between cores. The 1D thermal transport model assumed no radial heat transfer between coolant channels. Additional details can be found in Supplementary Note 6.

4.3. Reactivity Coefficients The core-average Doppler coefficient for the fuel was computed by uniformly perturbing the temperature of the fuel in the reactor by +300 K, and by recomputing the neutron multiplication factor. All other variables remained the same in order to isolate the effect of Doppler broadening. The coefficient was computed according to Reference [28]:

1 ∆k = aDoppler 2 . (4) k ∆Tf uel

The core-average reactivity coefﬁcient for the sodium coolant was computed by uniformly perturbing the coolant temperature by +300 K, by adjusting the sodium densities accordingly, and by recomputing the neutron multiplication factor:

1 ∆k = acoolant 2 . (5) k ∆Tcoolant

The core-average void coefﬁcients were computed using Reference [28] by uniformly reducing the coolant density by 90%, and by recomputing the neutron multiplication factor. All other variables remained the same to isolate the effect of coolant voiding:

1 ∆k = avoid 2 , (6) k ∆xcoolant where x is the core-average void fraction. Reactivity coefficients were computed at the beginning, middle (t = 4.4 years), and end of life. In all cases, this was done by sampling the materials from the burnup simulation at 0 years, 4.4 years, and 8.4 years, by performing the relevant temperature or density perturbation, and by recomputing the neutron multiplication factor using 432 million total neutron histories. The uncertainties of keff values of Figure9 were computed by repeating the Monte Carlo simulation five times, each with a different random number seed, and by calculating the mean and standard deviation of the results. The uncertainties in the reactivity coefficients were computed using the error propagation formula on the standard deviations for k, quoted by the Monte Carlo code. The errors were, in many cases, extremely small, as they reflected the statistical uncertainties in the Monte Carlo neutron transport. Additional details can be found in Supplementary Note 8.

5. Conclusions While the impact of spectral softening on both metal- and oxide-fueled long-life fast-reactor cores is significant, this paper only provided a demonstration of unoptimized core designs. Although the mixed-oxide-fueled, beryllium-doped core with beryllium reflectors presented in this paper achieved good performance, the unexplored parameter space for long-life fast-reactor design using a softened neutron spectrum is large. Additionally, the fuel densities in the metal-fueled cores, as well as in the mixed-oxide-fueled core without beryllium, were reduced in order to enable an even comparison with the mixed-oxide core containing beryllium. In reality, the fuel densities of the three higher-damage designs would be greater, which would give a bonus to neutron economy. This suggested that the optimal design of maximizing burnup while keeping dpa low might be achieved using a metal-fueled core with additional beryllium. Finally, a key goal would be to find the fuel-loading strategy Energies 2018, 11, 1507 15 of 16 which enables long-life cores to be operated using recycled actinides from their own discharged fuel. This would significantly reduce the net production of transuranics, which are the nuclides responsible for giving nuclear waste its long-lived signature.

Supplementary Materials: The following are available online at http://www.mdpi.com/1996-1073/11/6/1507/ s1. Supplementary-Document.pdf. Author Contributions: Data curation, Andrew Osborne; Formal analysis, Mark Deinert; Investigation, Andrew Osborne; Methodology, Mark Deinert; Project administration, Mark Deinert; Resources, Mark Deinert; Software, Andrew Osborne; Writing-original draft, Andrew Osborne and Mark Deinert; Writing-review & editing, Andrew Osborne and Mark Deinert. Funding: This research was funded by The US Nuclear Regulatory Commission grant number NRC-38-08-946. Acknowledgments: We thank Jaakko Leppanen and Tuomas Viitanen of the VTT Technical Research Centre of Finland for their helpful advice on the Serpent Monte Carlo code. We also thank Nathaniel Mendoza and Chris Hempel of the Texas Advanced Computing Center for their assistance with parallelization of Monte Carlo codes. Conﬂicts of Interest: The authors declare no conﬂict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

References

1. Kim, T.K.; Taiwo, T.A. Feasibility study of ultra-long life fast reactor core concept. In PHYSOR 2010; American Nuclear Society, Inc.: Pittsburgh, PA, USA, 2010. 2. Osborne, A.G.; Deinert, M.R. Comparison of neutron diffusion and monte carlo models for a ﬁssion wave. In Proceedings of the International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering, Sun Valley, ID, USA, 5–9 May 2013; American Nuclear Society: Sun Valley, ID, USA, 2013. 3. Cheon, J.S.; Lee, C.B.; Lee, B.O.; Raison, J.P.; Mizuno, T.; Delange, F.; Carmack, J. Sodium fast reactor evaluation: Core materials. J. Nucl. Mater. 2009, 392, 324–330. [CrossRef] 4. Byun, T.S.; Lewis, W.D.; Toloczko, M.; Maloy, S.A. Impact properties of irradiated HT9 from the fuel duct of FFTF. J. Nucl. Mater. 2012, 421, 104–111. [CrossRef] 5. Toloczko, M.B.; Garner, F.A.; Eiholzer, C.R. Irradiation creep and swelling of the US fusion heats of HT9 and 9Cr-1Mo to 208 dpa at ~400 ◦C. J. Nucl. Mater. 1994, 212, 604–607. [CrossRef] 6. Maloy, S.A.; Toloczko, M.; Cole, J.; Byun, T.S. Core materials development for the fuel cycle R&D program. J. Nucl. Mater. 2011, 415, 302–305. 7. Osborne, A.G.; Deinert, M.R. Neutron damage reduction in a traveling wave reactor. In Proceedings of the PHYSOR 2012: Advances in Reactor Physics, Knoxville, TN, USA, 15–20 April 2012. 8. Camarano, D.M.; Mansur, F.A.; Santos, A.M.M.; Ferraz, W.B.; Ferreira, R.A.N. Effects of beryllium and compaction pressure on the thermal diffusivity of uranium dioxide fuel pellets. Int. J. Thermophys. 2017, 38, 137. [CrossRef] 9. Bailey, H.S.; Lu, Y.S. Nuclear performance of liquid-metal fast breeder reactors designed to preclude energetic hypothetical core disruptive accidents. Nucl. Technol. 1979, 44, 76–82. [CrossRef] 10. Greebler, P.; Hutchins, B.A.; Sueoka, J.R. Calculation of Doppler Coefﬁcient and Other Safety Parameters for a Large Fast Oxide Reactor; General Electric Company: Boston, MA, USA, 1961. 11. Palmiotti, G.; Hill, R.; Wade, D. Physics of Plutonium Recycling; Organisation for Economic Cooperation and Development: Paris, France, 1995; Volume iv. 12. Waltar, A.E.; Todd, D.R.; Tsvetkov, P.V. Fast Spectrum Reactors; Springer: Heidelberg/Berlin, Germany, 2012. 13. Ahlfeld, C.; Burke, T.; Ellis, R.; Hejzlar, P.; Weaver, K.; Whitmer, C.; Gilleland, J.; Cohen, M.; Johnson, B.; Mazurkiewicz, S.; et al. Conceptual design of a 500 MWe traveling wave demonstration reactor plant. In Proceedings of the International Congress on Advances in Nuclear Power Plants, Nice, France, 2–6 May 2011. 14. Qvist, S.; Greenspan, E. An autonomous reactivity control system for improved fast reactor safety. Prog. Nucl. Energy 2014, 77, 32–47. [CrossRef] 15. Qvist, S.A.; Hellesen, C.; Gradecka, M.; Dubberley, A.E.; Fanning, T.; Greenspan, E. Tailoring the response of autonomous reactivity control (ARC) systems. Ann. Nucl. Energy 2017, 99, 383–398. [CrossRef] Energies 2018, 11, 1507 16 of 16

16. Merk, B.; Fridman, E.; Weiß, F.-P. On the use of a moderation layer to improve the safety behavior in sodium cooled fast reactors. Anna. Nucl. Energy 2011, 38, 921–929. [CrossRef] 17. Merk, B.; Weiß, F.-P. Analysis of the inﬂuence of different arrangements for ZrH moderator material on the performance of a sfr core. Ann. Nucl. Energy 2011, 38, 2374–2385. [CrossRef] 18. Fei, T.; Driscoll, M.J.; Shwageraus, E. Uranium startup fast reactors with metal fuel using once-through fuel cycle. Fuel Cycle Manag. 2013, 186, 378–389. [CrossRef] 19. Chen, Y. Irradiation effects of HT-9 martensitic steel. Nucl. Eng. Technol. 2013, 45, 311–322. [CrossRef] 20. Leppanen, J. Serpent—A continuous-energy monte carlo reactor physics burnup calculation code. User's Manual. 2013. Available online: http://montecarlo.vtt.ﬁ/download/Serpent_manual.pdf (accessed on 6 June 2018). 21. Pelowitz, D. MCNPX User’s Manual Version 2.7.0; LA-CP-11-00438; Los Alamos National Laboratory: Los Alamos, NM, USA, 2011. 22. MacFarlane, R.E.; Muir, D.W. The NJOY Nuclear Data Processing System Version 91; LA-12740-M; Los Alamos National Laboratory: Los Alamos, NM, USA, 1994. 23. Recktenwald, G.D.; Deinert, M.R. Effect of burnable absorbers on inert matrix fuel performance and transuranic burnup in a low power density light-water reactor. Energies 2013, 6, 2291–2304. [CrossRef] 24. Incropera, F.P.; Dewitt, D.P.; Bergman, T.L.; Lavine, A.S. Fundamentals of Heat and Mass Transfer, 6th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2007. 25. Lamarsh, J.R.; Baratta, A.J. Introduction to Nuclear Engineering, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2001; 783p. 26. Karahan, A. Modeling of Thermo-Mechanical and Irradiation Behavior of Metallic and Oxide Fuels for Sodium Fast Reactors; Massachussetts Institute of Technology: Cambridge, MA, USA, 2009. 27. IAEA. Status and Trends of Nuclear Fuels Technology for Sodium Cooled Fast Reactors; NF-T-4.1; International Atomic Energy Agency: Cambridge, MA, USA, 2011. 28. Nuclear Regulatory Commission Technical Training Center. In Power Plant Engineering Course Manual. Available online: https://www.nrc.gov/docs/ML0230/ML023020604.pdf (accessed on 6 June 2018).

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