2.1 Energy gain

The energy gain (G) of an EA is defined as the ratio of the total energy produced in the system to the power of the accelerator beam ^[27]. The total energy produced by EA can be expressed as

where Z stands for the number of source neutrons generated per incident proton, $i_{\text{p}}$ is the required beam intensity, ${\overline{E}}_{\text{f}}$ is the average recoverable energy released in a fission, $\overline{\nu }$ is the average number of neutrons produced per fission, and φ* is the neutron source efficiency, which represents the average importance of external source neutrons over the average importance of fission neutrons ^[28]. It is defined as

where $R_{\text{f}}$ is the fission reaction rate, and S is the total production of neutrons by the external source.

By rearranging Eq 1, the required beam intensity can be expressed as

where $E_{\text{p}}$ is the proton energy. The power of accelerator beam is $P_{\text{acc}}=i_{\text{p}}{E}_{\text{p}}$. Therefore, the expression of the energy gain is

Equation 4 shows that the closer k_eff is to 1, the larger the energy gain. In addition, it is also advisable to enhance the energy gain by increasing Z and φ*. For the molten salt spallation target used in the TMSFEA, increasing the content of HM in the molten salt can effectively improve Z and φ*, thus enhancing the energy gain, decreasing the required proton beam intensity.

2.2 Conversion ratio

2.2.1 Background of GVR

The Th-U breeding performance is quantitatively descripted by the conversion ratio (CR). The CR can be expressed as

$CR=\frac{R_{\text{c}}^{\text{fertile}}}{R_{\text{a}}^{\text{fissile}}},$ (5)

where $R_{\text{c}}^{\text{fertile}}$ represents the neutron capture reaction rate of fertile nuclide, and $R_{\text{a}}^{\text{fissile}}$ stands for the neutron absorption reaction rate of fissile nuclides. For the starting fuel of TMSFEA, the only fissile and fertile nuclides are ²³²Th, ²³⁹Pu, and ²⁴¹Pu. Therefore, equation 5 can be simplified as

$CR=\frac{R_{\text{c}}({}_{}{}^{232}\text{Th}+{}_{}{}^{240}\text{Pu})}{R_{\text{a}}({}_{}{}^{239}\text{Pu}+{}_{}{}^{241}\text{Pu})},$ (6)

CR can also be analyzed from the neutron balance point of view. The equation 7 gives the neutron balance in a critical fast reactor ^[29].

$CR=\eta +\epsilon -1-(l_{\text{A}}+l_{\text{L}}),$ (7)

where η refers to the number of neutrons produced by fissions in fissile nuclides (per absorbed neutron), $\epsilon$ is the average number of excess neutrons produced by fissions in fertile nuclides, $l_{\text{A}}$ is the number of neutrons absorbed in non-fuel materials, and $l_{\text{L}}$ is the number of neutrons lost by leakage. For the subcritical facility, the contribution of external source neutrons has to be considered, and equation 7 becomes

$CR=\eta +\epsilon +\mu -1-(l_{\text{A}}+l_{\text{L}})，$ (8)

where $\mu$ is an external source of neutrons normalized to neutron absorption in fissile nuclides. By applying equation 2, $\mu$ can be expressed as

$\mu =\frac{S}{R_{\text{a}}^{\text{fissile}}}=\frac{\overline{\nu }(1-k_{\text{eff}})}{k_{\text{eff}}(1+\alpha ){\phi }^{*}}，$ (9)

where $R_{\text{a}}^{\text{fissile}}$ refers to the neutron absorption reaction rate in fissile nuclides, and $\alpha$ is the capture-fission ratio. By inserting equation 9 into equation 8, the expression of CR becomes

$CR=\eta +\epsilon +\frac{\overline{\nu }(1-k_{\text{eff}})}{k_{\text{eff}}(1+\alpha ){\phi }^{*}}-1-(l_{\text{A}}+l_{\text{L}})$ (10)

Equation 10 shows that k_eff and CR are two competitive parameters. It is feasible to improve CR by increasing the sub-criticality. However, the cost is the necessity to increase the proton beam intensity and decrease the energy gain. It is also noted that η plays an essential role in realizing the breeding (CR>1). From the breeding point of view, η must be large enough (significantly greater than 2). Fig. 1 shows theη ratio of four fissile nuclei, namely, ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu, in the thermal, resonance, and fast regions. It can be seen that η of both nuclei rapidly increases with increasing energy in the fast range (E > 100 keV), and ²³⁹Pu and ²⁴¹Pu have the largest η in this region. Therefore, ²³⁹Pu and ²⁴¹Pu have better breeding performance if the average energy of neutron spectrum is higher than ~100 keV. As a result, the conversion ratio increases with increasingly harder neutron spectrum. To harden the neutron spectrum, the density of light nuclei in the core should be restricted.

Figure 1.Full-screen View

(Color online)η ratio as a function of the neutron energy for different fissile nuclei

3.1 TMSFEA system

The TMSFEA is an accelerator-driven subcritical single-fluid molten salt system. Instead of achieving a self-sustained chain fission reaction in a critical reactor, TMSFEA uses a subcritical facility to generate energy by introducing extra neutrons produced in the spallation reaction. The general layout of TMSFEA is displayed in Fig. 2. A proton-driver accelerator is coupled to a subcritical facility, and a beam of energetic protons from the accelerator is directly injected into the molten salt, which is held in the reflector, and spallation neutrons are produced to sustain the fission reaction in the subcritical facility. The entire system could be shut-down simply by switching off the proton beam rather than inserting any control rods, which makes TMSFEA a much safer system than conventional fission reactors. The fission heat produced in the subcritical core is transferred to the secondary heat exchanger through the primary heat exchanger and then transmitted to the turbine/generator for electricity generation, which can partially be used to run the accelerator.

Figure 2.Full-screen View

(Color online) Schematic diagram of TMSFEA

3.2 Molten salt fuels

One of our primary design goals for TMSFEA is the use of thorium fuel. In a fast-neutron subcritical facility, the net ²³³U production rate from Th with Pu as the starting fuel is much higher than that with U as the starting fuel, because ²³⁹Pu and ²⁴¹Pu have much larger η under the fast spectrum, as described in the previous section. The molten fluoride salt system has been extensively studied for many years at Oak Ridge National Laboratory as the fuel salt of a molten salt reactor ^[30] and the target salt of an accelerator-driven transmutation system ^[7]. However, the solubility of PuF₃ in the fluoride salt is limited, which will seriously affect the loading capacity of Pu in fluoride salts. Moreover, fluorine has a smaller mass number, which is a disadvantage for the neutron spectrum hardening, and thus reduces η of ²³⁹Pu and ²⁴¹Pu. In addition, as a spallation target salt, the fluoride salts have a lower spallation neutron production performance, because the solubility of HM in the fluoride salt is not high enough.

Therefore, a ternary chloride-based molten salt system containing thorium and plutonium was chosen in our design. Molten salts containing thorium and plutonium halides have been widely studied in the nuclear industry since the 1970s, and much research has been conducted regarding their physicochemical characteristics. The results, especially melting diagrams, are the basis of our design. Among these melting diagram results, the NaCl-PuCl₃-ThCl₄ and the ThCl₄-KCl-PuCl₃ molten salt systems have a more reasonable melting point for our design. In addition, sodium has a smaller neutron absorption cross section than potassium; therefore, NaCl-PuCl₃-ThCl₄ was selected as the fuel in our TMSFEA design. Fig. 3 illustrates the ternary phase diagram ^[31] for NaCl-PuCl₃-ThCl₄ (in mol%), and the expected melting temperature for any mixture of NaCl-PuCl₃-ThCl₄ is shown. The eutectic point E1 (46.5 mol% NaCl, 18.5 mol% PuCl₃, and 35.0 mol% ThCl₄) has the lowest melting temperature of 325 ^oC. Therefore, the range of specific NaCl-PuCl₃-ThCl₄ fuel salt implementations for any given melting point between 325 ^oC and 700 ^oC is allowed to be investigated. Of particular interest are mixtures having a melting point of less than 550 ^oC, which are illustrated in the gray region of the ternary phase diagram. Various specific compositions of NaCl-PuCl₃-ThCl₄ fuel salt in the gray region can be used in the TMSFEA. The choice of the final composition depends on a variety of factors, including the spallation neutrons production performance, the energy gain, and the Th-U breeding performance.

Figure 3.Full-screen View

Phase diagram of NaCl-PuCl₃-ThCl₄

As for the material compatibility problem, the majority of chloride-based salts are not aggressively corrosive to stainless steels or nickel-based alloys at a reactor temperature without the presence of oxygen ^[32]. However, long-term operation requires a high degree of material compatibility, and considerable future research of chloride salts is needed, since most of the available information relates to fluoride-based salts with uranium dissolved inside. In addition, as described in the ORNL report: ^[33] " in-core testing of material compatibility of structural materials exposed to fuel-bearing chloride salts has yet to be performed, so materials compatibility cannot yet inform the salt selection".

3.3 Subcritical facility modeling

The schematic of the subcritical facility of TMSFEA is shown in Fig. 4. To simplify the core structure, a single-fluid type design is adopted. The average operating temperature of fuel salt is 963 K, which refers to the design of REBUS-3700 ^[34]. The fuel salt is poured into a cylindrical core barrel which is made of Hastelloy-N with a 2 cm thickness. A reflector was employed to reduce the leakage of neutrons. A 5 cm thick Hastelloy-N vessel is placed outside the reflector. A 1GeV proton beam is transported in a 1 cm thick Hastelloy-N tube and is injected into the core. The outer diameter of the beam tube is 20 cm, which refers to the beam tube design of Rubbia’s EA ^[1]. In addition, initially, the plutonium in the ternary molten salt system consists of about 1.9wt% ²³⁸Pu, 58.6wt% ²³⁹Pu, 23.8wt% ²⁴⁰Pu, 10.2wt% ²⁴¹Pu, and 5.5wt% ²⁴²Pu, which is considered to be a typical isotopic composition of plutonium recovered from PWR spent fuel ^[35].

Figure 4.Full-screen View

(Color online) Schematic diagram of the subcritical facility model

5.1 Optimization of TMSFEA design parameters

According to Eq. 4, the amount of energy gain depends on the proton energy $E_{\text{p}}$, performance of the spallation target (i.e., Z and φ*), and the characteristics of the subcritical facility (i.e., $k_{\text{eff}}$, ${\overline{E}}_{\text{f}}$, and $\overline{\nu }$). The value of ${\overline{E}}_{\text{f}}$ is mainly determined by the fission nuclides of the fuel and can be considered constant when the proton energy varies ^[39]. For the starting fuel of TMSFEA, the only fissile nuclides are ²³⁹Pu and ²⁴¹Pu. The values of ${\overline{E}}_{\text{f}}$ for ²³⁹Pu and ²⁴¹Pu are 199.88 MeV and 201.88 MeV, respectively ^[40]. In this work, the compromise value of 200 MeV was chosen for the energy gain calculation.

For a fixed core structure and molten salt composition, the variation of energy gain with proton energy was calculated. In Fig. 7, the energy gain is plotted as a function of the energy of the incident proton. The results show that the energy gain increases rapidly with increasing proton energy up to about 1000 MeV and reaches the maximum value at about 1600 MeV. Table 1 gives the specific values ​​of Z, φ*, $\overline{\nu }$, and energy gain G. As shown in Table 1, the value of $\overline{\nu }$ is hardly changed with the proton energy, since $\overline{\nu }$ is determined by the fuel composition. When the proton energy is below 1000 MeV, φ* increases with the proton energy enhancement, which is due to the larger production of high-energy neutrons, which can produce more fission neutrons. However, when the proton energy is greater than 1000 MeV, φ* does not vary with the proton energy, which is mainly due to the increasing cross section of pion generation and the difficulty of producing higher energy neutrons. It is also noted that when the proton energy is increased from 1000 MeV to 1600 MeV (an increase of 60%), the energy gain increases only by 8%. Therefore, from the economic perspective, the proton energy of 1000 MeV is chosen in our design.

Figure 7.Full-screen View

Energy gain as a function of proton energy for proton beams

The sub-criticality factor k_eff also has a significant impact on the energy gain. The closer the neutron multiplication factor to 1, the greater the energy gain. However, the choice of k_eff is mainly based on the consideration of safety. A reasonable sub-criticality margin should be considered to avoid a critical accident at a subcritical facility. Therefore, the initial k_eff value was set to 0.98 in our design.

In addition to the energy gain, another concern of TMSFEA is the Th-U breeding performance. As shown in equation 10, the Th-U breeding performance can be enhanced by reducing the leakage ratio. There are two ways to reduce the leakage ratio: one is to arrange a reflector, and the other is to increase the core size. To assess the influence of the reflector on CR, the initial CRs of different layouts of reflector are calculated and shown in Fig. 8. Unlike the monotonic increase in CR for a lead reflector, with an increase of the graphite reflector thickness, CR tends to increase first and then to decline when the thickness is larger than 10 cm. As shown in Fig. 9, the tendency to increase CR for the graphite reflector is due to the larger positive contribution of l_L. After 10 cm, a significant increase in l_A leads to a decrease in CR, since the softened spectrum increases the neutron absorption by non-fuel nuclides. Furthermore, CR for both reflectors becomes almost steady when the thickness is larger than 60 cm.

Figure 8.Full-screen View

(Color online) Initial CR and k_eff for different reflector arrangements

Figure 9.Full-screen View

(Color online) Neutron parameters as the functions of the thickness of different reflectors

Fig. 10 shows the neutron spectra of the core for different reflector arrangements; the layout of reflectors can significantly soften the neutron spectrum, especially for graphite. It can be noticed that even 10 cm of graphite has a softer spectrum compared to lead with the thickness of 90 cm, which explains the variation trend of the neutron parameters, as shown in Fig. 8. Considering that the breeding of thorium in a plutonium-started core needs a harder neutron spectrum, a lead reflector is more suitable for our design.

Figure 10.Full-screen View

(Color online) Neutron spectrum for different reflector arrangements

An increase in the core size can also reduce leakage ratio. The height-diameter ratio of the core is fixed to 1. Fig. 11 shows that CR and k_eff change as the core size is increased. The larger the core, the lower the neutron leakage and, consequently, the higher the k_eff and CR. Furthermore, enlarging the core barely has any effect on the spectrum or spectrum related parameters (i.e., η, l_A, μ, and ε).

Figure 11.Full-screen View

(Color online) Initial CR and k_eff for different core sizes

The composition of NaCl-PuCl₃-ThCl₄ fuel is also optimized to meet the criticality and the CR requirements for long-term operation of the EA system. Based on the above analysis, a 60 cm liquid lead reflector was chosen. In order to keep a subcritical state and achieve breeding, k_eff is set to 0.98, and CR should be above 1. These conditions can be used to determine a relatively reasonable fuel salt composition. Fig. 12 shows the variation of CR for different fuel salt compositions. There is no adaptive area in Fig. 12(a), which means that the 120 cm radius core does not satisfy the breeding condition. When the radius of the core is set to 150 cm in Fig. 12(b), the contour line of k_eff = 0.98 intersects with the region CR >1 satisfying the breeding condition of the EA. The composition marked with a star in Fig. 12(b) is chosen in the molten salt EA design because it has a reasonable CR and melt temperature (~530 ^oC).

Figure 12.Full-screen View

CR as a function of fuel salt composition. (a) The radius of the core is 120 cm; (b) the radius of the core is 150 cm

After the optimization, the main parameters of TMSFEA are listed in Table 2.

5.2 Burn-up analysis

The evolution of TMSFEA k_eff and beam intensity with three different kinds of thermal power is displayed in Fig. 13. A 300 MW_th facility has the steadiest k_eff, and thus a relatively gentle evolution of the beam intensity compared to the other two kinds of cases with higher power. At a rated power, TMSFEA is expected to have a steady energy gain, namely, it meets the stable beam intensity requirement during the operation. As can be seen from Equation 3, the beam intensity of TMSFEA is mainly affected by k_eff, Z, ${\overline{E}}_{\text{f}}$, $\overline{\nu }$, and φ*. Except for k_eff, variations in other terms are small.

Figure 13.Full-screen View

(Color online) Time evolution of k_eff, CR, and beam intensity. (a) Evolution of k_eff and CR; (b) evolution of beam intensity

The k_eff evolution is associated with CR, which must be higher than approximately 1.03 to maintain stability in cases of Pu-Th loading, since η for Pu is about 1.03 times higher than that of ²³³U in this spectrum. Taking the thermal power of 1 GW as an example, the value of k_eff monotonically decreases when CR is below 1.03 after 5 years. In addition, due to the effect of Pa, k_eff shows a rapid decline in a few days after the beginning, which results in an increase in beam intensity at the beginning of the operation. It is also worth noting that the CR of 1 GW thermal power drops to a minimum value (~0.89) and then has a certain increase, which is due to the increased contribution of the external neutron source.

The evolution of the Pu and ²³³U at thermal powers, 300 MW_th and 500 MW_th, is presented in Fig. 14. As the contribution of ²³³U to the total fission rate increases, the net production rate of ²³³U will gradually decrease. The net ²³³U production at 500 MW_th thermal power starts decreasing after 45 years, since ²³³U contributes about 85% of the total fission rate, which can greatly increase the consumption of ²³³U.

Figure 14.Full-screen View

(Color online) Evolution of Pu and ²³³U. (a) Evolution of Pu; (b) evolution of ²³³U

Under different beam intensity restrictions, the production of ²³³U and the amount of Pu transmutation are given in Table 3. The results indicate that the life time is much longer for 300 MW_th. For a maximum beam intensity limit of 6 mA, the life time of TMSFEA for 300 MW_th and 500 MW_th is 51 and 24 years, respectively. At the end of the service life, about 57% of Pu and 16% of Th were consumed at 300 MW_th, compared with 50% and 13% at 500 MW_th. Furthermore, the net production of ²³³U is 7% higher at 300 MW_th than at 500 MW_th. With the maximum beam intensity limit of 4 mA, the life time of TMSFEA at 300 MW_th is 39 years, and the maximum fission rate contribution of ²³³U is 70.9%, which indicates the efficient thorium utilization. At the end of the life cycle at 300 MW_th with a 4 mA limit, the energy gain is 76, while that with a 6 mA limit is 53. Therefore, TMSFEA is recommended to operate at 300 MW with a 4 mA beam intensity limit, because the energy gain can be maintained at a higher level.