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Spherical Tokamak reactor (STR) is attractive due to its inherent capabilities such as disruption avoidance, natural elongation, natural divertor and high beta capability, apart from a smaller size, with presumably lower costs [ 1, 2]. There has been an extraordinary evolution from the early concepts like SMARTOR [ 3] with devices like START, NSTX, MAST, GLOBUS-M and a number of others with the HTS based future devices like STEP [4]. Given the pace of development of the new superconducting materials [5,6] and the new divertor concepts [7,8,9], the STRs represent a rapidly developing front and may very well be realized not far in the future. Following an elegant paper by Peng et al. in 1986, a range of compact reactor designs ($R$ and $P_f$) has emerged, e.g. FNS-ST (0.5m, 10 MW), DTST (1.1m, 30-60 MW), ARC (3.3m, 525 MW), SlimCS (5.5m, 2950 MW), ARIES-ST (3m, 2980 MW) with a variety of objectives like, neutron source, component-test-facility (CTF) and power plant [10,11,12,13,14]. However, while the high neutron loads are welcome for reactor economics, the size reduction comes at a penalty of extreme heat loads on the divertor with concomitant engineering challenges [15]. Several designs of STRs are currently being developed around the world with scoping studies and available data from currently operating tokamaks as well as other experimental/dedicated test facilities and insights from experts [16]. This paper brings out the role of constraints arising from steady-state power balance and core-radiation. It is argued that the core-radiation plays a crucial role in the reactor design, as it not only restricts the accessible parameter-space but also determines the limits on impurity accumulation [17]. A comprehensive physics-design study [18] shows that about 50$\%$ of the heating power needs to be lost by core-radiation. Such considerations can impact stability as well [19]. In the following, the ST-parameter space ($R-B_t$) is analyzed to elucidate the limits posed by the various constraints. For $T_i$ from 6 to 20 keV, the fusion power (MW) may be approximated for analytic purposes as:

$$P_{F}=0.026 \frac{(S_n+S_T+1)^2}{(2S_n+2S_T+1)} \frac{\kappa {\beta_{N}}^2 {S_{\kappa}}^2}{q^2 A^4} R^3 {B_t}^4$$ where $q=5 R B_t S_k/(A^2 I_p)$ is the safety factor, $I_p$ is the plasma current in MA, $A$ is the aspect ratio and $S_k$ is the shape factor. $\beta_N = \beta a B_t/I_p$ and $S_n$, $S_T$ are the exponents for the parabolic profile of the density and temperature respectively. The stored energy in MJ can be expressed as: $$W_\beta = \frac{\pi}{8}\frac{\kappa S_\kappa}{q A^3} \beta_N R^3 {B_t}^2$$ In steady-state, where the power from $\alpha$-particles and the externally injected power are balanced by the transport losses, the power-balance is given by $W_\beta = P_L \tau_E $, where $P_L$ (defined as $P_H (1-f)$) is the power reaching the edge, after a fraction $f$ of the power deposited $$P_H = P_\alpha + P_{ext}= P_F (1/5+1/Q)$$ is radiatively lost from the core region. It is assumed that the ITER-IPB(98,y2) scaling holds good, although it is likely to be more favorable in reality [20]: $$\tau_E = 0.0562 H_h {I_p}^{0.93} {B_t}^{0.15} {n_{19}}^{0.41} R^{1.97} \kappa^{0.78} \epsilon^{0.58} M^{0.19} P_L^{-0.69}$$ The power-balance can then be written as: $$Q_{LF} = (f_\alpha /5 + 1/Q )(1-f)$$ where $f_\alpha$ is the fraction of $\alpha$-particles which transfer their energy to the plasma. The $Q_{LF}$ is actually the ratio $P_L/P_F$ and is an involved expression with fractional powers of plasma parameters. To understand its dependencies, it is best approximated as: $$\frac{\beta_N \ A^{14/5}\ q^{6/5}}{{B_t}^{92/35}\ H_h^3 \ f_G^{6/5}\ S_k^{16/5} \ M^{3/5}\kappa^{2/5} \ R^{9/5}}$$ where, the nearest integer ratios are used to approximate the exponents in the expression for $\tau_E$. The radiated power fraction $f$ can be expressed in terms of $Q_{LF}$. Its role in accessibility constraints in the $R$-$B_t$ space has been shown in Fig.1, where, the contours of constant $P_f$ are shown along with the limits on achievable $B_t$ assuming either copper or HTS peak current-density in the center-stack. The constant fusion contours intersect increasingly high divertor load curves as one makes the reactor more compact. The dotted curves ($f$=0, 0.5 and 0.94) correspond to the power balance constraint. The $f=0$ curve shows the limit of 'no core-radiation' and thus represents the lower boundary of physically acceptable solutions. Thus, for a given set of parameters as an example ($q$=3, $\kappa=2.5$, $\delta=0.3$, $\beta_N=5$, $Q=5$), there exists an upper limit on the value of $R$ (3m). The two $Q_{LF}$ curves that 'bracket’ the fusion power curve, define the accessible space until the limit on achievable $B_t$ is encountered. An example of a design point (R=1.25 m, Bt=2.8 T, Pf = 200 MW) has been shown (red dot). It may not be possible to meet it unless almost 60$\%$ of the heating-power is radiated from the core. Such constraints make it necessary to examine how much core concentration of impurities would be acceptable. Fig.2 shows impact of $Q$ in the parameter space -- higher values reduce the available space in the lower left-hand corner. This has implications for the reactors which may operate at modest values of $Q$ (CTF or fusion-fission hybrid, fissile material converters or radioactive waste processing, or just fusion-science devices). At the same time, the higher $Q$ demand from power reactors (to remain cost-competitive and investment-attractive), eliminates a large space and pushes accessibility points further up. An important consequence of the power balance constraint is that the divertor heat load (transported power) $P_{div} \approx B_t^{3/2}/R^{4/5}$. The gradients of $P_{div} \approx$ constant are in dramatic contrast to those of constant neutron load contours, so while the neutron load per unit area varies slowly as one moves towards the top left-hand corner, the divertor load builds up rapidly. Three case studies will be presented ($R$=1.75, 1.25 and 2.25m for $P_f$=100, 200 and 900 MW respectively) in detail. Fig.3 shows how the power balance constrains the $\kappa- \beta$ space for the case $R$=1.25m, $P_F$ = 200 MW. It can be seen that higher $\beta$ cases will need a higher $\kappa$. The sensitivity to different $\tau_E$ scaling, as well as impurity transport, the effects of neutron and particle loads on the center-stack, first-wall and divertor will be presented in detail.

References:

1. Y-K.M Peng et al., Nucl. Fusion 26 769 (1986)

2. Gi et al., Nucl. Fusion 55 063036 (2015)

3. D. Jassaby et al., Plasma. Phys. Control. Fusion 3, 151, (1977)

4. L. A. El-Guebaly et al., Fusion Sci. and Tech., 74:4, 340-369 (2018)

5. A. Sykes et al., Nucl. Fusion 58 016039 (2018) and references therein

6. Dennis Whyte, Phil. Trans. R. Soc. A 377 20180354 (2018)

7. M. Kotschenreuther et al. Nucl. Fusion 50 35003 (2010)

8. R. J. Goldston et al., Phys. Scr. T167 014017 (2016)

9. V. A. Soukhanovskii, IEEE Trans. Plasma Sci. 44 3445 (2016)

10. B. V Kuteev et al., Nucl. Fusion 51 073013 (2011)

11. Y-K.M. Peng et al., Nucl. Fusion 40 583 (2000)

12. B.N. Sorbom et al., Fusion Eng. and Design 100, 378–405 (2015)

13. Tobita K. et al., Nucl. Fusion 49 075029 (2009)

14. Najmabadi F. et al., Fusion Eng. Des. 65 143 (2003)

15. E. Surrey, Phil. Trans. R. Soc. A 377: 20170442 (2019)

16. M. Tillack et al., Nucl. Fusion 53 027003 (2013)

17. H. Takenaga et al., Nucl. Fusion 45 1618 (2005)

18. S. C. Jardin et al., Fus. Eng. Des. 65 165 (2003)

19. P. Kaw et al., Phys. Rev. Lett. 65 2873 (1990)

20. P F Buxton et al.,Plasma Phys. Control. Fusion 61 035006 (2019)

Country or International Organization | India |
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Affiliation | Institute for Plasma Research, Bhat, Gandhinagar, 382428 |