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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.

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