Issue 
EPJ Nuclear Sci. Technol.
Volume 2, 2016



Article Number  8  
Number of page(s)  12  
DOI  https://doi.org/10.1051/epjn/e2015500326  
Published online  07 March 2016 
https://doi.org/10.1051/epjn/e2015500326
Regular article
Sustainability of thoriumuranium in pebblebed fluoride saltcooled high temperature reactor
^{1}
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Jialuo Road 2019#, Jiading District, 201800 Shanghai, P.R. China
^{2}
Key Laboratory of Nuclear Radiation and Nuclear Energy Technology, Chinese Academy of Sciences, Jialuo Road 2019#, Jiading District, Shanghai, P.R. China
^{3}
University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing, P.R. China
^{⁎} email: xuhongjie@sinap.ac.cn
Received:
11
May
2015
Received in final form:
25
October
2015
Accepted:
18
December
2015
Published online:
7
March
2016
Sustainability of thorium fuel in a PebbleBed Fluoride saltcooled High temperature Reactor (PBFHR) is investigated to find the feasible region of high discharge burnup and negative Flibe (2LiFBeF_{2}) salt Temperature Reactivity Coefficient (TRC). Dispersion fuel or pellet fuel with SiC cladding and SiC matrix is used to replace the tristructuralisotropic (TRISO) coated particle system for increasing fuel loading and decreasing excessive moderation. To analyze the neutronic characteristics, an equilibrium calculation method of thorium fuel selfsustainability is developed. We have compared two refueling schemes (mixing flow pattern and directional flow pattern) and two kinds of reflector materials (SiC and graphite). This method found that the feasible region of breeding and negative Flibe TRC is between 20 vol% and 62 vol% fuel loading in the fuel. A discharge burnup could be achieved up to about 200 MWd/kgHM. The case with directional flow pattern and SiC reflector showed superior burnup characteristics but the worst radial power peak factor, while the case with mixing flow pattern and SiC reflector, which was the best tradeoff between discharge burnup and radial power peak factor, could provide burnup of 140 MWd/kgHM and about 1.4 radial power peak factor with 50 vol% dispersion fuel. In addition, Flibe salt displays good neutron properties as a coolant of quasifast reactors due to the strong ^{9}Be(n,2n) reaction and low neutron absorption of ^{6}Li (even at 1000 ppm) in fast spectrum. Preliminary thermal hydraulic calculation shows good safety margin. The greatest challenge of this reactor may be the decades irradiation time of the pebble fuel.
© G. Zhu et al., published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The sustainability of nuclear energy resources has aroused great interest and attention since the Generation IV International Forum. A reactor system with breeding capability is very essential to extend the sustainability of nuclear fuel resources. Liquid metalcooled fast reactor is the preferred choice to achieve a high breeding ratio. However, it has some obstacles due to safety concerns associated with a positive void reactivity.
Thorium seems an attractive option of nuclear resources mainly due to its abundance, the opportunity to reduce the need for enrichment in the fuel cycle, the high conversion ratios (to ^{233}U) achievable in a thermal neutron spectrum, and also due to other neutron and thermal physical properties studied early in the development of nuclear power [1]. Due to the high effective number of neutrons for each ^{233}U fission in a thermal and epithermal neutron spectrum, thorium breeding is feasible in most existing and prospective reactor designs (including LWRs [2,3], HWRs [4–8], HTGRs [9] and molten salt reactors [10–12]), and it can provide the negative void reactivity coefficient due to the softer neutron spectrum than that of fast reactor. However, the thorium breeding gain in these reactors is far lower than fast reactor's. From an economical view, it is better to maintain fissile selfsustainability and to improve burnup for decreasing reprocessing mass per electricity.
This work focuses on sustainability of thorium in a PebbleBed Fluoride saltcooled High temperature Reactor [13–16] (PBFHR), to find its feasible region of high burnup and negative void reactivity coefficient. Expectant advantages of Flibe salt (2LiFBeF_{2}) as breeder reactor coolant [17] are that heatcarrying capacity and boiling point are both high; weak neutron slowingdown power will allow more coolant volume ratio than HWRs; and it may provide more negative temperature reactivity coefficient due to strong (n,2n) reaction of ^{9}Be in the fast spectrum. Furthermore, PBFHR is neutron saving with refueling online, and ^{233}Pa has the chance to decay away when thorium fuel is periodically removed from the core. However, one disadvantage is that Flibe salt in a flowing pebble bed will occupy about 40% volume of core, which enhances the moderation of Flibe and decreases the fuel inventory, as a result, a critical design should be required in fuel system of breeder PBFHR.
The system of tristructuralisotropic (TRISO) coated fuel particles embedded in massive graphite matrix in thermal spectrum PBFHRs is not adaptable to breeder reactor concepts due to its low fissile loading, the high irradiation swelling behavior of graphite in a quasifast spectrum, and the excessive moderation due to the large graphite/fuel ratio. Two kinds of fuel system [18] are developed for gascooled fast reactor (GFR) in order to increase fuel loading and improve radiation resistivity, which could be applied to PBFHRs. One is pintype GFR fuel with refractory cladding material (Fig. 1a); another one is dispersion fuel (or composite fuel or spherepac fuel) consisting of a distribution of discrete fuel particles embedded in a nonfuel matrix (Fig. 1b). Usually, fuel loading in dispersion fuel can reach 50 vol%, and in pintype fuel is beyond 75 vol% [19]. Buffer layers are both designed in pintype fuel and dispersion fuel to provide volume for fission gas and provide volume for fuel particle swelling. SiC is a good candidate cladding material or matrix material because of the good irradiation swelling behavior of SiC [20–25], the large irradiation behavior database, and the experience in use of SiC as a component in TRISO fuel. In addition, SiC has excellent oxidation resistance due to rapid formation of a dense, adherent SiO_{2} surface scale on exposure to air at elevated temperature, which offers protection from further oxidation. SiC is effective for retention of the solid fission products [26], but the migration of Ag in polycrystalline SiC can occur. Middle metallic liner designed in pintype fuel and SiC matrix in dispersion fuel ensure such fission product confinement within the fuel system.
For a preliminary concept design, the fuel system of thorium fuel selfsustainability in PBFHRs is considered as dispersed fuel particle filled in a sphere cladding. In order to simplify neutron calculation, an equivalent fuel system with only thorium fuel region and SiC region (Fig. 1c) is used because the weak moderation of SiC makes the space selfshielding effect insignificant. Oxycarbide thorium fuel is chosen in this work due to stable fission products bound by oxygen, low internal pressure for low product of free oxygen and compatibility with SiC material.
In order to simplify refueling scheme, homogeneous system with one kind of ^{233}U/^{232}Th pebble is carried out, in which mixing flow pattern and directional flow pattern are both performed. For neutronic analysis of thorium fuel selfsustainability, neutron spectrum is adjusted by fuel loading variable V _{ f }, which is defined as fuel volume dividing the volume of fuel system. In addition, graphite reflector is compared with SiC reflector to evaluate the moderation effect of reflector. Reactor model and refueling scheme are introduced in Section 2. Equilibrium calculation method of fissile selfsustainability is represented in Section 2.2. In Section 3, we show the results and discussions, in which achievable burnup of thorium fuel selfsustainability, temperature reactivity coefficient of Flibe, radial power distribution and preliminary thermal hydraulics are analyzed. Conclusions are drawn in Section 4.
Fig. 1 Fuel system: a. pintype fuel with SiC/SiC cladding; b. dispersion fuel filled with two kinds of coated fuel particle; c. equivalent fuel used for neutron calculation in this work. 
2 Model and calculation method
2.1 Reactor model and refueling scheme
Reactor model is simplified to a cylinder (Fig. 2). The core is divided into five radial annular flow channels with the same crosssectional area. Each channel is uniformly segmented into seven axial layers. In all, 35 burnup regions are used for neutronic calculation. Graphite or SiC is chosen as the material of both the axial and the radial reflector. Vacuum boundary condition is assumed outside the reflector. The layout of control rods and the B_{4}C shielding layer are outside the scope of this article, and are omitted in the equilibrium calculation. Dimensions of reactor are shown in Table 1. The diameter of pebble is chosen as 6 cm, but it may be changed for thermal hydraulic considerations. ^{233}U/^{232}Th pebbles are loaded in the core with a volumetric filling fraction of 0.6.
V _{ f } in pebble is varied by changing the packing factor. Usually, packing factor for the binary size particles is higher than unary size particle, in this paper, the packing factor in pin filling model is 0.73 calculated by equation from literature [27] (in sphere filling model, it will be lower than 0.73), fuel loading in particle could reach 78%, thus, the limiting V _{ f } is 0.73 × 0.78 = 0.57. However, for neutronic analysis, V _{ f } beyond 0.57 is also performed. Material properties of reactor are listed in Table 2. ^{6}Li in Flibe salt is assumed to be 22 ppm referred to literature [28], while the equilibrium concentration of ^{6}Li will be analyzed in the following section. Fresh ^{233}U/^{232}Th ratio (UTR) is automatically adjusted in the equilibrium calculation for fissile selfsustainability.
Multiplepassagethroughthecore (ten passage chosen in this work) with two kinds of flow patterns is simulated to flatten the axial power distribution. The mixing flow pattern is defined as that where pebbles, unloaded from each channel and not reached the limit of discharge burnup, are mixed with a batch of fresh pebbles and then are randomly recycled into five channels. The directional flow pattern is defined as that where a batch of fresh pebbles is recycled 10 times in channel 1, and then 10 times in channel 2, and so forth until discharged from channel 5. It is noteworthy that the radial position of pebbles in the core is determined by their inlet position [29], which implies that the directional flow could be easily achieved by only setting four baffles in the inlet. The outpile residence time of pebble is supposed to be equal to inpile residence time.
For the reprocessing of discharge fuel, only ^{233}U and ^{232}Th are extracted, while other uranium isotopes such as ^{234}U, ^{235}U, ^{236}U, are omitted in the calculation due to the long equilibrium cycle. ^{233}Pa from discharge fuel is regarded as ^{233}U, and will be returned to core. In the general model, average power density is 10 MW/m^{3} (corresponding to 980 MW total power), which will be changed in the analysis of ^{233}Pa effect. According to the refueling scheme and reflector material, 4 cases are analyzed, as shown in Table 3.
Fig. 2 Schematic view of the reactor geometry used during the neutronics calculation. On the left is the vertical view of the middle layer of the right horizontal view. Arabic numbers represent radial channels. 
Dimensions of reactor.
Material properties of reactor.
Core cases.
2.2 Equilibrium calculation method of thorium fuel selfsustainability
Equilibrium calculation of thorium fuel selfsustainability involves searching the fuel feed rate (or inpile residence time) and UTR to keep k _{ eff } convergent to 1 and to keep the ^{233}U fed into the core equivalent to ^{233}U from discharge fuel under different energy spectra. Convergence methods are analyzed below.
Ignoring the chain of ^{233}Pa and ^{233}Th, the evolution equations of ^{232}Th and ^{233}U can be shown as:(1) (2) N _{ Th } is the concentration of ^{232}Th, and A _{ Th } is a function of fluxes and onegroup capture crosssections of ^{232}Th in different regions. N _{ U3 } is the concentration of ^{233}U, and A _{ U3 } is a function as fluxes and onegroup absorption crosssections of ^{233}U in different regions. After inpile residence time T, the concentration of ^{233}U can be solved as follows:(3) N _{ U30 } is the fresh concentration of ^{233}U, and N _{ Th0 } is the fresh concentration of ^{232}Th. Ā is timeaveraged A. For fissile selfsustainability, N _{ U3 } (T) = N _{ U30 }. It can be deduced that:(4) UTR always can be determined by inpile residence time under specific A which is affected by neutron energy spectrum and can be adjusted by V _{ f }.
In addition, for simplified analysis, an equation can be established to connect k _{ eff } with T for fissile selfsustainability.(5) η is the effective number of ^{233}U fission neutrons, usually about 2.25 in epithermal spectrum. L is the sum of neutron absorption rate of structure material and leakage rate of core. Abs _{ fp } is the equivalent capture absorption rate of fission products and transuranic elements. For the differential equation (5),(6) V is feed rate of fresh fuel. Equation (6) can be changed into:(7)
Supposing L is equal to 2%, and k _{ eff } is 1, Abs _{ fp }·T can be obtained from equation (5). Equation (7) is changed into:(8)
Equation (8) describes a positive correlation between feed rate of fresh fuel and k _{ eff }, and is used to modify feed rate of fresh fuel with previous k _{ eff }. Constant 10 in equation (8) does not affect the accuracy but determines the rate of convergence.
Therefore, two loops are necessary for equilibrium calculation of thorium fuel selfsustainability. The outer loop modifies the feed rate of fresh fuel or inpile residence time to make k _{ eff } convergent, and the inner loop changes the UTR for fissile selfsustainability. It is notable that the neutron transportation calculation is only performed in the outer loop, which can obviously save computing time.
Equilibrium calculation of thorium fuel selfsustainability has been achieved in PBRE code [30], which is accurately verified by VSOP [31] code with the HTR10 model. PBRE is an equilibrium state searching code directly skipping the initial state and intermediate state. Method of PBRE is similar to literature [15,32,33]. The flow chart of PBRE with thorium selfsustainability is depicted in Figure 3. Guessing an equilibrium concentration of nuclides, MCNP code calculates equilibrium fluxes and onegroup crosssections of different regions. With the refueling scheme, equilibrium residence time in each region and pebble tracks are determined. Therefore, ORIGEN2 can give average concentrations in different regions and discharge concentrations. By modifying the UTR, fissile selfsustainability can be realized. Iteratively, average concentrations return to MCNP code until the k _{ eff } and concentrations are convergent. If the outer loop is diverging, it means that there does not exist the condition to meet fissile selfsustainability and reactor criticality.
Pebble tracks not only give the calculating order of different regions, but also contain the decay calculation when pebbles are unloaded from each channel. In addition, for mixing flow pattern, a mixing treatment for the same batch from different channels is performed.
Fig. 3 Flow chart of PBRE with thorium selfsustainability module. 
3 Results and discussions
Discharge burnup for thorium fuel selfsustainability and Temperature Reactivity Coefficient (TRC) of Flibe varied with V _{ f } are investigated in this section. For a further analysis of neutronic performance, properties of ^{6}Li, ^{233}Pa effect and radial power distribution are also studied. Finally, preliminary thermal hydraulic is analyzed to give the boundaries of power density and V _{ f }.
3.1 Neutron spectrum
Neutron spectrum provides a vital role for breeding or selfsustainability calculation. In the following analysis, V _{ f } is a main parameter to adjust neutron spectrum. As shown in Figure 4, neutron spectrum varies from quasifast spectrum to fast spectrum with the increase of V _{ f }. There are several dips around high energy range, corresponding to the main elastic scattering resonance of ^{7}Li and ^{19}F. In addition, there is a low peak at about 0.2 eV caused by thermal scattering of carbon from SiC and graphite reflector, but note that the peak is two or three orders of magnitude lower than the fast flux.
Fig. 4 Neutron spectrum dependent with V _{ f } (case 1). 
3.2 Discharge burnup for thorium fuel selfsustainability
In this section, the aim is to find the feasible region of thorium fuel selfsustainability and further to investigate the burnup characteristic of thorium fuel selfsustainability.
Discharge burnup for thorium fuel selfsustainability with different cases is shown in Figure 5. Discharge burnup is a function of V _{ f }. High V _{ f } can linearly improve the discharge burnup. For V _{ f }lower than 20%, the discharge burnup is near to zero, which implies that it may not be feasible to breed for thorium fuel in PBFHRs when V_{ f } is below 20%.
The mechanism of discharge burnup for thorium fuel selfsustainability variation with V _{ f }, can be understood in terms of the onegroup crosssection ratios of thoriumuranium and UTR (Fig. 6). The onegroup absorption crosssection ratio of thorium to uranium (XS(Tha)/XS(U3a)) reflects the conversion capability of thorium. The onegroup fissionabsorption ratio of ^{233}U (XS(U3f)/XS(U3a)) reflects ^{233}U fuel burning efficiency. As shown in Figure 6, a hard neutron spectrum provides high XS(Tha)/XS(U3a) and high XS(U3f)/XS(U3a), which show high conversion capability and high fuel burning efficiency. In the low V _{ f } region, UTR increases mainly due to the increase of XS(Tha)/XS(U3a) to keep reactor criticality, while in high V _{ f } region, higher fuel burning efficiency and lower neutron absorption crosssection of fission products will allow lower fresh UTR (Fig. 6). When XS(Tha)/XS(U3a) is higher than UTR, breeding of thorium is feasible and extra ^{233}U will be produced to improve the discharge burnup of thorium fuel. The evolution of ^{233}U in case 4 with 46.7% V _{ f } is shown in Figure 7. Concentration of ^{233}U will increase at low burnup and decrease to the initial concentration at the high burnup.
As shown in Figure 5, a directional flow pattern can provide higher discharge burnup than a mixing flow pattern, and a SiC reflector can provide higher discharge burnup than a graphite reflector. Reflector material has a more significant influence than flow pattern on discharge burnup in comparing case 2 and case 3. The neutron leakage rate and slowing down effect by reflector are the main contributions to differences among the four cases. In directional flow pattern, radial power fraction concentrates in the inner channel (Fig. 8) because of more ^{233}U, low fission products and consequent high flux in inner channel, which will decrease neutron leakage (Fig. 9) and lead to weak slowing down effect by reflector, and vice versa in mixing flow pattern. Additionally, power fraction in outer channel will be enhanced by graphite reflector due to the large fission crosssection caused by strong slowing down effect, which will further increase the neutron leakage (Fig. 9).
For 50% V _{ f } dispersion fuel, a discharge burnup of 63 MWd/kgHM, 103 MWd/kgHM, 140 MWd/kgHM and 165 MWd/kgHM can be achieved in case 1 to case 4, respectively. From a view of same discharge burnup, case 4 could provide smallest V _{ f } to reduce the manufacturing difficulty of fuel system. However, radial power peak factor case 4 is about 1.8. Case 3 is the best tradeoff between discharge burnup and radial power peak factor (about 1.4), and mixing flow pattern is the simplest refueling scheme.
Fig. 5 Discharge burnup for thorium fuel selfsustainability in different cases as a function of V _{ f }. 
Fig. 6 Onegroup crosssection ratio and concentration ratio of thoirumuranium (case 2). 
Fig. 7 Evolution of ^{233}U in case 4 with 46.7% V _{ f }. One wave represents a single passage caused by the delay of ^{233}Pa decay. 
Fig. 8 Radial power distribution with different V _{ f } and cases. Radial power distribution is tallied by TMESH card in column grid. In case 1 and case 2, power fraction in the outer channel increases because of strong slowing down effect by graphite reflector. 
Fig. 9 Neutron leakage rate as a function of V _{ f } with four cases. Neutron leakage rate is the escaped fraction outside the reflectors. Reflector with graphite material leads to more leakage than with SiC material. 
3.3 Thickness of reflector
To decrease the neutron leakage, thickness of reflector is analyzed. As shown in Figure 10, the thickness of graphite reflector has an apparent positive effect on the keff due to the strong slowing down power, which will lead to a lower breeding capacity or discharge burnup. However, neutron leakage rate almost does not vary with the thickness of graphite reflector (Fig. 11), which may be caused by offset between the enhanced power fraction in the outer channel and the enhanced reflectivity. As analyzed above, graphite reflector seems not suitable in this reactor. From Figures 10 and 11 , 50 cm thickness seems enough for SiC reflector to prevent neutron from escaping.
Fig. 10 keff with equilibrium concentrations as a function of reflector thickness and cases. 
Fig. 11 Neutron leakage rate as a function of reflector thickness and cases. Neutron leakage rate is the sum of escaped fraction outside the reflector and neutron absorption rate of reflectors. 
3.4 Flibe temperature reactivity coefficient
A negative Flibe TRC is necessary for PBFHR nuclear safety. The calculated Flibe TRC is shown in Figure 12. As the increase of V _{ f }, Flibe TRC increases. A positive Flibe TRC will happen when V _{ f } is beyond 62%, which shows the margin of inherent safety.
Comparing different cases, case 1 and case 2 show the more negative Flibe TRC than case 3 and case 4, which could be explained by softer spectrum in case 1 and case 2.
The mechanism of Flibe TRC is analyzed in models with 10% voided Flibe. The deviations of neutron absorption rate of main nuclides are shown in Figure 13. The ^{232}Th(n,γ) reaction and ^{233}U(n,f) reaction make great contributions to Flibe TRC. ^{232}Th(n,γ) makes Flibe TRC more positive, while ^{233}U(n,f) makes Flibe TRC more negative. With the increase of V _{ f }, the deviation of ^{233}U(n,f) reaction approaches zero, as can be explained with reference to Figure 14. In 33.0% V _{ f }, the ^{233}U(n,f) reaction in the resonance region has obvious shortfalls when slowing down by Flibe, while in 86.7% V _{ f }, the deviation in resonance region vanishes. The same situation happens with ^{232}Th(n,γ). This indicates that some level of slowing down is required to keep a negative Flibe TRC and this could not be achieved for solid thorium fuel in a fast neutron spectrum.
Figure 13 also shows the contribution of neutron leakage, Flibe absorption rate and other reaction rates to Flibe TRC. Neutron leakage makes the Flibe TRC a little negative, while Flibe absorption rate and other reaction rates make the Flibe TRC a little positive.
Fig. 12 Flibe salt temperature reactivity coefficient in different cases. Temperature changes from 920 K to 1050 K, and density of Flibe changes from 1.96 to 1.91 g/cm^{3}. 
Fig. 13 Deviations of neutron absorption rate of main nuclides as a function of V _{ f } (case 4). 
Fig. 14 Deviations of ^{233}U (n,f) reaction (90%Flibe–100%Flibe) as a function of neutron energy. 
3.5 Equilibrium concentration of Li6 and production rate of H3
The absorption rates of each nuclide in Flibe are shown in Figure 15. The ^{9}Be(n,2n) reaction rate is predominant, which could help reduce contribution to positive Flibe TRC. The ^{19}F(n,γ) reaction is apparent for several capture resonance peaks in fast spectrum. Notably, different from thermal spectrum, ^{6}Li and ^{7}Li show the low neutron absorption characteristics in a quasifast spectrum. The discharge burnup and Flibe TRC variations with concentration of ^{6}Li are shown in Figure 16. With the increase of ^{6}Li, discharge burnup decreases, while Flibe TRC does not change until beyond 3000 ppm. But it notes that discharge burnup only has a 6 MWd/kgHM drop when ^{6}Li increases from 22 ppm to 500 ppm, which shows that 99.95 at.% ^{7}Li at least is compatible for sustainability of thoriumuranium in PBFHR. This indicates that the cost of Flibe in quasifast reactor can be sharply reduced by lower enrichment of ^{7}Li.
In fact, the equilibrium concentration of ^{6}Li in a quasifast spectrum is very much larger than in a thermal spectrum. This can be calculated by equation (9), by assuming that the concentration of ^{9}Be in the core is constant.(9)
As shown in Figure 17, the equilibrium of ^{6}Li increases as the increase of V _{ f } due to the faster decline of onegroup absorption crosssection of ^{6}Li than that of ^{9}Be. 500 ppm of ^{6}Li can be achieved for 33% V _{ f }, which implies that enriching the ^{7}Li to more than 99.95 at.% level for improving the discharge burnup is unnecessary.
The product rate of ^{3}H in equilibrium state can be estimated by equation (10). Number of ^{233}U fission neutron is assumed to be 2.5, fission energy of ^{233}U is assumed to be 200 MeV. The product rate of ^{3}H is equal to the (n,α) reaction rate of ^{9}Be. Figure 17 shows that the product rate of ^{3}H in equilibrium state decreases as the increase of V _{ f }, which is in keeping with the (n,α) reaction rate of ^{9}Be shown in Figure 15. Since the (n,α) reaction rate of ^{9}Be in quasifast spectrum is low, the product rate of ^{3}H is only about 30–40 g/GW/year, which is not proportional to the concentration of ^{6}Li.(10)
Fig. 15 Absorption rate of each nuclide in Flibe as a function of V _{ f } (case 4). 
Fig. 16 Discharge burnup and Flibe temperature reactivity coefficient as functions of ^{6}Li concentration (30 MW/m^{3} power density with 46.7% V _{ f } in case 4). 
Fig. 17 Equilibrium concentration of Li6 and production rate of H3 as functions of V _{ f } in case 4. 
3.6 Effect of ^{233}Pa
In the conversion process of thorium, some of the ^{233}U will be lost by the irradiation of ^{233}Pa. This effect can be enhanced by a high neutron flux. As shown in Figure 18, discharge burnup has a 30 MWd/kgHM drop when power density increases from 10 to 30 MW/m^{3}. In PBFHRs, ^{233}Pa has opportunity to decay away by periodically removing pebbles from the core. Figure 19 shows that the discharge burnup in high power density condition can be improved by increasing the number of times each pebble is cycled through each channel. However, this effect becomes weak when number of cycles in each channel extends beyond 20.
Because of the low power density and high fuel loading, the residence time of each pebble in this reactor is very long (Fig. 20). It is necessary to reduce the residence time by increasing the power density and decreasing the fuel loading or V _{ f }. For 46.7% V _{ f } in case 4, if the core power density is 40 MW/m^{3}, the discharge burnup may drop from 148 to about 100 MWd/kgHM for the effect of ^{233}Pa, the residence time of a pebble will be 17 years.
Fig. 18 Discharge burnup as a function of power density (46.7% V _{ f } in case 4). 
Fig. 19 Discharge burnup as a function of pebble cycle number in each channel. 30 MW/m^{3} power densities with 46.7% V _{ f } in case 4. 
Fig. 20 Total residence time of pebble as a function of V _{ f } and cases (10 MW/m^{3} power density). 
3.7 Thermal hydraulic analysis
In this section, V _{ f } and power density will further be limited by thermal hydraulics considerations. For dispersion fuel system, a onedimensional sphere geometry with equivalent thermal conductivity is used. By reference to HTGRs [34], the limit temperature of fuel in normal conditions is assumed to be 1250 °C, and the limit temperature of fuel in accident conditions is supposed to be 1600 °C.
The maximum kernel temperature can be deduced from the maximum temperature of mixed fuel region:(11) T _{ f } is maximum temperature of mixed fuel region, P is the average power density, f is the total power peak factor (assumed as 1.4 × 1.4 ≈ 2 in the following calculation), ε is porosity of pebble bed, k _{ fuel } is thermal conductivity of thorium–uranium fuel, k _{ buffer } is thermal conductivity of buffer, k _{ SiC } is thermal conductivity of SiC cladding, r _{1} is the radius of fuel kernel, r _{2} is the outer radius of buffer and r _{3} is the outer radius of SiC cladding.
The maximum temperature of mixed fuel region can be obtained by:(12) T _{ s } is the surface temperature of pebble; R is the radius of pebble; k is equivalent thermal conductivity, in this paper, it is the volume average thermal conductivity, which will vary with V _{ f }.
The surface temperature of pebble can be obtained by heat convection equation:(13) T _{ c } is the average temperature of Flibe, k _{ c } is the thermal conductivity of Flibe, Nu is nusselt number cited from Wakao [35], Re is Reynolds number, and Pr is Prandtl number.(14) ρ is the density of Flibe, μ is dynamic viscosity, C _{ p } is heat capacity, U is superficial velocity of Flibe. U can be calculated by:(15) h is the height of core, T _{ outlet } is outlet temperature of Flibe, and T _{ inlet } is inlet temperature of Flibe.
The physical property parameters are listed in Table 4, the thermal conductivity of SiC is very high even after a long period of irradiation time, the thermal conductivity of Th_{2}CO_{3}/U_{2}CO_{3} is referred from that of ThO_{2}, which shows a little higher thermal conductivity than UO_{2}.
The results are shown in Table 5. As the increase of V _{ f }, the equivalent thermal conductivity decreases, as a result, the maximum temperature of fuel increases. However, even in 60% V _{ f }, the maximum temperature of fuel is still below 1250 °C for 6 cm pebble under 10 MW/m^{3} power density. On the other hand, the allowable power density for 6 cm pebble will not extend beyond 21 MW/m^{3} if the maximum temperature of fuel is below 1250 °C. Reducing the diameter of pebble is an effective means of improving the power density, as shown in Table 5, 60 MW/m^{3} power density is allowable in 40% V _{ f } for 3 cm pebble.
As analyzed above, thermal conductivity is sensitive to the maximum temperature of the fuel. ThC may be a good candidate ceramic fuel due to the high density and high thermal conductivity.
Loss of Forced Cooling (LOFC) and Anticipated Transient Without Scram (ATWS) are the most important accidents for PBFHRs. The decay heat removal system with Pool Reactor Auxiliary Cooling (PRAC) heat exchangers (PHX) modules in the PBAHTR could be applied to this work. In an LOFC accident, even under 40 MW/m^{3} power density, the outlet temperature of Flibe will not rise by as much as 50 °C, and the temperature of fuel will quickly drop to the level of Flibe [36]. In an ATWS accident with a 1000 pcm reactivity insertion, the temperature of the fuel will not rise by as much as 200 °C to 1450 °C, which is still lower than 1600 °C (–5 pcm/K of fuel TRC is calculated in 46.7% V _{ f }, case 4). In addition, the negative Flibe TRC is more effective to decrease the outlet temperature than a more negative fuel TRC, and the outlet temperature in this case will not rise by 200 °C [36].
Constants for thermal hydraulic calculations.
Temperature distribution in pebble.
4 Conclusions
This work investigated the sustainability of thorium fuel in PBFHR. Dispersion fuel with SiC cladding and SiC matrix was used to increase the fuel loading. A novel equilibrium calculation method of thorium fuel selfsustainability was developed to analyze discharge burnup. The mechanism of breeding and the characteristic of Flibe salt temperature reactivity coefficient are both performed.
Some preliminary findings are as follows:

more than 20 vol% fuel loading in fuel system is necessary to keep thorium fuel sustainable, and less than 62 vol% fuel loading is required for negative Flibe TRC. The allowed maximal discharge burnup for thorium fuel selfsustainability and negative Flibe TRC is about 200 MWd/kgHM;

case 4 with directional flow pattern and SiC reflector displays superior burnup characteristics due to having the hardest neutron spectrum and lowest neutron leakage. While case 3 with mixing flow pattern and SiC reflector shows the best tradeoff between discharge burnup and radial power peak factor. For 50% V _{ f } dispersion fuel, case 3 could provide 140 MWd/kgHM burnup and about 1.4 radial power peaking factor;

the ^{232}Th(n,γ) reaction and ^{233}U(n,f) reaction are main contributions to Flibe TRC. It indicates that some level of slowing down is required to keep a negative void reactivity coefficient, which provides a new insight for coolant in quasifast reactor. Flibe salt shows good neutron properties as coolant of quasifast reactor. The equilibrium concentration of ^{6}Li in fast spectrum is around 1000 ppm, which decreases the cost of enrichment, and the neutron absorption of ^{6}Li is still low. 99.95% ^{7}Li is compatible for sustainability of thoriumuranium in PBFHR. In addition, the production rate of ^{3}H in quasifast spectrum is about 30–40 g/GW/year, usually lower than in thermal spectrum;

effect of ^{233}Pa is significant in the high power density condition. A 30 MWd/kgHM drop in discharge burnup is obtained when power density increases from 10 to 30 MW/m^{3}. Increasing the number of time each thorium pebble is cycled through each channel can increase discharge burnup. The greatest challenge of this reactor is the very long irradiation time of the pebble fuel. Increasing power density can apparently decrease the irradiation time, but discharge burnup will also obviously decrease, and as a result, the reactor may not be competitive;

thermal hydraulic calculations show good safety margin. 20 MW/m^{3} is allowable for 6 cm pebble, and 60 MW/m^{3} is allowable for 3 cm pebble. V _{ f } affects the thermal conductivity, and a value lower than 50% is recommended.
In further analysis, we will focus on the high power density case, investigate how to reduce the effect of ^{233}Pa, and also perform a detail thermal hydraulic analysis.
Acknowledgments
This paper is supported by the “Strategic Priority Research Program” of the Chinese Academy of Sciences (Grant No. XDA02010200), and Science and Technology Commission of Shanghai Municipality (Grant No. 11JC1414900). Thanks for the suggestions from David W. Dean and reviewers.
Nomenclature
PBFHR: PebbleBed Fluoride saltcooled High temperature Reactor
V _{ f } : Fuel volume dividing the volume of fuel system
UTR: Fresh ^{233}U/^{232}Th ratio
XS(Tha)/XS(U3a): Onegroup absorption crosssection ratio of thoriumuranium
XS(U3f)/XS(U3a): Onegroup fission crosssection of ^{233}U over onegroup absorption crosssection of ^{233}U
TRC: Temperature Reactivity Coefficient
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All Tables
All Figures
Fig. 1 Fuel system: a. pintype fuel with SiC/SiC cladding; b. dispersion fuel filled with two kinds of coated fuel particle; c. equivalent fuel used for neutron calculation in this work. 

In the text 
Fig. 2 Schematic view of the reactor geometry used during the neutronics calculation. On the left is the vertical view of the middle layer of the right horizontal view. Arabic numbers represent radial channels. 

In the text 
Fig. 3 Flow chart of PBRE with thorium selfsustainability module. 

In the text 
Fig. 4 Neutron spectrum dependent with V _{ f } (case 1). 

In the text 
Fig. 5 Discharge burnup for thorium fuel selfsustainability in different cases as a function of V _{ f }. 

In the text 
Fig. 6 Onegroup crosssection ratio and concentration ratio of thoirumuranium (case 2). 

In the text 
Fig. 7 Evolution of ^{233}U in case 4 with 46.7% V _{ f }. One wave represents a single passage caused by the delay of ^{233}Pa decay. 

In the text 
Fig. 8 Radial power distribution with different V _{ f } and cases. Radial power distribution is tallied by TMESH card in column grid. In case 1 and case 2, power fraction in the outer channel increases because of strong slowing down effect by graphite reflector. 

In the text 
Fig. 9 Neutron leakage rate as a function of V _{ f } with four cases. Neutron leakage rate is the escaped fraction outside the reflectors. Reflector with graphite material leads to more leakage than with SiC material. 

In the text 
Fig. 10 keff with equilibrium concentrations as a function of reflector thickness and cases. 

In the text 
Fig. 11 Neutron leakage rate as a function of reflector thickness and cases. Neutron leakage rate is the sum of escaped fraction outside the reflector and neutron absorption rate of reflectors. 

In the text 
Fig. 12 Flibe salt temperature reactivity coefficient in different cases. Temperature changes from 920 K to 1050 K, and density of Flibe changes from 1.96 to 1.91 g/cm^{3}. 

In the text 
Fig. 13 Deviations of neutron absorption rate of main nuclides as a function of V _{ f } (case 4). 

In the text 
Fig. 14 Deviations of ^{233}U (n,f) reaction (90%Flibe–100%Flibe) as a function of neutron energy. 

In the text 
Fig. 15 Absorption rate of each nuclide in Flibe as a function of V _{ f } (case 4). 

In the text 
Fig. 16 Discharge burnup and Flibe temperature reactivity coefficient as functions of ^{6}Li concentration (30 MW/m^{3} power density with 46.7% V _{ f } in case 4). 

In the text 
Fig. 17 Equilibrium concentration of Li6 and production rate of H3 as functions of V _{ f } in case 4. 

In the text 
Fig. 18 Discharge burnup as a function of power density (46.7% V _{ f } in case 4). 

In the text 
Fig. 19 Discharge burnup as a function of pebble cycle number in each channel. 30 MW/m^{3} power densities with 46.7% V _{ f } in case 4. 

In the text 
Fig. 20 Total residence time of pebble as a function of V _{ f } and cases (10 MW/m^{3} power density). 

In the text 
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