2.1 Simulation model specification

The reactor model of MSR used in the study has a simplified cylindrical geometry (Fig. 1) and is composed of an active zone and a surrounding reflector. The active zone was filled with graphite-salt lattices with a salt volume fraction (the volume of fuel salt over that of graphite) of approximately 12% corresponding to an under-moderated core. Both the diameter and the height of the reactor core are equal to 300 cm while the thicknesses of radial and axial reflector are 50 cm and 20 cm, respectively. The geometry values selected in the study were only based on considering the neutron economy (the effective multiplication factor (k_eff). The density of the graphite for moderator and reflector was set as 1.86 g/cm³.

Figure 1:Full-screen View

(Color online) The schematic diagram of the reactor core of MSR.

The fuel salt composition included the carrier salt and heavy metals. FLiBe salt (2LiF-BeF₂) was selected as carrier salt by considering its excellent neutron economy and chemical & physical properties. As mentioned above, the proportion of heavy metals (HM proportion) in the fuel salt had a direct and significant impact on neutron spectrum and subsequently on neutronic characteristics such as breeding capability and temperature coefficient of reactivity. The reduction in the HM proportion may have entailed an increase in the melting point of the fuel salt [23]. Furthermore, a higher melting point indicated that a higher operation temperature was required and led to a challenge with respect to the heat and irradiation resistance of the structural materials. Conversely, a higher proportion of carrier salt may cause higher neutron capture. Therefore, the minimum HM proportion was set as 2 mole%. Moreover, the upper limit of the HM proportion was set as 12 mole% by considering the solubility of heavy metals in fluoride salt and the feasibility to control oxygen. The mean operating temperature of the reactor was 880 K. Additionally, the ²³⁵U enrichment was under 20%, given the non-proliferation while the ⁷Li enrichments of 99.95% and 99.995% were analyzed for comparison purposes.

2.2 Calculation of temperature coefficient

The analysis of the temperature coefficient (in this case, FSTC) was based on the six-factor formula as follows:

where ε denotes the fast fission factor, p denotes the resonance escape probability, η denotes the thermal reproduction factor, f denotes the thermal utilization factor, and Λ denotes the total non-leakage probability that corresponds to the product of the fast (P_FNL) and the thermal (P_TNL) non-leakage probability. Thus, five factors, i.e., ε, p, η, f, and Λ, were adopted to calculate FSTC.

The FSTC (α_T) is a measure of the variation in k_eff (Δk) with the temperature of fuel salt (ΔT) as follows:

The change in the fuel salt temperature can lead to a variation in neutron spectrum, and subsequently an alteration of k_eff. Thus, each factor in the six-factor formula is related to the temperature and is used to clarify the FSTC. Therefore, the FSTC was separated into five components (αT(ε), αT(p), αT(η), αT(f), and αT(Λ) by taking the derivative of Eq. (1) with respect to temperature as follows:

where C₁=pηfΛ, C₂=εηfΛ, C₃=εpfΛ, C₄=εpηΛ, C₅=εpηf. Each factor with the superscript of left-hand diagonal (${\alpha }_{\text{T}}^{\prime }(\epsilon )$, ${\alpha }_{\text{T}}^{\prime }(p)$, ${\alpha }_{\text{T}}^{\prime }(\eta )$, ${\alpha }_{\text{T}}^{\prime }(f)$, ${\alpha }_{\text{T}}^{\prime }(\Lambda )$) was estimated successively for two temperatures as follows:

where F denote ε, p, η, f, or Λ.

2.3 Calculation tool

The MCNP5 code was adopted to perform all the neutronic calculations including the criticality and reactivity coefficients. Specifically, the concept of a universe was used to model the hierarchical geometry for MSR. The tally multiplier card, FMn, was employed to calculate the cross sections and subsequently to calculate the factors in the six-factor formula. In order to perform accurate calculations with the uranium based fuels, an ACE (A Compact ENDF) format cross section library with continuous energy was chosen based on the ENDF/B-VII library. In order to improve the accuracy of the results, each criticality calculation was scheduled to skip 50 cycles and run a total of 200 cycles with nominally one million neutrons per cycle. The typical computing time of a criticality calculation was approximately 7 h with 16-point parallel computing.

3.1 Six-factor contributions to FSTC

As mentioned previously, the FSTC is expressed as the sum of the five components via the six-factor formula (Eq. (3)). Therefore, further research on the contributions (positive or negative) of the five components to FSTC is crucial for understanding how the fuel salt compositions affect FSTC for the variants of operation temperature. In order to quantitatively describe the neutron spectrum of a reactor, the energy of the average lethargy causing fission (EALF) was introduced, and it exhibited a higher value for a harder neutron spectrum and vice versa. Furthermore, the FSTC is a combination of the effect of the Doppler coefficient and density coefficient. Therefore, the contributions of the five components to the two effects were discussed in sequence as follows:

• The temperature coefficient α_T(ε) is significantly affected by the neutron spectrum. When the fuel salt temperature increases, the Doppler broadening leads to an increase in the resonance absorption and subsequently a decrease in the thermal neutron range, and thus the neutron spectrum becomes harder corresponding to a higher EALF. Subsequently, the fast fission increases, and thus a positive α_T(ε) is shown for the Doppler coefficient. Conversely, the increases in fuel salt temperature lead to a reduction in the fuel salt density and a softer neutron spectrum corresponding to a smaller EALF due to the growing graphite-to-heavy metal ratio. This causes a decreased fast fission and leads to a negative α_T(ε) for the density coefficient.

• The temperature coefficient α_T(p) is also significantly dependent on the neutron spectrum. As discussed in the previous paragraph, for the Doppler coefficient, an increase in fuel salt temperature indicates that the neutron spectrum becomes harder and leads to a reduction of the resonance escape probability, and subsequently a negative α_T(p) is shown. Conversely, for the density coefficient, the sign of α_T(p) is positive.

• The temperature coefficient α_T(η) is mainly related to the fission reaction and the capture reaction of heavy metals. With respect to the uranium based fuel, η is expressed as follows:

where ν denotes the average number of neutrons released per fission, ∑_f,th denotes the macroscopic thermal fission cross section, ∑_a,th denotes the macroscopic thermal absorption cross section, and the subscripts 5 and 8 denote ²³⁵U and ²³⁸U, respectively. With the increases in fuel salt temperature, the Doppler broadening effect makes the neutron spectrum harder and weakens the thermal fission cross section and the absorption cross section. The variation in the thermal fission cross section is typically faster than that of the thermal absorption cross section mainly due to the fertile nuclide (²³⁸U for the uranium based fuel, see Fig. 2). Thus, a reduction in η is introduced and a negative α_T(η) is displayed for the Doppler coefficient. Conversely, the increases in fuel salt temperature reduce the fuel salt density and increase the C/HM (the atom ratio of graphite and heavy metals). Subsequently, the Maxwell’s spectrum becomes wider and its peak increases, and thus the neutron spectrum becomes softer. In this case, the fission cross sections increase faster than the absorption cross sections (Fig. 2). Therefore, a higher η is introduced and a positive α_T(η) is exhibited for the density coefficient.

Figure 2:Full-screen View

(Color online) The cross sections of ²³⁵U and ²³⁸U.

• The temperature coefficient α_T(f) depends on the effect of the atomic density of fuel salt and graphite over that of neutron spectrum. With respect to a graphite-moderated MSR, f was calculated as follows:

where the numerator denotes the thermal absorption of heavy metals (HM) while the denominator denotes the thermal absorption in total including the heavy metals, FLiBe salt, and graphite (labeled C). When the fuel salt temperature increases, the Doppler coefficient makes the neutron spectrum harder and weakens the thermal absorption for all elements while the density coefficient leads to an opposite tendency. The relative change in the thermal absorption of heavy metals is comparable to that of FLiBe, and thus the variation in f is mainly dependent on the relative variation in graphite and heavy metals. The calculations show that the Doppler coefficient causes the thermal absorption of heavy metals to decrease faster than graphite while the density coefficient causes the thermal absorption of graphite to increase faster than heavy metals. Subsequently, a reduction in f is introduced and displays a negative α_T(f) either for the Doppler coefficient or the density coefficient.

• The temperature coefficient α_T(Λ) is closely related to neutron diffusion. With respect to the Doppler coefficient, when the fuel salt temperature increases, the widening neutron spectrum enhances the resonance absorption. Hence, the slowing down time of fast neutrons increases and leads to a decrease in the fast non-leakage probability (P_FNL). Furthermore, for the Density coefficient, the reduction in the fuel salt density weakens the collision probability between thermal neutrons and heavy metals, and thus the thermal neutron diffusion length increases and decreases the thermal non-leakage probability (P_TNL). Therefore, the total non-leakage probability (Λ) decreases and a negative α_T(Λ) is shown for both the Doppler coefficient and the density coefficient.

Furthermore, the main contribution to Λ corresponds to P_TNL due to the significantly longer diffusion time relative to the slowing-down time. An increase in the softness of the neutron spectrum increases the significance of the variation in k_eff with a small change in Λ, and thus a stronger α_T(Λ) is introduced. In an under-moderated core of MSR, when the ²³⁵U enrichment and/or HM changes, the neutron spectrum is still relatively hard, and subsequently the change in α_T(Λ) is insignificant when compared with that of α_T(p) and α_T(ε) and is not shown in Section 3.2 and Section 3.3. However, the neutron spectrum tends to be softer with the variation in ⁷Li enrichment, and the contribution of α_T(Λ) to FSTC should be increasingly focused and is discussed in Section 3.4.

Based on the above discussions, Table 1 shows the sign of the five components. Here, "+" and "-" denote a positive and negative temperature coefficient, respectively. With respect to the Doppler coefficient, the sign of α_T(ε) is positive while the other four are negative. With respect to the density coefficient, the sign of α_T(p) and α_T(η) are positive while the other three are negative.

3.2 Effect of ²³⁵U enrichment on FSTC

With respect to the uranium based fuel, the ²³⁵U enrichment impacts the neutron spectrum and subsequently the microcosmic and/or macroscopic cross section of heavy metals. Fig. 3 shows the trends of EALF with ²³⁵U enrichment for different HM proportions. As shown in the figure, an increase in the HM proportion and/or an increase in ²³⁵U enrichment increases the EALF corresponding to a harder spectrum. Furthermore, the value of EALF is more sensitive to the ²³⁵U enrichment for a higher HM proportion due to its harder spectrum.

Figure 3:Full-screen View

(Color online) EALF as a function of ²³⁵U enrichment for different HM proportions.

The variation in the FSTC with the ²³⁵U enrichment for different HM proportion is shown in Fig. 4. The FSTC is always negative for all ²³⁵U enrichments and HM proportions. With respect to a fixed HM proportion, when the ²³⁵U enrichment increases, the magnitude of FSTC initially increases and subsequently decreases, and the turning point corresponds to a higher ²³⁵U enrichment for a lower HM proportion. It should be noted that the difference in the FSTC between any two HM proportions is almost independent of the ²³⁵U enrichment after the turning point.

Figure 4:Full-screen View

(Color online) FSTC as a function of ²³⁵U enrichment for different HM proportions.

In order to further understand the above phenomena, Fig. 5 shows the variations in FSTC and its two separate effects with ²³⁵U enrichment for HM=4 mole% as an example. First, the Doppler coefficient is always negative for any ²³⁵U enrichment. When the ²³⁵U enrichment increases, the magnitude of the Doppler coefficient initially increases and subsequently becomes almost saturated due to the gradually hardening spectrum that is more favorable to the resonances of ²³⁸U (main resonances located between 6.7 eV and 20 keV while those of ²³⁵U are between 4.8 eV and 2.2 keV). Second, when the ²³⁵U enrichment increases, the density coefficient changes from negative to positive. In an under-moderated core, a decrease in fuel salt density allows for increased moderation, and thus softens the neutron spectrum that leads to a higher fission rate. Subsequently, a positive density coefficient is typically observed. Furthermore, a negative density coefficient at low enrichment possibly implies that the core is near over-moderation. In this case, a decrease in the fuel salt density can lead to increased parasitic absorptions in the graphite that decrease the fission rate, thereby introducing a negative coefficient. Thus, a combination of the above two effects initially strengthens the FSTC and subsequently weakens it when the ²³⁵U enrichment increases.

Figure 5:Full-screen View

(Color online) FSTC, Doppler and Density coefficient as a function of ²³⁵U enrichment for HM=4 mole%.

Based on Eq. (4), the variations in the five components for FSTC with the ²³⁵U enrichment are shown in Fig. 6. The results indicate that the sign of α_T(ε) is negative, and its magnitude increases when the ²³⁵U enrichment increases. Furthermore, the sign of α_T(p) changes from less negative to more positive. Additionally, the signs of α_T(f) and α_T(Λ) are negative and they both weaken with increase in the ²³⁵U enrichment. Although their variations with respect to the ²³⁵U enrichment are significantly less than those of α_T(ε) and α_T(p), their magnitudes cannot be neglected. Moreover, the magnitude of α_T(η) is almost zero for all the ²³⁵U enrichment. Therefore, as a whole, the magnitude of FSTC (black square in Fig. 5) initially mainly increases due to α_T(ε) and subsequently mainly decreases due to α_T(p).

Figure 6:Full-screen View

(Color online) Variations in α_T(ε) and α_T(p) for FSTC with ²³⁵U enrichment.

Furthermore, the values of α_T(ε) and α_T(p) for FSTC are separated into the Doppler and density contributions as shown in Fig. 7. Thus, when the ²³⁵U enrichment increases, the absolute value of α_T(ε) and α_T(p) increases for both the Doppler coefficient and the density coefficient due to the hardening neutron spectrum. As interpreted in section 3.1 (Table 1), the sign of α_T(ε) for the Doppler coefficient and that of α_T(p) for the density coefficient are positive while that of α_T(ε) for the density coefficient and that of α_T(p) for the Doppler coefficient are negative. In conjunction with Fig. 6 and Fig. 7, the magnitude and the sign of α_T(ε) and α_T(p) for FSTC that varies with the ²³⁵U enrichment is primarily due to the density coefficient.

Figure 7:Full-screen View

(Color online) Variations in α_T(ε) and α_T(p) for Doppler and Density with ²³⁵U enrichment.

In summary, the effect of the ²³⁵U enrichment on FSTC is mainly dependent on the density coefficient especially for its α_T(ε) and α_T(p). In order to obtain an sufficiently negative FSTC, a lower ²³⁵U enrichment is recommended to suppress the positive α_T(p) for the density coefficient.

3.3 Effect of HM proportion on FSTC

With respect to a liquid-fueled MSR, the proportion of heavy metals (HM proportion) in the fuel salt phase diagram is closely related with the neutron spectrum. A change in the HM proportion, accordingly, shifts the neutron spectrum, and subsequently, a high variation in the microscopic cross sections of heavy metals is introduced that alters the FSTC. In order to understand the effects of the HM proportion on FSTC, the ²³⁵U enrichment and the ⁷Li enrichment are set as constants with values of 19.75% and 99.95%, respectively.

The variations in FSTC and its two separate effects with the HM proportion are shown in Fig. 8. When the HM proportion increases, the Doppler coefficient strengthens and always remains negative while the density coefficient varies from less negative (nearly zero) to more positive. With respect to the aforementioned two effects, the FSTC is mainly attributed to the effect of the density coefficient, and its value is negative and weakens with increases in the HM proportion.

Figure 8:Full-screen View

(Color online) FSTC, Doppler, and Density coefficient as a function of the HM proportion with ²³⁵U enrichment of 19.75% and ⁷Li enrichment of 99.95%.

Fig. 9 shows the variations in α_T(ε) and α_T(p) for FSTC, Doppler coefficient, and the density coefficient with increases in HM proportion with a constant ²³⁵U enrichment. As shown in the left figure, the signs of α_T(ε) and α_T(p) are opposite, and their magnitudes increase when the HM proportion increases. The other three factors (α_T(η), α_T(f), and α_T(Λ)) for FSTC are not shown due to their significantly lower sensitivity to the HM proportions. Furthermore, as shown in the right figure, the signs of α_T(ε) and α_T(p) for the two separate effects are consistent with those in Table 1. In a manner similar to Fig. 7, the magnitudes of α_T(ε) and α_T(p) for both the Doppler coefficient and the density coefficient increase when the HM proportion increases because the neutron spectrum becomes harder. Given the above considerations, α_T(ε) and α_T(p) for FSTC are mainly affected by the density coefficient.

Figure 9:Full-screen View

(Color online) Variations in α_T(ε) and α_T(p) with the HM proportion for FSTC (left) and its two separate coefficients (right) with fixed ²³⁵U enrichment.

In summary, recommendations include lowering HM proportion to suppress the possible density coefficient and to achieve an sufficiently negative FSTC. Additionally, it is important to note that lowering the HM proportion is conducive to oxygen control although it results in a higher melting point for the fuel salt mixture.

3.4 Effect of ⁷Li enrichment on FSTC

The carrier salt FLiBe exhibits good compatibility with structure material, including nuclear grade graphite and hastelloy. This is verified by the molten-salt reactor experiment (MSRE) [24]. Enriched ⁷Li is recommended given its smaller thermal neutron absorption cross section that is important in improving the neutron economy and suppressing the undesired tritium production [25]. Furthermore, the enrichment of ⁷Li is highly correlated with neutronic properties such as the neutron spectrum, the critical ²³⁵U enrichment, and FSTC. The ²³⁵U enrichment used in this section is less than 10 wt% to achieve criticality for all the HM proportions.

The neutron spectra of two types of ⁷Li enrichment for different HM proportion are quantified as EALF and shown in Fig. 10. The results indicate that a higher ⁷Li enrichment exhibits a smaller EALF corresponding to a softer neutron spectrum due to reduced thermal absorption from ⁶Li.

Figure 10:Full-screen View

(Color online) Variation in EALF with the HM proportion for two ⁷Li enrichments.

Fig. 11 shows the effects of ⁷Li enrichment on FSTC (left) and its two separate effects (right) with the HM proportion. As shown in the left figure, a higher ⁷Li enrichment corresponds to a softer spectrum, and thus even a small change in the ⁷Li enrichment can cause a significant variation in k_eff, and subsequently, a more negative FSTC. Furthermore, an increase in the ⁷Li enrichment makes the variation of FSTC between different HM proportions increasingly significant. As shown in the right figure, the results indicate that a higher ⁷Li enrichment leads to a more negative Doppler coefficient for all HM proportions and more negative and less positive density coefficients for the lower HM proportions and higher HM proportions, respectively. Furthermore, a decrease in the HM proportion increases the proportion of FLiBe, and this increases the difference between FSTC for the two ⁷Li enrichments, which is mainly due to the density coefficient.

Figure 11:Full-screen View

(Color online) Variation in FSTC (left) and its two separate effects (right) with the HM proportion for different ⁷Li enrichment.

In order to further understand the effect of the ⁷Li enrichment on FSTC, Fig. 12 shows the variations in α_T(f), α_T(Λ), α_T(p) and α_T(ε) along with the HM proportion for the two ⁷Li enrichments. The results indicate that for all the HM proportions, a higher ⁷Li enrichment leads to a more negative α_T(f), α_T(Λ) and α_T(p) especially for the former two cases due to their more sensitive dependence on the shift of neutron spectrum. In parallel, a lower ⁷Li enrichment leads to a more negative α_T(ε) benefited from a harder spectrum. It should be noted that although the variation in α_T(η) along with the HM proportion exhibits a similar trend as α_T(f) for two ⁷Li enrichments, its value is almost excessively low to be considered. In summary, enhancements in ⁷Li enrichment are recommended to gain a more negative FSTC.

Figure 12:Full-screen View

(Color online) Variations in α_T(f), α_T(Λ), α_T(p) and α_T(ε) with the HM proportion for different ⁷Li enrichment values.