Conductor and strand parameters

The TF-magnet system comprises 16 TF coils, each containing 126 turns. It carries a nominal current of 55 kA and provides a 6.5 T magnetic field at the plasma center to confine the plasma. The WP is enclosed in a stainless-steel case. The geometry of the cable-in-conduit conductors (CICCs) is shown in Fig. 3(a). The WP can be divided into two sub-WPs: high-field (HF) and low-field (LF). The HF WP comprises six double pancakes (DPs), in which a high-Jc Nb₃Sn strand is selected, whereas the LF adopts an ITER-like Nb₃Sn strand. The structure of the conductor includes an insulator, jacket, cable, and helium channel, corresponding to gray, white, yellow, and black, respectively, in Fig. 3(a). SS316LN was used for the TF case and jacket, and G10 was used as an insulator [34].

Fig. 3Full-screen View

(Color online) (a) A half-horizontal profile of the TF coil at the inboard leg half. The numbering of the pancakes and the turns adopted in this work are shown. (b) Cooling scheme of high field

Figure 3(b) shows the horizontal profile of the cooling scheme for the HF WP. Six parallel cooling circuits are included, with two pancakes set as one cooling circuit. The length of each cooling circuit was approximately 80 . Helium flowed in opposite poloidal directions in two adjacent cooling circuits. The primary parameters of the conductor and strands are listed in Table 1.

The critical current in the Nb₃Sn composites was calculated according to the scaling law of [35]. The scaling parameters of the Nb₃Sn strand are listed in Table 2.

Neutronics model

Considering the symmetry of the fusion reactor, a sector spanning 22.5 degrees enclosed by reflective planes was used for the Monte Carlo neutronics simulations. The model encompasses components such as the blanket, divertor, vacuum vessel, thermal shield, TF coils, PF coil, central solenoid, ports, and cryostat.

The blanket, divertor, and vacuum vessel collectively contribute to neutron shielding, with the blanket assuming the primary role of shielding neutrons in the magnet system. As illustrated in Fig. 4, the TF coils were divided into 45 poloidal sectors arranged counterclockwise from the inboard leg. Sectors 12–23 are positioned at the rear of the divertor, whereas the remaining sectors are situated at the rear of the blanket. Sector 19 houses the helium inlet, and Sector 18 contains the outlet, as shown in the diagram.

Fig. 4Full-screen View

Poloidal sectors of TF coils

Owing to the intricate nature of the WP and the substantial number of torus surfaces, significant effort has been invested in neutron modeling using the constructive solid geometry (CSG) method. The complexity of modeling in this area is typically addressed by employing a hierarchical structure or homogeneous model. In line with the design of the TF coils, this study modeled a heterogeneous parametric WP model with a cable, stainless-steel jacket, and insulator structure.

However, the simulation faces limitations owing to the constrained capabilities of the Monte Carlo computing code MCNP [36, 37] when applied to a multitude of torus surfaces. To maximally enhance model accuracy, an “approximate rectangular cross-section" was employed as a substitute for the “circular cross-section," determined based on material-volume proportions of the turns.

Based on Fig. 5, both a homogeneous WP and a more intricate WP configuration were constructed to facilitate a comparison for the precision enhancement.

Fig. 5Full-screen View

(Color online) Horizontal cross section of the inboard neutronics model

The probability distribution for sampling 14.1 MeV neutrons is defined as follows: $s(a)={\left[1-{\left(a/A\right)}^2\right]}^P\mathrm{,0}\le a\le A\mathrm{,}$ (1) where the value of P is determined through physical models and assumptions, typically ranging between 3 and 4. The parameter A corresponds to the minor plasma radius, and parameter a defines the plasma contour. The representation of the magnetic surface in polar coordinates is expressed as follows: $\{\begin{array}{c}R=R_0+a\cdot \cos \left(\alpha +\delta \cdot \sin \alpha \right)\\ +esh\left[1-{\left(a/A\right)}^2\right]\\ Z=El\cdot a\cdot \sin \alpha \end{array}$ (2) Here, esh denotes the Shafranov factor, indicating the displacement from the magnetic-surface center to the low-field side. R₀ is the major plasma radius. El and δ are the elongation ratio and triangularity of the plasma, respectively [38].

During this phase, the principal parameters of the conductor and strand are initially established, and the critical current in the Nb₃Sn composites is computed using scaling-law equations. Subsequently, a heterogeneous parametric neutronics model and neutron source were constructed. These models were based on the reference CAD model and plasma parameters, respectively.

Nuclear heating

Nuclear heating was evaluated using the Monte Carlo particle-transport code MCNP v5.1.6 with the FENDL 3.1d nuclear-data library. Assuming a fusion power of 40 MW, the corresponding D-T neutron-emission rate was set as 1.42×10¹⁹ s^-1. A bi-dimensional (radial and poloidal) discrete neutron source was employed to characterize the spatial distribution of D-T neutrons. By employing the neutronics model above, the nuclear-heating distributions within the TF coils were calculated based on 10¹⁰ neutron histories. To enhance the statistical efficiency of the Monte Carlo simulations for shielding calculations, an importance-based variance-reduction technique was implemented to ensure satisfactory results and statistics, which generally remained below 0.5%.

Table 3 outlines the nuclear-power deposition in the TF coils. The overall TF-coil deposition amounts to 17.86 kW, with the case contributing 51.7% and WP providing 48.3% of this total. Additionally, a comparison was made between the nuclear-power deposition at the rear of the blanket and divertor with the aim of evaluating the neutron-shielding effectiveness. Notably, the total nuclear heat of TF coils obtained through the homogenized model reached 17.46 kW, exhibiting a relative error of 2.2% compared with the detailed heterogeneous model. Considering the statistical-count error (below 0.5%), the influence of the model accuracy on the comprehensive nuclear heat cannot be dismissed.

To further assess the precision enhancement of the heterogeneous model in terms of TF nuclear heating, a comparison of two typical cross-sections is shown in Table 4: one at the inboard middle plane (IB) and the other at the outboard middle plane (OB). The differences in the nuclear-heating densities of the cable, jacket, and insulator in the IB were -16.2%, 7.3%, and 2.7%, respectively. Significant differences were observed in the OB. This is because in the heterogeneous model, only neutrons passing through the case, jacket, and insulation interacted with the cable material, leading to the observed variations. In the thermal-stability analysis of the TF-magnet system, the precision of the cable in the heterogeneous model improved by more than 12%. This improvement is crucial for subsequent heat-loading and temperature-margin assessments in the thermal-hydraulic nuclear context.

Figure 6 shows the polar distribution of the WP nuclear-heating density within HF conductor layer 1. The average nuclear-heating densities of the cables, jackets, and insulators were 3.35×10^-4 W/cm³, 4.34×10^-4 W/cm³, and 1.63×10^-4 W/cm³, respectively. For the nuclear-heating density, the insulator displayed the lowest value, the cable registered approximately 2.06 times the value of the insulator, and the jacket demonstrated a nuclear-heating density approximately 2.67 times greater than that of the insulator. The highest value was observed on the upper side of the IB (Sector 5). Furthermore, the conductor (Sector 16–18) at the rear of the divertor exhibited an elevated nuclear-heating density, suggesting its potential for divertor-shielding enhancement. Neutron fluxes in proximity to the middle plane and port tended to be high, accompanied by corresponding nuclear-heating densities on the rear side of TF coils.

Fig. 6Full-screen View

Nuclear-heating density poloidal profile in first layer of WP

Figure 7 illustrates the exponential decay of the nuclear-heating density along the radial direction for the WP in Sectors 5 (IB) and 30 (OB). Comparing the rear layer in the HF with the front layer at the IB, the value decreased by 3.56 times in the HF and by 16.26 times in the LF. Similar comparisons with the OB front layer showed a 3.14 times decrease in the HF and only a 4.99 times decrease in the LF. Notably, the nuclear-heating density in the HF surpassed that in the LF by an order of magnitude, highlighting the preference for the HF in subsequent analyses.

Fig. 7Full-screen View

Nuclear-heating density radial profile of the WP in Sector 5 (IB) and Sector 30 (OB)

The statistical assessment of the nuclear-heating toroidal profile in the HF is important for selecting the optimal cooling circuit for thermal-hydraulic analysis. As shown in Fig. 8, a comparison between Sectors 1 and 5 revealed that, in Sector 1, the cable, jacket, and insulator values in HF-6 (central pancakes) were 13%–16% higher than those in HF-1 and HF-2 (side pancake). A similar trend was observed in Sector 5, with an increase of approximately 5%–8%. As expected, the central turn exhibited a higher nuclear deposition than the corner turn; however, the discrepancy was not notable.

Fig. 8Full-screen View

Nuclear heating toroidal profile of the WP in Sector 1 and Sector 5

Heat transferred from case to coil

The TF case comprised U-shaped enclosures (AU and BU) and cover plates (AP and BP). Both AU and BU featured 22 cooling channels, whereas AP and BP had eight cooling channels each. The mass-flow rate for each cooling channel was 2.7 g/s, with an inlet pressure of 6 bar and inlet temperature of 4.2 K. During operation with a pulse duration of 1000 s, a steady state for heat transfer between the case and conductor was achieved. However, in the case of a pulse duration of 10 s, heat transfer was disregarded. Ansys 19.0 was utilized to conduct a 2D heat-transfer analysis, relying on the nuclear-heating results obtained from MCNP for the case at 40 MW. Owing to the inward compression resulting from the centering force on the WP, a millimeter-sized gap was formed between the inner side of the WP and the case [16]. Consequently, the heat transfer between the case and plasma-side of the WP became limited. An ideal heat transfer between the case and other sides of the WP is assumed considering the isotropic material properties. The heat-transfer coefficient was calculated using the following equation: $h_{\text{tra}}=0.027\times R{e}^{0.8}\times P{r}^{0.4}\times \lambda /D_{\text{h}}\mathrm{,}$ (3) where Re and Pr represent the Reynolds and Prandtl numbers, respectively. λ denotes the thermal conductivity, $D_{\text{h}}$ is the hydraulic diameter, and $h_{\text{tra}}$ represents the heat-transfer coefficient.

The finite-element model of the TF case for the heat-transfer calculation was discretized into 45 2D sectors in the poloidal direction. As indicated in Table 5, the heat flux transferred from the TF case to the conductors was calculated. Notably, corner turns exhibited a significantly higher heat flux than central turns. In the toroidal direction of Sector 5, the heat-flux value in HF 1-1 was approximately 2.5 times greater than those in HF 2-1 and HF 6-1. In the radial direction, the heat-flux values in HF 1-1, HF 2-1, and HF 6-1 surpassed those in the turns of the second layer by a substantial margin.

These outcomes highlight the substantial variation in heat transfer from the case to the coil across different turns, and these values constituted a significant portion of the overall heat load.

Total heat load on conductor

In the subsequent phase, the cumulative heat load absorbed by the conductors along the cooling circuits is quantified. This process can be divided into two steps: first, the average nuclear-heating flux for each turn is calculated based on the material proportion and cross-sectional area of the turn; second, the nuclear heat on HF-1 and HF-6 is determined based on the arrangement of the helium cooling circuits. The calculation of the average nuclear-heating flux is given by Eq. (4), where q and V symbolize nuclear-heating density and the volume proportion of the cable in each turn, respectively. A represents the cross-sectional area of the turn. $\begin{array}{r}\hfill N{H}_{\text{flux}}=(q_{\text{cable}}\cdot V_{\text{cable}}+q_{\text{jacket}}\cdot V_{\text{jacket}}\\ \hfill +q_{\text{insulator}}\cdot V_{\text{insulator}})\cdot A_{\text{turn}}\end{array}$ (4)

In this scenario, the helium inlet and outlet were strategically positioned in Sectors 19 and 18, respectively. Consequently, the heat load along the cooling circuits within each of the four layers (each approximately 4×20 m) was initiated at the heat flux originating from Sector 19 and concluded in Sector 18.

Figure 9 shows the heat load on the conductor along the cooling circuits. HF-6 HT (heat transferred from the case to HF-6) was approximately half that of HF-6 NH (nuclear heating on HF-6). While the nuclear heating on the side pancake (HF-1 NH) was marginally lower than that on the central pancake (HF-6 NH), the heat load transferred from the case (HF-1 HT) significantly surpassed that of the central pancake (HF-6 HT). Thus, HF-1 experienced the highest heat load, followed by HF-6.

Fig. 9Full-screen View

(Color online) Nuclear heating (NH), heat transferred (HT) and total heat load (Total) along HF-1 and HF-6 cooling circuits

Notably, the analysis did not consider the impact of AC losses on the heat load. Owing to the unique cooling scheme and variations in the heat-transfer coefficient and WP structure, the heat load of the central and side pancakes may differ. Thus, a comprehensive analysis of the maximum heat load that considers both the distribution of nuclear heating and conduction heat of the coil box is imperative.

Furthermore, the conductor heat-loading method can be applied to alternative arrangements of helium cooling circuits. Of particular significance are the TF sector numbers, where the helium inlet and outlet are located, and the poloidal direction of the helium flow.

Steady-state mode

In the steady-state mode, both nuclear heating and heat transferred from the case to the coil were considered. Figure 11 illustrates the temperature evolution of the superconducting strand. The temperature history at the helium-flow outlet and pressure decrease are shown in Fig. 12. The conductor-load temperature sequentially increased with distance from the helium inlet owing to the heat load, and the average temperature continued to increase over time. By the end of the 1000 s plasma pulse, the conductor temperature at the end of the helium-flow channel reached 5.502 K, indicating an increase of 1.3 K from the initial value of 4.2 K. The most substantial temperature increase occurred in the first turn (0 m–20 m distance from the helium inlet) of HF-7, contributing approximately 45%–50% of the total temperature increase. A milder increase was observed in the last turn (60 m–80 m distance from the helium inlet). The conductor temperature at the helium outlet reached its peak at 550 s, started to decrease after 1000 s, and reached its minimum value after1550 s. The outlet pressure was 5.536 bar. The initial temperature margin under the aforementioned peak magnetic field and strain conditions was 3.0 K. After subtracting other heat loads apart from nuclear heating (such as hysteresis loss, coupling currents, thermal radiation, and conduction, which amount to approximately 0.45 K), a remaining temperature margin of 1.25 K was achieved, satisfying the design requirement of > 1.0 K.

Fig. 11Full-screen View

Temperature evolution of the superconducting strand in steady-state mode

Fig. 12Full-screen View

Conductor temperature and pressure history at the helium outlet

Transient mode

In the transient mode, the nuclear heat directly absorbed by the conductor remained constant throughout the 10-s pulse. However, the heat transferred from the case was not included in the analysis. Unlike the extended plasma-pulse mode, where the conductor and coolant can approach a thermal-equilibrium state, the 10-splasma pulse prevents them from achieving such an equilibrium. Consequently, in this mode, the total heat load can be approximated as 4.5 times the nuclear heating load in the steady-state mode.

Figure 13 shows the temperature evolution of the superconducting strand over a 10-s pulse. In general, the conductor temperature followed an increasing trend that aligned with the nuclear-heating load. The first turn of HF-7, which experienced the greatest nuclear heating, exhibited the most significant temperature variation. At the end of the pulse, a peak temperature of 4.583 K was reached, indicating an increase in 0.38 K over the inlet temperature. After deducting the other thermal loads, the remaining temperature margin was approximately 2.17 K, which satisfied the design requirements. The outlet pressure was maintained at 5.536 bar. Compared with the steady-state mode, the transient mode afforded a larger temperature margin.

Fig. 13Full-screen View

(Color online) Temperature evolution of the superconducting strand in transient mode