Core Model

The core model adopted in this paper is a liquid-thorium-based MSR with a thermal neutron spectrum and rated thermal power of 150 MW [44]. The fuel salt comprises a fluoride-lithium-beryllium eutectic containing thorium and enriched uranium; its thermophysical parameters are provided in Table 1. Graphite serves as the moderator. The reactor comprises essential components such as the core active region, control rod assembly, graphite reflector, upper and lower plenums, downcomer annulus, internal support structures, and primary reactor vessel (Fig. 1). The core configuration features a hexagonal close-packed arrangement with an equivalent diameter of 3.0 m and active region height of 3.2 m, consisting of 241 uniformly distributed hexagonal graphite blocks, each with a side length of 18 cm. Each graphite block contains a central flow channel with a diameter of 6 cm, forming the primary path for fuel-salt circulation. The graphite array is encased in a graphite reflector 20 cm thick. The core barrel encloses the reflector assembly, whereas the downcomer annulus is formed between the core barrel and primary reactor vessel, which houses the entire reactor structure. The core’s principal design parameters are summarized in Table 1.

Fig. 1Full-screen View

(Color online) Cross (left) and longitudinal (right) section diagrams of the core

Six control-rod guide tubes are symmetrically located at the fourth radial ring of the graphite assemblies (denoted by “#” in Figure 1), ensuring physical isolation between the fuel salt and control rod assemblies while facilitating the vertical movement of the control rods with complete separation from the fuel salt. The control rod system comprises six individual rods, each serving a specific function. These rods are actuated using electric drive mechanisms, which facilitate precise positioning via electrical signal control. In pump coast-down scenarios, the emergency shutdown system is automatically activated when the core flow rates fall below predefined safety thresholds. However, reliance on electrical signal transmission in the control-rod actuation mechanism introduces potential failure modes, including signal loss during station blackout conditions or mechanical seizures of the control rods despite valid signals. Such failure mechanisms can compromise the capacity of the reactor to execute emergency shutdown after a pump failure [45].

Building on the preceding analysis, this paper introduces an innovative passive fluid-driven control rod (PFCR) design concept, specifically developed to mitigate the reactivity insertion risks during anticipated transient-without-scram scenarios in pump coast-down events. The PFCR system incorporates a flexible core deployment strategy based on the reactivity compensation requirements. For minimal interference with the active control system, PFCR units are positioned at the midpoints of the hexagonal ring segments (with active control rods located at the vertices of the hexagonal rings), as indicated by the “*” and “&” markers in Fig. 1. The deployment strategy for PFCR units is determined by specific reactivity control requirements. In scenarios with lower reactivity control demands, PFCR units are positioned radially outward from the core centerline with minimal population density, as exemplified by a single PFCR installation at the sixth radial ring position (indicated by one of the green “&” markers in Fig. 1). Conversely, for enhanced reactivity control requirements, PFCR units are positioned closer to the core centerline with an increased population density, as demonstrated by the installation of three PFCR units at the second radial ring position (denoted by blue “*” markers in Fig. 1).

Structure and working principles of fluid-driven control rods

The primary objective of the PFCR system is to compensate for the insertion of positive reactivity following a reduction in the core flow, thereby effectively preventing temperature excursions during pump seizure events. Based on the distinct operational principles, PFCR systems are categorized into two main types: FCR and SCR.

The FCR features a cylindrical configuration that comprises essential components—namely, a cladding, neutron absorber, weighting module, and gas plenum—vertically integrated into the molten salt circulation channel. The cladding isolates the molten salt, whereas the neutron absorber captures thermal neutrons. The weighting module facilitates precise density and center-of-gravity adjustments, whereas the gas plenum performs the dual functions of density modulation and fission gas containment. The material specifications include a nickel-based alloy (density: 8900 kg/m³) for the weighting module, Gd₂O₃ (density: 5870 kg/m³) for the neutron absorber, and helium gas (density: 0.17 kg/m³) in the plenum chamber. Precise density control is achieved by optimizing the volume ratios between the weighting module and the gas plenum, with the neutron absorber occupying 80% of the total rod length. The detailed FCR configuration is shown in Fig. 2(a). The mean density of the FCR is engineered to be lower than that of the molten salt, resulting in partial rod protrusion above the liquid level and a buoyant suspension within the fuel salt. FCR motion is governed by the equilibrium between the gravitational, buoyant, and hydrodynamic forces. Increased core flow enhances hydrodynamic forces, which induce upward rod movement and reduce the buoyancy until a dynamic equilibrium is established, thereby stabilizing the FCR at a specific elevation. Conversely, reduced core flow results in a downward movement toward a new equilibrium position.

Fig. 2Full-screen View

(Color online) Schematic diagram of the FCR (a) and SCR (b) structures

As depicted in Fig. 2(b), the SCR shares similar components and functions with the FCR but features a truncated conical geometry characterized by a wider upper section and narrower base. Correspondingly, the flow channel is designed as a vertically oriented, upwardly expanding conical graphite conduit. The mean density of the SCR exceeds that of the molten salt, resulting in complete submersion and suspension within the fuel salt. Similar to the FCR, SCR operation relies on the equilibrium between gravitational, buoyant, and hydrodynamic forces. Fuel-salt circulation through the conical conduit establishes a pressure differential across the upper and lower surfaces of the SCR. This generates hydrodynamic forces. When these hydrodynamic forces exceed the resultant of gravitational and buoyant forces, the upward rod movement commences. The upward SCR movement increases the annular flow area between the rod and conical conduit, thereby reducing the fuel-salt velocity and consequently decreasing the hydrodynamic forces until a new equilibrium position is established at a specific elevation. The SCR position exhibits a positive correlation with the core flow rate: an increased core flow induces upward SCR movement, whereas a decreased flow results in a corresponding downward displacement.

The motion of the FCR is governed by the dynamic equilibrium between the buoyancy and hydrodynamic forces. As the FCR ascends, the volume of the rod exposed above the liquid surface increases, which in turn reduces its buoyancy. This reduction in buoyancy necessitates greater hydrodynamic forces to preserve the equilibrium, thereby requiring a significant force for effective FCR actuation. Furthermore, heightened hydrodynamic forces generate larger pressure differentials across the FCR, which reduces the flow allocation within its channel. This effect may potentially cause flow obstruction, thereby severely compromising the thermal–hydraulic performance of the channel.

In contrast, the SCR remains fully submerged in the molten salt, maintaining consistent gravitational, buoyant, and hydrodynamic forces. The magnitude of the hydrodynamic force can be regulated precisely by optimizing the differential between the gravitational and buoyant forces. Consequently, the SCR can be actuated with lower hydrodynamic forces, thereby minimizing its impact on the flow distribution in the channel. Moreover, the displacement of the SCR is determined by the height-dependent variation in the surrounding annular flow area. This characteristic enables the precise calibration of the relationship between the rod position and flow rate via optimization of the SCR channel geometry, thereby facilitating an effective compensation for flow-induced reactivity perturbations. The inherent design characteristics of the SCR render it particularly well-suited for mitigating flow-induced reactivity oscillations, thereby demonstrating its superior engineering applicability.

Under static fuel-salt conditions, the SCR maintains a stable equilibrium at the base of the conical conduit because its mean density exceeds that of the molten salt. During fuel-salt circulation, increased core flow rates exacerbate DNP loss, consequently reducing the core reactivity. Concurrently, elevated core flow rates induce the upward movement of the SCR, releasing sufficient reactivity to compensate for the reduction in reactivity induced by DNP loss. Under nominal core flow conditions, the SCR reaches equilibrium at its maximum elevation, with the released reactivity exactly offsetting the reactivity deficit caused by DNP loss. In the pump coast-down scenarios, when fuel-salt circulation terminates, the SCR descends to the base of the conical conduit, introducing a negative reactivity that precisely neutralizes the positive reactivity resulting from DNP accumulation in the core. Throughout its operational trajectory, the SCR remains within the active region of the core. Under normal operating conditions, the SCR is positioned at the top of the active region of the reactor core. As shown in the Fig. 3, the fast neutron flux (0.05–20 MeV) at the SCR location is approximately four times lower than that at the core center. Therefore, the neutron-absorbing material is expected to have significantly longer irradiation lifetime. In addition, the SCR does not bear mechanical loads, irradiation does not generate gases, and no internal pressure exists, resulting in minimal overall stress. Therefore, its service life should be comparable to that of the structural materials of the reactor, such as the upper support plate.

Fig. 3Full-screen View

(Color online) Distribution of fast neutron flux (0.05–20 MeV)

The SCR is characterized by a straightforward structural design that does not require complex external drive mechanisms or electrical systems. Consequently, its manufacturing cost is significantly lower than that of conventional electrically driven control rod systems, which often relies on high-reliability motors, signal transmission networks, and multiple redundant components. Furthermore, the SCR employs materials that are resistant to high temperatures, irradiation, and molten salt corrosion and adopts a fully mechanical structure without the requirement for precision components. This design enables reliable long-term operation of MSRs under extreme conditions, thereby significantly reducing the maintenance demands and extending the inspection intervals. As a fully passive system, the SCR can automatically insert control rods under extreme accident scenarios. such as complete power failure or signal loss. This significantly enhances the inherent safety of the reactor core, mitigates the risks of unplanned shutdowns, equipment damage, and personnel hazards, and indirectly reduces long-term operational and insurance expenses.

Existing MSR control rod systems isolate the control rods from the molten salt by installing thin-walled alloy sleeves in the active core region. This avoids radioactive sealing and isolation challenges for the drive mechanisms. This solution was implemented in the US MSRE [46] and China’s 2 MW TMSR-LF1 [47] reactors. However, for large-scale MSRs, this approach faces the risk of alloy sleeves experiencing helium embrittlement and rupturing under high-neutron-flux irradiation. This could result in the failure of the primary circuit boundary. In contrast, the SCR is in direct contact with the molten salt and does not require any drive mechanism. This effectively eliminates concerns related to alloy sleeve rupturing and radioactive isolation from the drive system. Moreover, the SCR is fully integrated within the active core region, thereby preventing control rod ejection accidents.

In summary, the SCR system offers not only outstanding economic advantages and reduced manufacturing and maintenance costs but also improved safety, which further supports a reduction in long-term operational and insurance expenditures.

SCR reactivity worth calculation

The neutronics calculations in this study were performed using OpenMC code. Because its development by the Computational Reactor Physics Group at the Massachusetts Institute of Technology in 2011, OpenMC has benefited from continuous enhancements and significant contributions from the open-source community. OpenMC is a Monte Carlo neutron transport code that provides essential capabilities, including fixed-source calculations, effective multiplication factor determination, and subcritical system analysis. The program features an extensive Python API that facilitates functionalities, such as input file generation, data post-processing, and result visualization [48].

Leveraging OpenMC, we developed a comprehensive computational core model to evaluate the effective multiplication factor (k_eff) of the SCR at various insertion depths under different design parameters—including geometric configurations, density, and deployment strategies. The core reactivity (ρ) was computed using Eq. (1), whereas SCR reactivity changes (Δρ) at various positions were determined using Eq. (2). Subsequently, the resulting data were processed using polynomial fitting to generate the SCR position-reactivity worth curve.(1)(2)where ρ(h) is the reactivity of the SCR at position height h, and is the reactivity of the SCR at the initial position.

DNP flow-induced reactivity

During MSR operation, fuel-salt circulation induces fluctuations in the effective delayed neutron fraction, which results in core reactivity oscillations. Accurately determining these flow-induced reactivity changes theoretically requires coupling the neutron transport with fluid dynamics simulations—a level of complexity that exceeds the scope of this study. Comprehensive methodologies for addressing this issue are detailed in Ref. [49]. In this investigation, a point kinetics model that neglects spatial effects is employed to approximate fuel flow-induced reactivity changes (Δρ_fluid) [50], with the associated estimation formulation expressed as(3)Among these parameters, β represents the effective delayed neutron fraction in the absence of fuel-salt flow in the core, βi denotes the fraction of delayed neutrons in each group, λi is the decay constant for each group of DNP, τ_c indicates the residence time of the fuel salt in the core, and τ_el represents the residence time of the fuel salt outside the core. The parameters τ_c and τ_el are closely related to the core flow rate: As the core flow rate increases, τ_c decreases, whereas τ_el increases. With a further increase in the core flow rate, the fuel salt is recycled back into the core, causing the rates of change of τ_c and τ_el to decrease until they eventually stabilize. The fractions and decay constants for each group of delayed neutrons are listed in Table 2 [44].

SCR Dynamic Equilibrium Modeling

The movement of the SCR is dictated by the interplay between gravitational, buoyant, and hydrodynamic forces, with its equilibrium position under varying core flow conditions as determined by their dynamic balance. Figure 4 presents a schematic of the force balance analysis for the SCR.

Fig. 4Full-screen View

(Color online) SCR force analysis

Based on the force analysis, the corresponding motion equation for the SCR is derived as follows:(4)where M_s represents the mass of the SCR, h is the height of the SCR’s top from the bottom of the conical pipeline, and t represents the time. G, F_b, and F_d denote the gravitational, buoyancy, and fluid dynamic forces acting on the SCR, respectively. These forces are calculated using the following relationships:(5)(6)Here, V_s, A_s, L_s, and D_s denote the SCR volume, upper base area, length, and upper base diameter, respectively; θ represents the conical taper angle; ρ_s indicates the SCR mean density; ρ_f signifies the fuel-salt density; and g is the gravitational acceleration. Δp_s denotes the flow-induced pressure differential across the SCR. As the SCR is embedded in the active region of the core—with only minor differences in the cross-sectional areas between its top and bottom—and under the assumption of a uniform core temperature distribution, the lift and acceleration pressure drops are neglected, accounting solely for the friction and local pressure drops. The friction and local pressure drops are computed using the following expressions:(7)(8)(9)where v denotes the mean flow velocity of the fuel salt through the SCR conduit. The parameters λ and ζ represent the friction factor and local loss coefficient, respectively. The friction factor λ is determined using the Reynolds number (, where D_e is the equivalent diameter of the pipe, and ρ, v, and μ denote the fluid density, velocity, and viscosity, respectively. As Δpλ is relatively minor, the influence of fluid viscosity variations on Δps is negligible. Consequently, the effect of temperature on the fuel-salt viscosity is disregarded in this paper, and the viscosity of the fuel salt at 650 °C, μ = 0.0072895 Pa·s, is used throughout the analysis.). It is also affected by the relative roughness ε (, where Δ, defined as the absolute roughness, represents the average height of the surface asperities; in this study, Δ = 10^-4 m). The local loss coefficient ζ is primarily affected by the geometric configuration of the conduit. In practice, the precise theoretical determination of λ and ζ is challenging, necessitating an estimation through experimental measurements or empirical correlations. Computational approaches for determining λ and ζ across various flow regimes are systematically summarized in Tables 3 and 4, respectively [51].

To determine the friction loss coefficient for a section with a noncircular cross-section, We multiply the friction loss coefficient for a circular pipe by the corresponding shape influence factor [52]:(10)where λ represents the friction loss coefficient for circular pipe sections under identical Reynolds numbers, λ_n denotes the friction loss coefficient for the non-circular pipe sections under the same Reynolds number, and k_n is the cross-sectional shape correction coefficient.

For laminar flow, the correction factor for a circular annular tube depends on the ratio of its inner to outer diameters [53]:(11)Here, d and D₀ denote the inner and outer diameters, respectively.

For turbulent flow, the friction loss coefficient λ_n for a circular annular tube can be calculated using the following equation [54]:(12)By combining Eqs. (4) and (5), at the steady state (i.e., when ), the pressure drop acting across the SCR remains constant, that is,(13)Furthermore, combining Eqs. (9) and (13) yields(14)Equation (14) indicates that when the SCR reaches a steady equilibrium, the average flow velocity of the fuel salt through the SCR, , remains essentially constant and independent of its position. Under these circumstances, the inlet SCR channel flow rate, , is given by(15)Assuming remains constant, the flow rate through the SCR channel is primarily determined by the annular area, A_an, which varies with the SCR position height, h (i.e., A_an = f(h)). For a conical conduit, the relationship between A_an and h is expressed as(16)Consequently, under steady-state conditions, a definitive relationship exists between the SCR position height and channel flow rate:(17)This demonstrates that higher instantaneous flow rates through the SCR channel correspond to larger annular areas and, consequently, higher SCR equilibrium positions. Lower flow rates result in smaller annular areas and lower SCR positions. Moreover, the instantaneous flow rate through the SCR channel is intrinsically linked to the core flow distribution characteristics. Flow distribution is evaluated using a closed parallel-channel model based on two assumptions: (1) uniform flow distribution across all fuel channels, and (2) frictional pressure drop as the only influencing factor. For a reactor with 241 channels, the average core pressure drop, ΔP_av, is equal to the pressure drop across an individual fuel channel, ΔP_f,fuel, and is expressed as(18)where L_f denotes the fuel channel length, D_f is the fuel channel diameter, and Q_f denotes the core mass flow rate.

The total pressure drop (ΔP_CR) across the SCR channel consists of both frictional and local pressure drop components. As shown in Fig. 3, the SCR channel comprises several successive segments from bottom to top: circular pipe, gradual expansion, sudden contraction, annular, sudden expansion, second gradual expansion segment, second sudden contraction, and final circular pipe segments. Accordingly, the overall pressure drop can be expressed as(19)where ΔP_tube, ΔP_grad,exp, ΔP_s, and ΔP_sud,exp represent the pressure drops across the circular pipe, gradual expansion, SCR, and sudden expansion segments, respectively. The frictional and local pressure drop components for each segment are determined using Eqs. (7) and (8), respectively. By adjusting the SCR channel flow rate such that its pressure drop matches the average pressure drop in the core, we can determine both the channel flow rate and axial pressure drop distribution.

Optimization Methodology

To vary the core flow conditions, we first employ an iterative computational approach to determine both the SCR channel flow rate and equilibrium position height. Subsequently, an iterative optimization process is conducted to identify the parameters that satisfy the reactivity control specifications, with the iterative calculations executed using Python. The detailed iterative computational workflow is shown in Fig. 5.

Fig. 5Full-screen View

(Color online) Flowchart of rod position-reactivity calculation for an SCR at different core flows

The computational procedure commences with parameter initialization, followed by an evaluation of the average core pressure drop (ΔP_av) and SCR channel pressure drop (ΔP_CR) at the current flow rate. Subsequently, the relative error between the pressure drops is computed. If , the core flow distribution is assumed to converge, thereby yielding the SCR channel flow rate Q_s. Otherwise, Q_s is adjusted and the process is iterated. After the flow distribution is computed, the relative error between the SCR pressure drop (Δp_s) at the current position height and steady-state SCR pressure drop (Δp_s,stable) is evaluated. If the condition is not satisfied, the SCR position height is adjusted, and the flow distribution computation is repeated. When convergence is achieved, the core flow rate (Q_f), Q_s, and position height (h) are output. Subsequently, the reactivity worth (Δρ_rod) provided by the SCR and flow reactivity loss (Δρ_fluid) at the nominal core flow rate are evaluated. If , then the reactivity, , introduced by SCR across varying core flow rates is computed and output, thereby completing the iterative process and obtaining the optimal geometric parameters for the current configuration. Otherwise, the upper base diameter, D_s, and the conical taper angle, θ, of the SCR are adjusted, and the flow distribution computation is repeated.

DNP Flow-induced Reactivity Perturbations

The reduction in the effective delayed neutron fraction caused by DNPs flow is closely related to the residence time of the DNPs both within and outside the core. At lower average fuel flow rates, a reduced proportion of DNP decays outside the core, resulting in a smaller decrease in the effective delayed neutron fraction in the core and, consequently, lower reactivity loss. As the average fuel flow rate increases, the fraction of DNP decaying outside the core increases, causing a greater decrease in the effective delayed neutron fraction within the core and a proportional increase in reactivity loss. When the average fuel flow rate is increased further, DNP recirculates into the active region of the core, resulting in a decrease in the eventual saturation of the fraction of DNP decaying outside the core, at which point the reactivity loss attains its maximum value and then stabilizes.

Figure 6 shows the variations in flow reactivity across different core flow conditions, revealing that at lower core flow rates, the flow reactivity changes considerably with flow variations. At higher flow rates, these changes are notably smaller. Consequently, based on the flow reactivity-core flow variation curve, SCR design should aim for a greater variation in the rod position height or introduce reactivity at lower core flow rates and a smaller variation at higher core flow rates. At the rated core flow, the SCR is expected to release approximately 275 pcm of reactivity.

Fig. 6Full-screen View

(Color online) Relationship between DNP flow-induced reactivity changes and core flow rate

SCR Position–Core Flow Rate Correlation

The correlation between SCR rod position height and core flow directly affects its reactivity insertion characteristics. Thus, a detailed analysis of the key influencing factors is essential for optimizing the SCR structural design and enhancing its performance. As analyzed in Sect. 2.1, this correlation is primarily determined by two key parameters: the SCR stable pressure drop (Δp_s,stable) and flow channel annular area (A_an(h)). Δp_s,stable governs the response of the SCR to flow variations and influences the initial start-up flow threshold. As presented in Eq. (13), Δp_s,stable is a function of the density difference between the SCR and fuel salt, SCR volume, and frontal area. In this context, the SCR volume is determined by the upper base diameter (D_s), conical taper angle (θ), and length (L_s), whereas the frontal area depends solely on D_s. According to Eq. (16), the variation in the flow channel annular area, A_an(h), is determined by D_s and θ.

This section uses a controlled variable approach to examine the effect of individual parameters on the SCR position-flow rate relationship. The analysis is conducted using the following initial parameters: L_s=30 cm, D_s=7 cm, θ=2°, ρ_s=2800 kg/m³, and ρ_f=2710 kg/m³. The calculation procedure shown in Fig. 5 is employed to assess the influence of different parameters on the displacement of the SCR under various core flow rates, as well as the SCR channel flow at the rated core flow rate. The results of these calculations are shown in Figs. 7 and 8.

Fig. 7Full-screen View

(Color online) Impact of SCR mean density (a), length (b), upper base diameter (c), and conical taper angle (d) on the SCR position–core flow rate correlation

Fig. 8Full-screen View

(Color online) Impact of SCR mean density (a), length (b), upper base diameter (c), and conical taper angle (d) on the SCR channel flow rate

The average density (ρ_s) is the most critical parameter that affects SCR motion. As illustrated in Fig. 7(a), an increase in ρs results in a significant increase in the start-up flow threshold, a rightward shift of the core flow–rod position curve, and a marked decrease in the rod position height at the same core flow rate. Specifically, for every 40 kg/m³ increase in ρ_s, the start-up flow threshold increases by approximately 10%, whereas the rod position at the rated flow decreases by an average of 10.8 cm. Additionally, the extent of the reduction in the rod position height diminishes as the density increases. Simultaneously, an increase in ρ_s significantly increases the flow resistance within the SCR channel, resulting in a significant decrease in the flow fraction allocated to the SCR. As indicated in Fig. 8(a), when ρ_s increases from 2720 to 2880 kg/m³, the flow fraction decreases from approximately 85% to 53%.

An increase in the SCR length (L_s) similarly increases the start-up flow threshold, shifts the core flow–rod position curve to the right, and reduces the maximum displacement. Specifically, when L_s increases from 20 to 40 cm, the start-up flow threshold increases by 12%, and the rod position at rated flow decreases by 2.4 cm, as illustrated in Fig. 7(b). Moreover, a longer L_s increases the frictional resistance within the channel, further diminishing the flow allocated to the SCR. Within the considered range, the SCR channel flow changes by about 6%, as shown in Fig. 8(b).

Increasing the upper base diameter (D_s) also results in a higher start-up flow threshold and lower rod position height at the rated flow. However, its effect is minimal and can be neglected, as shown in Fig. 7(c). Regarding SCR channel flow, a larger D_s reduces the flow fraction. However, this reduction is limited, as shown in Fig. 8(c), indicating a relatively weak effect on the overall SCR performance.

The effect of the conical taper angle (θ) on the SCR rod position–flow relationship is distinct from that of the aforementioned parameters. As θ increases, the start-up flow threshold decreases, whereas the maximum displacement decreases, as shown in Fig. 7(d). This occurs because θ influences the SCR volume and determines the annular flow area at various rod positions, which directly affects the correspondence between the rod position and flow rate. An increase in θ can increase the flow fraction in the SCR channel. For example, as θ increases from 1° to 3°, the flow fraction increases from approximately 71% to 73%, as shown in Fig. 8(d). A larger cone angle aids in expanding the annular flow passage area and reduces the local flow resistance, thereby enhancing channel flow. However, if θ becomes excessively large, it can increase the heat generation rate within the SCR channel, adversely affecting its heat transfer safety.

In summary, the average density exerts the most significant influence on the start-up flow threshold, core flow–rod position relationship, and and channel flow; thus, it is the most critical parameter in SCR structural design. The effects of SCR length and upper base diameter are relatively minor, rendering them secondary control parameters. In contrast, the conical taper angle has a significant impact on the core flow–rod position relationship and may serve as an effective tuning parameter for its optimization. Consequently, the optimization of SCR structural parameters should prioritize controlling the average density, appropriately selecting the SCR length and upper base diameter, and adjusting the conical taper angle to achieve the desired SCR performance for reactivity control.

Impact of Fuel Density Variations on SCR Performance

Variations in the fuel-salt temperature, online fueling, and reprocessing all result in changes in the fuel-salt density. This, in turn, affects the SCR position height at the rated core flow. Assuming a constant SCR average density, a decrease in the fuel-salt density causes a corresponding reduction in the SCR position height. However, if the fuel-salt density exceeds the SCR average density, the SCR remains buoyant at its highest position and does not descend, even in flow-loss scenarios. Therefore, given the dynamic variations in fuel-salt density, the mechanisms by which the SCR responds to such density changes should be investigated.

The analysis considers only the effect of temperature variation, assuming a uniform temperature distribution within the reactor core. Owing to the thermal expansion effect, temperature changes lead to changes in material density. Given the low thermal expansion coefficients and small size of the constituent materials of the SCR, the thermal expansion effect for the SCR is negligible. In contrast, the fuel salt exhibits a high thermal expansion coefficient and significant thermal expansion. Therefore, only the density change of the fuel salt resulting from the temperature variation is considered, whereas the change in the density of the SCR due to temperature variation is neglected. Within a temperature range from a maximum outlet temperature of 750 °C to a cold shutdown temperature of 550 °C, the fuel-salt density changes from about 2665 to 2755 kg/m³ [55]. If the SCR average density is below 2755 kg/m³, after a cold shutdown, the resulting decrease in temperature may cause the SCR to float upward, thus posing a risk of reactor re-criticality. Thus, the SCR average density should be maintained at above 2755 kg/m³ to avoid such scenarios. When only the effects of online fueling and reprocessing are considered, the fuel-salt density is expected to increase to approximately 2800 kg/m³ over the lifetime of the fuel, as heavy metals accumulate in the molten salt [44]. Thus, an SCR with a ρ_s of 2805 kg/m³ is selected, with the other parameters identical to those in the previous section, to study the effect of changes in the fuel-salt density on the SCR. The calculations reveal the SCR rod position heights under varying fuel-salt densities and core flow conditions, as illustrated in Fig. 9.

Fig. 9Full-screen View

(Color online) SCR position height correlation with core flow rate and fuel-salt density

The results show that a decrease in the fuel-salt density elevates the SCR start-up flow threshold while simultaneously reducing its position height at the rated core flow. In other words, an increase in temperature—which leads to a lower density—results in a lower SCR rod position height, thereby introducing a negative reactivity and enhancing the negative temperature reactivity coefficient of the reactor. Furthermore, increasing the fuel-salt density does not change the rod position height at the rated flow because the SCR is already at its uppermost position, restrained at top of the channel. However, higher fuel-salt densities alter the SCR’s response under flow-loss conditions, causing it to respond more slowly. When the fuel-salt density is 2800 kg/m³, the SCR only begins to descend when the core flow decreases to approximately 50% of the rated value. Accordingly, to improve the SCR’s resilience against disturbances due to density variations, its average density must be further increased.

Based on conservative estimates that incorporate temperature fluctuations and the effects of online fueling and reprocessing on fuel lifetime, the SCR average density must be maintained above 2855 kg/m³ under cold conditions at the end of fuel life to prevent the fuel-salt density from surpassing the SCR average density. However, if the SCR average density is extremely high, the pressure drop necessary to drive the SCR increases, which reduces the channel flow and adversely affects the heat transfer performance of the SCR channel. As shown in Fig. 10, if the SCR average density exceeds 2855 kg/m³ and the fuel-salt density remains at a nominal value of 2710 kg/m³, the channel flow drops to less than 60% of the fuel channel flow, which is unacceptable for heat transfer safety. However, as the fuel-salt density increases, the average core pressure drop increases, resulting in a greater allocation of the flow to the SCR channel. This, in turn, improves the heat transfer capacity. Thus, as heavy metals accumulate in the molten salt and the fuel-salt density increases, the heat transfer capacity within the SCR channel is enhanced.

Fig. 10Full-screen View

(Color online) SCR-to-fuel salt channel flow rate ratio correlation with SCR mean density and fuel density

In summary, the influence of fuel-salt density variations on the SCR fundamentally depends on the difference between the SCR average and fuel-salt densities. A larger difference enhances the SCR’s resistance to density disturbances. However, this may also reduce the heat transfer performance of the channels. Therefore, an optimal balance should be obtained between the SCR’s ability to withstand density variations and its heat transfer capability.

SCR Reactivity Worth

Because the fluid-driven control and graphite components are integrated within the reactor core, their dimensions are limited by the specifications of the graphite components. Consequently, assessing whether the SCR can satisfy the reactivity control requirements, given these dimensional limitations, is vital. Because the SCR remains entirely within the active region of the core throughout its motion, its reactivity worth is predominantly determined by the neutron flux distribution within the core. Specifically, the larger the neutron flux gradient encountered within the SCR’s range of motion, the more significant the change in its reactivity worth, resulting in a higher amount of reactivity compensation. Because the variations in the upper base diameter, conical taper angle, and average density of the SCR are restricted to narrow ranges and exert minimal influence on its reactivity worth, this paper primarily focuses on analyzing the effects of SCR length, number of units, and spatial placement on reactivity, whereas the effects of the upper base diameter, conical taper angle, and average density are incorporated into the final iterative calculations.

The SCR is designed with a length range of 20–40 cm and deployment quantity of 1–3 units, employing a centrally symmetric arrangement. When deploying three SCR units, a tripartite symmetric arrangement — positioned at the midpoints of the three sides (as indicated by “*” in Fig. 1) — is used. For two SCR units, a bilateral symmetric arrangement at the midpoints is adopted; for a single unit, the midpoint of one side is selected. Using the core central axis (designated as the zeroth ring) as reference, the placement extends outward to the positions of the graphite components in the second, fourth, and sixth rings, as shown in Fig. 1. The core’s axial midplane is defined as the reference plane (0 position) for the SCR’s height. When calculating the reactivity worth of the SCR, the upper base diameter is fixed at 6 cm, the conical taper angle at 2°, and the average density at 2800 kg/m³ to ensure an adequate range of motion.

Two SCR units with lengths of 20, 25, 30, 35, and 40 cm, respectively, are deployed at the graphite components in the fourth ring. The reactivity introduced from their lowest to highest positions is computed, and the results are presented in Fig. 11. As the SCR length increases, the reactivity induced by movement exhibits an increasing trend, with the most significant reactivity introduced during the mid-range of motion and lower reactivity in the initial and final phases. Hence, to align with the DNPs flow reactivity–core flow curve shown in Fig. 6, we should configure the SCR’s range of motion in the mid-to-rear section. Among all the SCR lengths evaluated, only the 20 cm variant fails to satisfy the 275 pcm reactivity control requirement. To minimize any detrimental effects on neutron economy under normal operating conditions, the SCR length should be reduced as much as possible while still satisfying the reactivity control requirements.

Fig. 11Full-screen View

(Color online) Reactivity changes of SCRs with different lengths at different rod positions (two SCRs at the fourth ring component position)

SCRs with a length of 30 cm are deployed in varying quantities and at different locations within the core, and the induced reactivity changes at different heights are computed. The results are presented in Fig. 12. The results indicate that a minimum of two SCR units must be deployed to satisfy the 275 pcm reactivity control requirement. At equal deployment quantities, the reactivity change in the fourth ring exceeds that in the second ring owing to the higher axial neutron flux gradient in the fourth ring. In the sixth ring, with a reduced neutron flux distribution, the reactivity change for the same SCR movement is minimal.

Fig. 12Full-screen View

(Color online) Reactivity changes at different rod positions with different SCR deployment quantities and positions (L_s=30 cm)

Under normal operating conditions, the SCR is positioned at the top of the active region of the core, in contact with the upper support plate. The corresponding reactivity values for different deployment quantities and positions are summarized in Table 5. The analysis reveals that, for identical deployment quantities, the placement position has a relatively minor impact on the reactivity, with a difference of approximately 60 pcm between the second and sixth rings. However, for an identical deployment position, the number of SCR units significantly affects the reactivity, with three SCRs exhibiting approximately 100 pcm higher reactivity than that with two SCRs.

Therefore, the primary focus should be on determining the number of SCR units to be deployed, after which the optimal placement can be refined. For minimal adverse effects of SCR deployment on the neutron economy, core flow distribution, and system compatibility under normal operating conditions, the number of SCR units should be minimized, and their placement should be as far from the core’s central axis as possible. Thus, from a purely reactive perspective, deploying two SCR units in the fourth ring is the optimal solution. Under normal operating conditions, the variations in reactivity owing to changes in SCR deployment position and quantity are limited, with a maximum difference of only 148 pcm. Thus, by integrating the SCR rod position–core flow relationship, an optimal deployment scheme can be determined.

Correlation between SCR Reactivity Introduction and Core Flow Rate

Based on the preceding analysis, the SCR average density significantly affects the start-up flow threshold, channel flow, and its resistance to density disturbances, whereas the conical taper angle profoundly influences the relationship between the SCR rod position and core flow. In contrast, the SCR length and inlet diameter exert relatively minor impacts on SCR performance. Therefore, this paper omits the effects of SCR length and Upper base diameter — fixed at L_s=30 cm and D_s=7 cm — and, according to the procedure shown in Fig. 5, calculates and screens the conical taper angles that satisfy the reactivity compensation criteria. Subsequently, we analyze the reactivity compensation capabilities at different SCR average densities.

With increasing SCR average density, its displacement at rated flow decreases. Hence, a higher SCR average density necessitates larger reactivity insertion. As illustrated in Fig. 7(a), when ρ_s is 2840, 2800, and 2760 kg/m³, the displacements at rated flow are approximately 35, 40, and 50 cm, respectively. According to Fig. 12, to satisfy the reactivity control requirements, the three average density conditions correspond to the following SCR deployment schemes: Three SCR units in the fourth ring for ρ_s = 2840 kg/m³, two SCR units at the fourth ring for ρ_s = 2800 kg/m³, and two SCR units in the 6th ring for ρ_s = 2760 kg/m³. Figure 13 shows the reactivity variations induced by SCR motion across various core flow conditions for these scenarios.

Fig. 13Full-screen View

(Color online) Reactivity introduced by different parameter SCR under various core flow rates

The results reveal that a lower SCR average density produces a more pronounced reactivity compensation over a broader range of core flow variations, whereas a higher SCR average density enables a more rapid response to flow changes. For instance, at ρ_s = 2760 kg/m³, the flow reactivity fluctuations can be confined within 30 pcm over 55%–100% of the rated core flow, but the SCR only introduces -275 pcm of reactivity when the core flow falls below 40% of its rated value. In contrast, at ρ_s = 2840 kg/m³, the flow reactivity fluctuations are suppressed within 30 pcm only in the 80%–100% rated flow range, and -275 pcm reactivity is achieved only when the core flow drops below 68% of the rated value. As shown in Table 6, from the perspective of the neutron economy, the impact differences among these schemes are minimal, with a maximum variance of only 22 pcm. In terms of heat transfer performance, an SCR with ρ_s = 2840 kg/m³ exhibits a channel flow that amounts to only 61.46% of the fuel channel flow, potentially compromising the heat transfer safety of channels.

In summary, for improved matching between flow and reactivity changes, an SCR with a lower average density should be selected. In contrast, if the response speed of the SCR under loss-of-flow scenarios is of higher priority, a higher-average-density SCR should be selected. However, note that increasing the average density of the SCR decreases the flow within its channels. Variations in the primary loop flow are generally minimal during MSR operation. However, under accident conditions, the response speed of the SCR is a crucial factor; thus, prioritizing an SCR with a higher average density is more appropriate. Therefore, considering both the response speed and heat-transfer safety, the SCR with an average density of ρ_s = 2800 kg/m³ demonstrated a superior performance. However, attention should be paid to the increase in fuel density toward the end of the fuel cycle and under cold-reactor conditions. The average SCR density, timely separation of heavy metals from the fuel is required.