1 Introduction

As one of the six Gen-IV reactor types, the molten salt reactor (MSR) was first investigated in the 1950s to as a method to provide a compact high-temperature power plant for nuclear powered aircraft [1]. Subsequently, an aircraft reactor experiment was constructed at the Oak Ridge National Library, USA, in which the nuclear feasibility of operating a molten salt-fueled reactor at high temperature was demonstrated [2]. After discovering the high potential of the MSR for achieving low electric-power-generating costs in central power plants, a molten salt reactor experiment (MSRE) was performed successfully [3]. In addition, a critical experiment device (hereinafter the device) was established at the Shanghai Institute of Applied Physics (originally known as the Shanghai Institute of Nuclear Research), China, in the 1970s. Its main purpose was to reveal the physics of MSRs and thorium–uranium cycles. Although studies regarding the MSR has not been interrupted, an actual reactor has not been built [4,5,6].

After rediscovering that MSRs offer unique advantages in terms of inherent safety, economy, sustainable development of nuclear fuel, and nonproliferation, studies regarding MSRs has re-emerged and progressed since the 21st century [7,8,9]. To satisfy national strategic requirements in the field of nuclear energy, the Chinese Academy of Sciences launched a strategic and pioneering science and technology project known as “Future Advanced Fission Energy-Thorium-based Molten Salt Reactor Nuclear Energy System” in 2011. Subsequently, the Thorium Molten Salt Reactor (TMSR) Center was established; it is dedicated to the research and development of Gen-IV advanced fission reactor nuclear energy systems for achieving strategic goals such as diversifying nuclear fuel, preventing nuclear proliferation, and minimizing nuclear waste. The preliminary target of the project was to build a 2 MW thorium-based MSR [10, 11].

MSRs are a type of thermal neutron reactor that involves a molten salt mixture fuel cycle and is moderated by graphite to produce fission energy. Molten salt mixture fuel can be the fluoride of uranium, thorium, zirconium, etc., and is used as both a nuclear fuel and heat carrier under a high-temperature liquid without requiring solid fuel elements [12]. In general, the physical characteristics of the core have been investigated by varying the volume ratio between the moderator and the nuclear fuel in thermal reactors, such as pressurized water reactors [13, 14]. For an MSR, its lattice cell is a component of the traditional reactor. The cells are typically columnar graphite blocks, and the side length increases with the core size. For example, the side length of the regular quadrilateral graphite lattice cell used in the MSRE was 5 cm, whereas the side length of the hexagonal graphite lattice cell of the molten salt breeder reactor (MSBR) was 15 cm [15,16,17,18]. Despite two graphite cells having the same geometric shape and the same graphite-to-fuel salt volume ratio, the different sizes of the cells will change the neutron physical properties. The geometry of liquid nuclear fuel in MSRs is flexible, allowing MSRs to have fuel cells of various shapes. In addition, because the coefficient of thermal expansion between graphite and alloy differ significantly, the change in the core geometry owing to high temperature should not only be attributed to the thermal expansion of each component, but also the graphite cell displacement. New fuel salt channels can be formed between the graphite cells after the latter are displaced, and their geometric shapes differ from that of the original fuel salt channel [19]. Therefore, the neutron physical properties of different graphite cell sizes and fuel salt channel shapes must be investigated.

Based on the Monte–Carlo N particle transport code (MCNP), several types of graphite cell models of fuel salt channels with different sizes and geometric shapes are presented herein [20, 21]. In this study, the effects of the size and geometric shapes of the fuel salt channel on the infinite multiplication factor k∞ and neutron energy spectrum of the cell were investigated. In addition, the variation law of k∞ with the average chord length of the fuel salt channel under different fuel salt volume ratios and heavy metal molar ratios was calculated and analyzed. Finally, some practical application examples of this study are presented, and they can be used as a reference for the design of MSRs.

2 Models and parameters

MSRs are manufactured in many forms; however, liquid fuel salts typically flow through cores made of graphite with a critical geometry. Heat is produced by the fission of the fuel salt and then dissipated by the flowing fuel salt. Fuel is concentrated into blocks in an inhomogeneous reactor, such as cylinders, circles, spheres, and sheets (planks), which are placed in a moderator of a specific geometric form to form the core of the lattice structure. The elementary units that compose a lattice structure are generally known as cells [22]. The fuel salt channel in an MSR corresponds to the fuel block in the traditional reactor. For this study, only the variation trend of the k∞ of the cell was considered in this study. Subsequently, the neutron physics law of the corresponding cell type was obtained via k∞ value analysis. Therefore, the entire core need not be modeled. k∞ is related to the size and shape of the fuel salt channel, and the effects of the size and shape of the fuel salt channel on k∞ can be constrained by the average chord length of the fuel salt channel L. As shown in Formula (1), the average chord length of the fuel salt channel is defined as four times the volume of the fuel salt channel V divided by the surface area of the fuel salt channel S [22].

$$L{ = }\frac{4V}{S}$$
(1)

As shown in Fig. 1, several basic fuel salt channel models with different shapes were established using MCNP6 for comparative analysis in this study, including cylindrical, annular, spherical, and flaky [20, 21]. The central region of each cell in Fig. 1 represents the fuel salt and the surrounding region represents graphite. Typically, a cylindrical shape is selected for the fuel channel of the MSR, e.g., the MSBR for breeding purposes. The well-known fuel salt channel of the MSRE is the runway type and can be regarded as a combination of cylinders and flakes. After the displacement of the graphite cell, the new fuel salt channel formed between cells is generally flake shaped. Circular and spherical fuel salt channels were selected for result comparison. It is noteworthy that the fuel salt can flow up and down by gouging out a channel above and below the sphere. The cell is segmented into a fuel salt zone and a graphite moderator zone. For example, in the cylindrical fuel salt channel model, the fuel salt zone is the cylindrical zone, and the graphite moderator zone is located outside the cylinder. Key parameters such as the fuel salt composition and density used by the cell are shown in Table 1. The fuel salt composition is consistent with the MSRE; therefore, the trend of density change with temperature can be obtained from MSRE technical report data [3]. The density of the fuel salt was 2.34 g/cm3 at 550 °C. T220-graphite with a density of 1.86 g/cm3 at room temperature and a linear thermal expansion coefficient of 3.49 × 10−6/ °C at room temperature to 550 °C were used in this study [3]. The average temperature of the cell was 550 °C, which was used in the calculation and analysis of this study. Nuclear data library ENDF/B-VII.0 was used for the calculation performed in this study, and its applicability and advantages to MSRs have been verified by Liu Yafen et al. in 2018 [5, 23]. For result comparison, all the calculations were performed under the same condition, i.e., the number of source neutrons selected for each cycle were 100, 000, with 1050 active cycles and 50 abandoned cycles, resulting in a standard deviation of approximately 0.00005.

Fig. 1
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(Color online) Schematic diagrams of computational models

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Table 1 Parameters of graphite cell model
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Even if the fuel salt fraction and the fuel volume fraction (FVF) are the same, the k∞ of the cell is still related to the cell arrangement, an aspect not considered herein. When changing the average chord length, only a relatively convenient method was adopted in this study, and the difference in k∞ caused by other methods was not considered. The method is described in detail as follows: the average chord lengths of the cylindrical and spherical fuel salt channels were changed by changing the channel size. The average chord length of the circular fuel salt channel was changed by changing the outer diameter of the circular fuel salt channel. In addition, the average chord length of the flaky fuel salt channel was changed by changing the thickness of the flaky fuel salt channel. The FVF of the cell was maintained constant when the average chord length was changed in the cases above. First, the effects of the size and geometric shapes of the fuel salt channel on k∞ and the neutron energy spectrum of the cell was investigated. Subsequently, the variation law of k∞ with the average chord length of the fuel salt channel under different FVF values and heavy metal molar ratios was calculated and analyzed. Finally, some practical application examples of this study are presented.

3 Results and discussion

3.1 Effect of fuel salt channel geometry

As described in the previous section, the fuel salt channel geometric shapes considered in this section include cylindrical, circular, spherical, and flaky. The effects of different geometric shapes on k∞ and the neutron energy spectrum of the cell are analyzed in this section. In addition, the physical causes of those effects were analyzed along with the calculation results. Table 2 shows the graphite cell side length of the fuel salt channel with different geometric shapes under the same average chord length. The data in the table are important input parameters for the subsequent calculation in this study. It can be assumed that under the same average chord length, the graphite cell side length of the fuel salt channel with different shapes differs significantly.

Table 2 Side length of cell with different fuel salt channel geometric shapes
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Figure 2 shows the variation of k∞ with the average chord length of fuel salt channel under different fuel salt channel geometric shapes. As shown in the figure, when the average chord length was relatively small (L <7.5 cm), the k∞ of the fuel salt channel with different geometric shapes agreed well. This was because when the same L value was less than 7.5 cm, the energy spectrum of the fuel region under different geometric shapes was similar. As the average chord length increased to 7.5 cm, the difference in k∞ of the fuel salt channel with different geometric shapes began to be significant. In addition, in the relatively large range (7.5–40 cm) of the average chord length, the k∞ of the cylindrical and spherical fuel salt channel was greater than that of the flaky or circular fuel salt channel. By contrast, the k∞ of the flaky and circular fuel salt channels were similar. Meanwhile, the k∞ of the cylindrical fuel salt channel was similar to that of the spherical fuel salt channel.

Fig. 2
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(Color online) k∞ at different chord lengths for different fuel salt channel shapes

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Figure 3 shows the normalized neutron flux distributions of the cell (left) and fuel salt zone (right) with different geometric shapes. The FVF of each cell was 15%, and the average chord length was set to 16 cm. As shown from the figure, the neutron spectra of the cells with circular and flaky fuel salt channel shapes were consistent. In addition, the neutron spectrum of the cell with a cylindrical fuel salt channel was similar to that of a cell with a spherical fuel salt channel. As shown from the normalized neutron flux distribution in the cell (the left of Fig. 3), the neutron spectra of the cells with flaky and circular fuel salt channels were relatively thermal, and the thermal neutron utilization factor was small. Furthermore, the normalized neutron flux distribution in the fuel salt (the right of Fig. 3) indicates that the neutron spectra of the flaky and circular fuel salt channels were relatively hard, and the neutron share in the resonance absorption region increased correspondingly. Therefore, in the interval with a relatively large average chord length, the variation in k∞ appeared as shown in Fig. 2.

Fig. 3
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(Color online) Neutron spectrum of cell (left) and fuel salt zone (right) for different fuel salt channel shape with L = 16 cm

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Through the calculation and analysis above, it was discovered that although the FVF and fuel concentration (FC) of the cell were the same, the geometric shapes of the fuel and salt channels differed. This affected the neutron spectrum significantly, resulting in a significant discrepancy in k∞. The average chord length had a certain constraint on k∞. When the average chord length was relatively small (less than 10 cm), the k∞ of the fuel salt channel with different geometric shapes agreed well. In the interval with a relatively large average chord length (greater than 10 cm), the k∞ variation trends of the fuel salt channel with different geometric shapes were the same, but the specific k∞ values differed significantly. The next section provides the detailed reasons for these significant k∞ differences.

3.2 Effect of fuel salt channel size

A graphite cell with a circular fuel salt channel was selected for calculation and analysis in this study. Based on the material composition of the cell and an FVF value of 15%, the cell size was changed to calculate the variation in k∞. The variation curve of this value is shown in Table 3, and the reason for the change based on the four factors and the neutron energy spectrum of the cell is emphasized in this section. The method to calculate the four factors has been provided by Sun Jianyou et al. in 2014 [24, 25]. The results of these factors are also listed in Table 3, where ε is the fast neutron multiplication factor, p is the probability of escape resonance absorption, η is the effective number of fission neutrons, and f is the thermal utilization factor.

Table 3 Variations of four factors with radius of fuel salt channel
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As shown in Table 3, as the fuel salt channel radius increased, the k∞ value first increased and then decreased. The variation of the four factors is summarized as follows:

  • ε first decreased and then increased;

  • p first increased and then decreased;

  • η was unaffected by the change in the fuel salt channel radius;

  • f decreased as the fuel salt channel radius increased.

The results indicate that the variation in k∞ was primarily caused by the escape resonance absorption probability p and thermal neutron utilization coefficient f. Although the fast neutron multiplication factor ε affected k∞, the amplitude of its variation was much smaller than that of the escape resonance absorption probability.

Figure 4 shows the normalized neutron flux distributions in the cell (left) and fuel salt zones (right) with different fuel salt channel radii, where R is the radius of the fuel salt channel. With the increase in the fuel salt channel radius, the neutron spectrum of the cell tended to be thermal. This was because as the fuel salt channel radius as well as the distance between the channels increased, the graphite area expanded, increasing the probability of the fast neutron in the graphite to decelerate to less than the resonance energy. In other words, the share of thermal neutron flux in graphite increased, resulting in a decrease in the thermal neutron utilization of the cell. Resonance neutrons were primarily produced in graphite moderators and then incident on the fuel salts. Because of the large resonance absorption cross-section of the nuclei in the fuel, the resonance neutrons were first absorbed by the nuclei on the surface of the fuel salt channel and served as a shield to the nuclei inside the fuel salt channel. ∆R denotes the resonant absorption surface of the fuel salt channel, and R is the radius of the fuel salt channel. An increase in R will not cause a change in ∆R, and the area where neutrons are not absorbed by resonance will increase. In other words, the spatial self-screen effect becomes more prominent, whereas the resonance absorption decreases and the probability of escaping resonance absorption increases.

Fig. 4
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(Color online) Neutron spectrum of cell (left) and fuel salt zone (right) under different fuel salt channel radii

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However, as shown from the right side of Fig. 4, the neutron flux distribution of the fuel salt zone shows that the spectrum became harder as the radius of the fuel salt channel increased. In particular, the spectrum exhibited a clear hardening trend when the radius of the fuel salt channel exceeded 8 cm. This was primarily because the fuel salt contained moderated materials, such as lithium and beryllium. When the radius of the fuel salt channel increased, the probability of neutron reaction in the fuel salt increased; consequently, the probability of neutrons decelerating to the resonance region in the fuel salt increased. In other words, as the resonance absorption of the neutron increased, the probability of escaping from the resonance absorption decreased.

Based on the analysis above, when the FVF of the cell with a cylindrical fuel salt channel was set to 15%, the probability of escaping from the resonance absorption first increased and then decreased with the increase in the fuel salt channel radius. The thermal neutron utilization coefficient always decreased as the channel radius increased. Therefore, the overall effect was that as the fuel salt channel radius increased, k∞ first increased and then decreased. It is noteworthy that the fuel salt can in decelerate neutrons. Therefore, resonance neutrons were also produced in the fuel salt region, thereby affecting the resonance absorption of the cell.

3.3 Effects of FVF and FC

The moderator-to-fuel volume ratio and the composition of the nuclear fuel directly affected the physical properties of the core. The variation law of the k∞ value of the cell was obtained based on the fuel salt volume ratio and fuel salt heavy metal molar ratio. This section focuses on the analysis of the variation law of k∞ with the average chord length under different FVFs and FCs. The FVF value of the cell was changed by adjusting the size of the graphite portion, whereas the size of the fuel salt channel remained unchanged. The FVFs were set to 5%, 15%, and 30%. The effect of the FC on k∞ was also investigated, and the FCs were set to 1.55%, 4%, and 8%.

Figure 5 shows the k∞ variation of a cell with a cylindrical fuel salt channel under different FVFs and FCs with changes in the fuel salt channel radius. Although k∞ was significantly affected by the FVF, under different FVFs, it first increased and then decreased with the change in the fuel salt channel radius. In the case of FVF = 5%, when the fuel channel radius was less than 1.5 cm, k∞ increased with the fuel channel radius; k∞ decreased as the channel radius increased when the fuel channel radius was larger than 1.5 cm. In other words, k∞ was maximized at a fuel channel radius of 1.5 cm. Furthermore, k∞ was maximized at fuel channel radii of 4.5 and 8 cm in cases where FVF = 15% and 30%, respectively. When FC = 1.55% and the fuel channel radius was less than 4 cm, k∞ increased with the fuel channel radius. Furthermore, k∞ decreased as the channel radius increased when the fuel channel radius exceeded 4 cm. Additionally, k∞ was maximized at a fuel channel radius of 7.5 cm when FC = 4% and 8%. The k∞ variation trend of the fuel channel radius remained unchanged, and only the radius of the fuel salt channel corresponding to the maximum k∞ value changed. This was primarily due to the different degrees of neutron decelerating under different FVFs and FCs, resulting in different neutron energy spectra.

Fig. 5
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(Color online) Effect of fuel salt channel radius on k∞ for different FVFs (left) and FCs (right)

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Figures 6 and 7 show the variation in k∞ of the fuel salt channel with the average chord length of the fuel salt channel under different FVFs and FCs, respectively. PZ, Q, YZ, and YH represent the flaky, spherical, cylindrical, and circular fuel salt channels, respectively. It was observed that the variation law of k∞ with the average chord length was consistent under different FCs for the four fuel salt channels with different geometric shapes. In addition, the variation law of k∞ with the average chord length was consistent under different FVFs except for the case of a circular fuel salt channel with an FVF of 5%. Figure 8 shows the normalized neutron spectrum in the cell of the circular fuel salt channel. In the region where FVF was low and the average chord length was large, owing to the increase in the fast neutron multiplication coefficient, the spectrum became harder with the increase in the average chord length. The thermal neutron utilization coefficient increased correspondingly. Therefore, k∞ tended to increase again.

Fig. 6
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(Color online) Effect of average chord length on k∞ for different FVFs

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Fig. 7
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(Color online) Effect of average chord length on k∞ for different FCs

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Fig. 8
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Normalized neutron spectrum of circular cell

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Similarly, for the fuel salt channel cells with the same FVF and FC, the k∞ of the fuel salt channels with different geometric shapes agreed well in the interval with an average chord length of less than 10 cm. The k∞ of the fuel salt channel with different geometric shapes differed significantly in the range of the average chord length greater than 10 cm. Moreover, the variation laws of k∞ with the average chord length of the cells with flaky and circular fuel salt channels were similar, whereas those of the cells with cylindrical and spherical fuel salt channels were similar. In summary, although the FVFs and FCs differed, the variation law of the k∞ of the cell with the average chord length was universal.

4 Application examples

In the research and design of actual MSRs, the appropriate size and shape of the grid element must be selected. MSR cores can be large or small, and large graphite cells can be used for large cores to enhance the strength of the cell mechanical structure. Additionally, the characteristics of the MSR, namely its high-temperature state and the flow properties of the fuel salt, determine the geometric diversity of the fuel salt channel in the core. Therefore, investigations regarding the effects of different flow passage geometries on neutron physics can provide a reference for the selection of graphite cells.

4.1 Cell selection

The fuel assembly in the molten salt core is the graphite cell in the core. According to the calculation data in Table 3, when the FVF was fixed, i.e., for the same fuel salt load of the core, a maximum k∞ value occurred with the change in the graphite cell size. Therefore, for a specific FVF, the graphite cell size can be reasonably selected only if k∞ is considered in the optimal design of the cell.

As shown in Fig. 2, the fuel salt channel with different geometric shapes had the same k∞ for the same FVF when the average chord length was relatively small. As shown in Table 2, for the same average chord length, the graphite cell side length of the fuel salt channel with different geometric shapes varied significantly. Therefore, for the same k∞, the lengths of the graphite cells with flaky and circular fuel salt channels were much larger than those with cylinder and spherical fuel salt channels. The flaky or circular fuel salt channels can be used when larger graphite cells are required.

4.2 Effect of expansion displacement on cell

In an MSR, graphite is located on an alloy support plate. A significant difference exists in the expansion coefficient between the graphite and alloy, rendering the spacing between the graphite gate elements more conspicuous at high temperatures; hence, the graphite cell is displaced. Owing to the fluidity of the fuel salt, the gap between the graphite cells is filled with the fuel salt after the graphite cell displacement. Therefore, after the expansion and displacement of the graphite cells, not only did the FVF change, but also a new fuel salt channel was formed between the cells, and the geometric shape differed from that the original fuel salt channel. These changes will affect the physical properties of the core.

The reference fuel salt channel was assumed to be cylindrical and its FVF value 15%. The second row in Table 4 lists the k∞ values of this reference cell for different chord lengths. As the temperature increased from 550 to 700 °C (accident temperature), the graphite cells shifted owing to the difference in thermal expansion coefficient between graphite and Hastelloy-N. Areas that enlarged due to differences in the coefficient of thermal expansion were filled with the fuel salt. Consequently, the FVF increased from 15 to 15.4% and a flaky new fuel salt channel appeared around the graphite cell. Line 5 in Table 4 shows the deviations between the k∞ values of the cell after displacement and the corresponding reference k∞ values. To distinguish the effect of the FVF on the neutron physical properties of the cell, the fuel salt channel radius was fixed, and the side length of the graphite gate element was artificially reduced to obtain k∞ values by increasing the FVF to 15.4%. Line 4 shows the differences between these k∞ values and those of the corresponding cells after displacement. In summary, the difference shown in the third row in Table 4 reflects the effect of FVF increase on k∞. The difference in line 4 reflects the effects of different cell geometries (cylindrical and flaky) of a new fuel salt channel on k∞.

Table 4 Effects of different factors on k∞
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As shown by the comparison results in Table 4, the FVF changed and formed new fuel salt channels owing to the expansion displacement of the graphite cell, thereby affecting k∞. In addition, the effects of the two abovementioned factors on the k∞ of the cell changed under different cell sizes. Therefore, the effect of material expansion displacement should be considered in the core design. Alternatively, the cell can be bound to prevent the cell from moving with the support plate.

5 Conclusion

Several graphite cell models of fuel salt channels with different sizes and shapes were established in this study, and the effects of the size and shape of the fuel salt channel on k∞ and the neutron energy spectrum of the cell were investigated. Additionally, the variation laws of k∞ with the average chord length of the fuel salt channel under different fuel salt volume ratios and heavy metal molar ratios were calculated and analyzed. Some practical application examples of this study were illustrated toward the end. The results indicated that, owing to variations in the size and shape of the cell, the types and times of nuclear reactions between neutrons and the material in the cell from neutron generation to absorption changed, resulting in a change in the physical properties of the cell. The main conclusions were as follows:

  1. (1)

    Within a certain range of FVF and FC, k∞ first increased and then decreased as the average chord length increased. In addition, the difference in k∞ of the fuel salt channel with different geometric shapes increased with the average chord length.

  2. (2)

    The neutron spectrum in the cell became thermal, and the spectrum in the fuel salt region became hard as the average chord length increased. The neutron spectrum of the flaky fuel salt channel was similar to that of the circular fuel salt channel. Meanwhile, the neutron spectrum of the cylindrical fuel salt channel was similar to that of the spherical fuel salt channel. Therefore, the variation laws of k∞ with the average chord length of the cells with flaky and circular fuel salt channels were similar, whereas those of the cells with cylindrical and spherical fuel salt channels were similar.

  3. (3)

    For the fuel salt channel cells with the same FVF and FC, the k∞ of the fuel salt channels with different geometric shapes agreed well in the interval with an average chord length less than 10. The k∞ of the fuel salt channel with different geometric shapes differed significantly when the average chord length was greater than 10 cm.

  4. (4)

    The variation law of the k∞ of the cell with the average chord length was universal and independent of the FVF and FC.

The results of this study demonstrated the basic physical law of different geometrical fuel channel cells for MSRs and can provide a reference for selection, to some extent. The section toward the end of this article presents some application examples of this study. For the future, relevant studies regarding the hydraulic and thermal properties of cells with different shapes of fuel channels, e.g., the temperature distribution of the cell and the flow field distribution of the fuel channel, will be performed. Furthermore, the safety performance of the cores composed of these cells will be verified. Ultimately, the knowledge of neutron physics and thermal–hydraulic characteristics can provide a good foundation for the core selection optimization of MSRs. The results of this study provide a basis for future engineering applications.