Burning fraction, radial transport, and steady state profiles of multi-species particles in CFETR burning plasmas

The burning fraction of fuel particles is a crucial issue for future fusion reactors. In order to achieve the high tritium burning fraction required by China Fusion Engineering Test Reactor (CFETR) engineering design, fueling depths and quantities should be estimated by particle control analysis for different scenarios. Thus, in this paper, a multi-species fluid model of deuterium-tritium (D-T) fusion plasmas is applied to study radial transport and profile evolution with CFETR parameters under different fueling conditions. In the model, alpha particles are treated with a slowing down model and diffusion coefficients are introduced according to τE_98 . Then, in such a self-consistent burning plasma simulation, the results show that the fusion reaction and fueling parameters effect remarkably changes the shape of D/T profiles, while to alphas and helium ash however, the effect of the fueling parameters is much weaker. It is also seen that the burning fraction is increased substantially with the fueling depth, and significantly affected by particle confinement. Furthermore, by substantially raising the D:T ratio to the regime of above unity, the burning fraction can be increased notably, but with a cost of a certain level of fusion power reduction.


Introduction
For the next generation of fusion reactors, such as China Fusion Engineering Test Reactor (CFETR), particle control is a key issue to be addressed. With significant fueling and ash removal due to GW fusion power and high burning fraction, Original Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. sustaining preferred particle distribution profiles for various scenarios is a great challenge for steady state operation [1].
One of the most crucial issues in CFETR engineering design is the requirement for the fueling system. Different form previous tokamaks, the aim of CFETR is to achieve the steady-state burning plasma with possible tritium selfsustaining [2]. Abdou et al [3] proposed a model of the fuel cycle for D-T fusion reactors and pointed out that rising the burning fraction would effectively easy the requirement of Tritium breeding ratio (TBR). The figure 1 illustrate the fuel cycle in detail.As the D-T mixture pellets are fueled into the plasma, some of them participate in the D-T fusion reaction and produce neutrons and alpha particles, and the rest will be diffused and/or pumped out. While the neutrons provide the fusion power and the tritium breeding source, the alpha particles slow down or even thermalized to helium ash to be removed. Through purification, both of reproduced and recycled tritium particles are deposited in tritium repository for pellet injection. This fuel cycle directly relates the fueling condition with TBR and fusion performance, as shown in figure 1.
According to the definition of burning fraction, the ratio of burned to total tritium particles, the fueling condition is directly related to the fusion performance. By separating effects of edge and central NBI fueling contributions in JT-60 experiments, Takenaga et al [4] found that the confinement time of centrally fueled particles was about 3 times longer than that of edge fueled particles. Nevertheless, since significant NBI central fueling in CFETR is hardly to be realized, we thus in this paper refer the central fueling to the pellet injection. The fuel depth effect on ITER fusion performance was studied by Wang and Wang [5,6]. Their results showed that even a small deviation of the fuel depth was able to make a remarkable difference, e.g. a 0.1 minor radius deeper led to a 61% improvement of the particle confinement as well as a fusion performance change of 108%.
Not only is the fueling system of a fusion reactor a measure to keep the steady state fusion reaction but also an implement for burning control and the requirement of TBR. As a D-T plasma starts to burn, substantial fusion reactions take place and a significant number of alpha particles is generated and then slowed down to helium ash. Then the plasma enters a multi-species state totally different from the original burning-free state. Guazzotto and Betti [7] found that in such a multi-fluid situation, a higher Lawson product threshold was required for ignition than that in a single-fluid model. Also, Boyer et al [8] applied a zero-dimensional multi-species fluid model to study particle control for future fusion reactors with global fueling. Therefore, in order to make more precise prediction of operation scenario and control, it is necessary to consider a multi-fluid model with the burning plasma effect for the fuel system design of CFETR. Previously, the prediction for CFETR density profiles was made by TGYRO [9] under OMFIT framework [10] with the electron particle flux calculated by TGLF [11] in core plasmas and fixed density pedestal [12]. It nevertheless did not take burning effect and fuel system design consideration into account.
To address the issue of particle control in CFETR-like tokamaks, we in this paper propose a multi-species fluid model of deuterium-tritium (D-T) fusion plasmas for radial transport and profile evolution. The related numerical code is then developed based on BOUT++ [13]. Tritium/deuterium, helium ash, and alpha particle density profiles under the different fueling cases are then shown in numerical simulations. Different fuel depth and quantity and the ratio of deuterium to tritium (D:T) are also tested to investigate the effects on the burning fraction.
The layout of the paper as follows. Basic equations of the model, settings of fueling profiles, and boundary conditions are listed and discussed in section 2. Numerical results of particle density profiles, as well as burning fraction and fusion power, with different fueling depth and quantity, and D:T are shown in section 3. Finally, the conclusion is summarized and discussed in section 4.

Basic equations
The multi-species fluid model for particle transport and profile evolution in burning plasmas includes five particle species (deuterium, tritium, helium ash, alphas and electrons), corresponding to n D , n T , n α , n He , n e . For deuterium and tritium, regarded as background species, the fueling and burning loss will directly influence the density. Thus, the corresponding equations are written where Γ D , Γ T is the transport flux of deuterium and tritium, and ⟨σ DT v⟩ is the reaction rate of D-T fusion relying on temperature. The sources of S D and S T are assumed in the form of where S 0 is the amplitude of the fueling source, ψ is the normalized poloidal flux, ψ 0 is the fueling deposition position, and (w = 0.1) is the profile width of fueling. Alphas are created by D-T fusion reaction and then slow down to helium ash. While it is a reasonable approximation for the helium ash, the fluid treatment is not a good assumption for alphas with a shell-like distribution in the velocity space. Nevertheless, following the slowing down discussions in reference [14,15], we have the steady state relation of S α = n D n T ⟨σ DT v⟩ − nα τs , where τ s is the energy confinement time of alphas on the order of the slowing down time. Hereafter we then use the slowing down time to approximate the energy confinement time. Thus, one can assume a fluid-like treatment of the continuity for alphas where Γ α is the particle transport flux of alphas. Then we also can use the relation to distinguish the helium ash from alphas by using the alphas loss due to energy dissipation, nα τs , as the source of the ash. Therefore, one can also have the continuity for the helium ash where Γ He is the particle transport flux of the helium ash. According to the classical slowing down theory [16], which works pretty well in JET D-T fusion [17] and alpha slowing down experiments [18], the slowing down time can be expressed by: where τ se is the electron-alpha slowing down time, W c is the critical energy, T th,i is the ion temperature, and f α is the alpha particle energy distribution, with also and Z are the initial energy, atomic mass, charge, and effective charge of alphas, respectively. Then the electron density can be determined by the quasi-neutral condition n e = n D + n T + 2n He + 2n α (10)

Boundary conditions and code development
To simplify the problem, the temperatures of these species are assumed to remain the same (T e = T D = T T = T He ) in the simulation, with the central temperature slightly less than 30 keV to avoid overestimates of fusion power and burning fraction. For boundary conditions, the Neumann condition is applied for all species in the core. On the edge however, we set

Diffusion and pinch profiles
The particle flux in above basic equations for different species can be expressed where the subscript s stands for different species, with the diffusion coefficient D s , and the pinch velocity V s . Clearly, in steady states without fusion reaction, the diffusion term −D s ∇n s should be balanced by the pinch term to maintain the profile. Both D s and V s can then be obtained by theory or experiments. In this simulation, we assume that in the radial direction, diffusion coefficient and pinch velocity for approximately thermalized species (D, T, He) have the form of where χ s is the heat conduction coefficient, and a is the minor radius. Also, the same as in Corediv JT-60SA simulation [19], Note that the average density n,the average atomic mass M, and the heating power P will evolve with the simulation process, which directly affects the magnitude of the diffusion coefficient as shown in equation (14). And they are calculated consistently as M =´V 4n α + 4n He + 2n D + 3n T n α + n He + n D + n T Jdψdθdξ V Jdψdθdξ The diffusion and pinch of alphas is very different from the other species. While the fully kinetic treatment remains a great challenge, we in this simulation use a simplified approach based on the quasilinear model by Angioni et al for study of micro-turbulent transport of alphas and helium ash [20]. With fitting the results of the gyrokinetic simulation, they found that the alpha particle diffusion and pinch can be expressed as where L α n and L Te are the characteristic lengths of alpha particle density and electron temperature, respectively. The alpha particle pinch velocity is set to be 0 when very close to edge. To clearly describe the numerical model, the crucial profiles in the simulation, i.e.

Particle density profiles of various species
As the first step of our simulation, we compare the variation of density profiles under different fuel conditions. Four different fuel depths at ψ 0 = 0.3, 05, 0.7, 0.9 in equation (3), and four different fuel quantities of´V SJdψdθdξ = 9.51 × 10 21 , 1.05 × 10 22 , 1.26 × 10 22 .1.415 × 10 22 , are used in this simulation. The ratio of deuterium to tritium is fixed as 1:1 in this subsection. Then the steady state density profiles corresponding to different fueling cases are shown in figure 3.
Comparing the results of figure 3 with fusion term n D n T ⟨σ DT v⟩(the solid lines) in equations (1)-(2) and without it (the dashed lines), we can we can find that the fusion effect significantly re-shapes the density profiles. In the region of ψ < ψ 0 = 0.4, the density profile gets flattened due to the balance between fusion reaction and fueling. For a fixed fuel depth, i.e. the same color lines in figures 3(a)-(d), but with different fuel quantities, it is shown that the fuel quantity hardly affects the density profile shape, but only its magnitude. The more obvious effect is seen as the fuel position changes. The density is more significantly reduced as the fuel depth gets shallower, i.e. the fueling position ψ 0 moves out to the edge, from ψ 0 = 0.3 to ψ 0 = 0.9. For example, in the case of the fuel quantity of 9.51 × 10 21 , as shown in figure 3(a), the density profile significantly re-shapes and becomes higher in the The profiles are similar to that measured in TFTR D-T fusion experiments [21], with a very sharp peak at the center due to the small alpha diffusion coefficient in that region, as figure 2(d) shown. Nevertheless, such simulation results are only an upper limit of alphas density, due to the fact that certain important losses of the alphas, particularly the noteworthy first orbit loss due to the kinetic effect [21], are not included in the model. With the fuel quantities rising from 9.51 × 10 21 to 1.415 × 10 22 at fixed fuel position, the magnitude of the central density is only changed by~16%. However, when the fuel position ψ 0 change from 0.3 to 0.9, the magnitude of central density is changed almost 100%, which is consistent with the conclusion from figure 3 that the fuel depth strongly reshape the density of deuterium and tritium profile. The fraction of alphas is about 1.5%-3% of the D/T ions in the core, indicating a 3% burning fraction (with D:T = 1) under fueling conditions in the parameter regime of CFETR operations.
Helium ash density profiles in a steady state operation under corresponding fueling conditions are shown in figure 5. Unlike the D/T density profiles, the shape of the helium ash profile is not changed much as the fueling depth varies. A larger transport coefficient makes the helium ash density consistently higher as the fuel position ψ 0 increases from 0.3 to 0.9, i.e. the fueling depth (= 1 − ψ 0 ) decreases from 0.7 to 0.1,which is quite different from the alphas density profile. It is because that the distribution of the ash is not only due to alphas slowing down but also the self-diffusion of the helium ash. On the other hand, the tendency of helium ash variation with the fueling quantity is similar to the alphas since the fueling is the basic source of both. The central density of the ash is increased about~16% as the fueling quantity rising from 9.51 × 10 21 to 1.415 × 10 22 .

Fusion power and burning fraction under different fueling conditions
In this section, we focus on the relationship of burning fractions and fusion power under the different fueling conditions. The tritium burning fraction can be calculated as follows, Figure 6(a) shows the change of the fusion powers in different fuel cases. As be raised substantially by the fueling depth, the fusion power is also shown approximately proportional to the fueling quantity for a given fueling depth, e.g. the fusion power for 1.415 × 10 22 s −1 is approximately 1.5 times of that for 9.51 × 10 21 . It is due to the fact that the burning fraction is hardly affected by the fueling quantity, for a given fueling depth, as shown in figure 6(b). In other words, since the burn-  ing fraction is almost a constant for fixed fueling depth and temperature, the burning rate is then proportional to the fueling quantity, i.e. the particle source. Clearly, the burning rate is also proportional to the fusion power. Therefore, one sees the proportional relation of the fusion power and the fueling quantity. Figure 6(b) shows the change of the burning fraction in different fueling cases. It is found that clearly, the fuel quantity variation has little effect on improving the burning fraction. On the other hand however, as the fuel position change from ψ 0 = 0.3 to 0.9, the burning fraction can rise about 3 times.

Effects of diffusion and D-T rate
The diffusion coefficient D s = 0.1χ s and D:T = 1 are assumed in above simulations. In this subsection, we fix the fuel quantity at 1.   ψ 0 = 0.9, very easy to be achieved. However, if the confinement gets worse, the design goal of the 1 GW fusion power plus the >3% burning fraction requires a fueling depth of 0.7, i.e. a fueling position at ψ 0 = 0.3 or deeper, hardly to be reached. In fact, the scaling of the diffusion coefficient reduces the density linearly, causing a squarely falling of fusion power and burning fraction as shown in figures 7(a) and 7(b).
For engineering design, if the cost of deepening the fueling depth is infeasible, one may try to change the D:T radio to raise the tritium burning fraction. However, it is on the other hand with the cost of fusion power reduction. Then, one can make a balance between the two. To do the optimization, we plot the relation of burning fraction to fusion power vs. the D:T ratio in figure 8, with the fusion power in blue dots and the burning fraction in green triangles. Surprisingly, for the mixture ratio D:T = X:Y, the fusion power is not reduced by the factor of 4XY (X+Y) 2 . As shown in figure 8, the burning fraction is almost double with the ratio D:T = 4:1 and the fusion power is however only reduced by 20%. The reason is due to the diffusion coefficient scaling of τ E_98 ∝ P −0.69 . The diffusion coefficient is significantly reduced with the fusion power, leading to the increment of the particle density to prevent the further falling of the fusion power. It can also be seen in figure 8 that the plot of the fusion power is not symmetric on D:T = 1:1 with the points at the left is higher than that at the right, a similar effect due to the τ E_98 dependence on the mass ratio. These results then propose a new measure to achieve the high tritium burning fraction by choosing a proper D:T mixture ratio. To avoid a substantial the fusion power falling, more work is needed to optimize the operation scenarios.

Conclusion
In summary, in order to address the issue of particle control, including fusion reaction, fueling, and ash removal, in burning plasmas and to reach the goal of tritium self-sustaining, we develop a multi-species fluid model for burning plasmas including five different species (n D , n T , n He , n α , n e ). The classical slowing down model is applied for a fluid treatment of alphas, and empirical diffusion coefficients and a pinch model are also used.
The numerical simulation results are as follows.
(a) The fusion reaction effect is very significant in shaping the D/T ion density profiles. (b) The fuel depth remarkably changes the shape of D/T profiles, while to alphas and helium ash, the effect is much weaker. (c) While it is not affected by the fueling quantity much, the burning fraction is increased substantially with the fueling depth, e.g. three times as the fueling depth changes from 0.9 to 0.3. (d) The fusion power will linearly incresed with the fuel rate when the temperature fixed. Also, the fusion power monotonically increase with the fuel depth increasing. (e) The particle confinement has a great influence on the fusion power and the burning fraction. (f) For engineering design, if the cost of deepening the fueling depth is infeasible, one may try to change the D:T radio to raise the tritium burning fraction. By raising the D:T ratio to the regime of above unity, one can increase the burning fraction remarkably, but with a cost of a certain level fusion power reduction.