The effect of gas injection location on a lithium vapor box divertor in NSTX-U

The lithium vapor box divertor is a proposed divertor design that will minimize contamination of the upstream plasma in a fusion device, while also ensuring protection of the target. In this design lithium is evaporated near the target by high temperature lithium surfaces, dissipating the plasma heat flux. The lithium vapor box has been predicted via the fluid-kinetic code scrape off layer plasma simulator (SOLPS-ITER) to achieve low ( nLi / ne∼  0.05) upstream concentrations of lithium and low target heat fluxes. Here we compare two choices of deuterium gas puff location using SOLPS-ITER, the private flux region (PFR) and the common flux region (CFR), and find significant differences in the contamination level required to reach an acceptable target heat flux (defined here as q tarmax⩽ 10 MW m−2). Deuterium gas puffing from the PFR is seen to better reduce upstream lithium contamination. The difference in puffing location is seen to cause changes in the upstream flow of lithium ions. The PFR puff, having better access to the separatrix, can better reduce the upstream-directed flow of lithium near the separatrix which is the primary source of contamination due to a large thermal force in this region. Puffing from the CFR, partially due to inefficacy at reducing separatrix lithium flow, has higher lithium concentration within the plasma. Solutions that reduce the heat flux to below 10 MW m−2 have a range of lithium concentrations between nLi / ne ∼  0.01–0.12 depending on puff intensity, location, evaporator temperature and recycling at the various plasma facing components. The efficacy of the puffs is tested for sensitivity to deuterium recycling coefficient at the target, evaporators, and main chamber walls. A CFR located puff is found to be more dependent on the recycling coefficients used than a PFR located puff. regardless of the set of recycling coefficients chosen, PFR puffing achieves lower lithium contamination than CFR puffing for a given heat flux.


Introduction
Commercial tokamak fusion devices are predicted to face large heat fluxes at the target that, if left unmitigated, would be harmful to plasma facing components (PFCs) [1,2]. * Author to whom any correspondence should be addressed.
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Dissipating the power reaching the divertor target, a process known as detachment, appears to be necessary for these reactors. This dissipation has two requirements relevant to the analyses presented here: 1. Impurities reaching the main plasma, where fusion should occur, are kept to a minimum in order to reduce fuel dilution. 2. Radiation from these impurities should be kept below the X-point or else risk reductions in energy confinement time, τ E .
Recently, radiation at the X-point in ASDEX-Upgrade is seen to be well controlled such that reductions in energy confinement for nitrogen puffing were minimal (H 98 ∼ 0.95) [3,4]. However, nitrogen can form tritiated ammonia, potentially leading to unacceptably large retention of tritium in a burning plasma device making it unattractive for future reactors. Furthermore, stable X-point radiator solutions with other gaseous impurity species, such as neon, argon or krypton, proved difficult to achieve since these species dominantly radiate at higher temperatures and thus radiate further upstream [5]. The lithium vapor box divertor is a proposed divertor design which would achieve detachment by lithium evaporation near the target. Lithium radiates at temperatures lower than nitrogen and is predicted to radiate below the X-point, avoiding the need for controlling an X-point radiator. With its boxed divertor design, simulations of the lithium vapor box have shown it to be successful at preventing neutral lithium from ionizing upstream [6] while fuel puffing has been shown to be successful at increasing the friction force compressing the lithium ions into the divertor [7]. The combination of puffing and divertor geometry results in minimal lithium contamination of the upstream plasma while maintaining acceptable heat flux (sub-10 MW m −2 ) in a scenario with unmitigated heat flux to the lower outer target of 65 MW m −2 . The lithium concentration in the detached scenario was (n Li /n e ) OMP,sep ∼ 0.05 [6]. While sub-10 MW m −2 has been shown to be acceptable for a tungsten target, a lithium wetted capillary porous system is expected to survive similar heat fluxes, with up to 16 MW m −2 being reported as acceptable for lithium filled tungsten-CPS targets in the literature [8,9]. The 10 MW m −2 limit can be seen as a conservative limit on the acceptable heat flux for a lithiumwetted CPS divertor target.
A central difference between past articles on the vapor box is the location of the fuel puff used which has a consequential impact on the prediction of upstream lithium concentration [6,7]. Here, scrape off layer plasma simulator (SOLPS-ITER) [10] simulations of a lithium vapor box divertor are shown exploring how to optimize the fuel puff used to increase the friction force acting on lithium ion impurities. Fuel puff location is found to have consequential impact on lithium concentration predictions at similar target heat flux. Deuterium puffing from the private flux region (PFR) is found to be much more effective at containing lithium to the divertor region compared with puffing from the common flux region (CFR). Lithium is also known to reduce deuterium recycling at lithium covered surfaces depending on the temperature and deuterium particle flux to the lithium covered surfaces. Here, the target, evaporator and main chamber recycling coefficients are varied in order to estimate the effect of a reduction in deuterium recycling. It is found that with higher recycling coefficients, lithium concentration can drop to practically negligible levels, while the maximum lithium concentration with unfavorable recycling conditions is approximately (n Li /n e ) LCFS ∼ 0.12, where the LCFS superscript denotes the average value at the last closed flux surface.

Transport coefficients and core boundary conditions
The same artificial NSTX-U magnetic equilibrium and grid is used as a previous analysis of a lithium vapor box divertor and the reader is referred to that analysis for further context [6]. Though it will be said again that with the parameters of table 1, n Greenwald = 1.98 × 10 20 m −3 . The magnetic equilibrium is also noted here to have a low flux expansion (f x = (B θ R)u (B θ R)t = 5.7) thus allowing for reactor-relevant heat fluxes to be achieved even with the relatively low heating power of NSTX-U. The crossfield transport coefficients have been altered from the previous analysis to achieve the following more realistic characteristics: 1. The core particle transport coefficient, D n , is reduced in order to reduce the core-edge interface (CEI) deuterium particle flux to a more realistic level. 2. The SOL heat transport coefficients, χ e,i , are reduced such that fitting the unmitigated target heat flux with an Eich function yields a SOL width consistent with the heuristic drift model [11]. The Eich function fit to the previous analysis' target heat flux profile yielded a SOL power width of 3 mm while the current target heat flux profile yields 2 mm, in line with the heuristic drift model prediction of 2.1 mm [11]. The Eich regression model would predict a power width of 1.4 mm when using data from MAST and NSTX single-null H-modes (regression #15) [12].
The first change yields higher upstream lithium concentrations since reduced main ion flow from the core reduces the friction force acting on the lithium particles. The second change yields a higher target heat flux. The resulting transport coefficients are shown in figure 1. Several changes are also made to the simulation boundary conditions that mitigate the increased difficulty of the power exhaust scenario. In order to maintain the same separatrix density (n sep ∼ n GW /3), the CEI density boundary condition is raised from 1.3×10 20 m −3 to 1.6×10 20 m −3 . The increased CEI density results in increased particle flux from the core, as SOLPS must increase the main ion particle flux  The Eich function fit to the unmitigated target heat flux used for this analysis. This is unmitigated heat flux profile is given for the current NSTX-U PFC geometry (i.e no box divertor). λq is consistent with the predictions by the heuristic drift model. in order to satisfy the given density boundary condition at fixed D n . However, combined with the reduced core transport coefficients the CEI particle flux saw a net reduction from the previous analysis from Γ core D+ = 8.98 × 10 22 to Γ core D+ = 3.56 × 10 22 D + /s in the unmitigated case. Using a density at the Greenwald limit of 1.98 × 10 20 m −3 and taking the volume of NSTX-U as V = 2π 2 Ra 2 κ = 10.9 m 3 , the particle confinement time is calculated as Particle confinement studies would indicate τ p /τ E ≈2-4 for L-mode and quiescent H-modes [13][14][15]. Using the ITER H 98(y,2) scaling law [16] for energy confinement time and the values in table 1 predicts approximately 52 ms for the energy confinement time. Thus the core particle flux used here is still somewhat higher than expected based on the H 98(y, 2) scaling. However, energy confinement scaling based on lower current, lower power NSTX-U discharges would indicate an energy confinement time of 15-30 ms for a 2 MA and 10 MW discharge [17]. This is based on current and power scaling of I 0.38 p and P −0.66 which is similar in power scaling to the H 98(y,2) scaling but has a weaker dependence on current. Studies to quantify the effect of different core particle fluxes on lithium vapor box performance are the subject of future modeling efforts.
The previous analysis [6] also ignored power radiation from within the CEI and assumed a power from the CEI of the full 10 MW that NSTX-U can deliver to the plasma. 30% core radiation is assumed here and thus the CEI power boundary condition is 7 MW here compared to 10 MW in the previous analysis. The reduced core power combined with the reduced SOL transport coefficients results in the unmitigated target heat flux increasing from 65 MW m −2 to 91 MW m −2 . The unmitigated heat flux and resulting Eich function fit is shown in figure 2 [18].
The simulations presented here do not include drifts, which are expected to significantly impact the solution for particle flows with greatest impact on the inner divertor [19]. Including drifts in the present analysis would have required significantly more computational time, as convergence with drifts turned on can be slow. Simulations examining the effect of drifts on lithium vapor box performance are under way. Deuterium puff location is shown via the green dot, lithium evaporators are shown in red and the target (which has a non-unity recycling coefficient for deuterium, R D ) is shown in orange. The fuel puffs significantly increase the downstream lithium poloidal particle flux, which reduces upstream lithium concentration.

SOLPS momentum equation
SOLPS-ITER version 3.0.8 is used here and has the appropriate impurity forces, which are valid for Z eff >1.5 [20]. For ease of reference, the full parallel momentum equation solved for a species a by SOLPS is reproduced below: where Γ m ax is the momentum flux in the poloidal direction and Γ m ay is the momentum flux in the radial direction. h x,y = 1 |∇x,y| and h z = 2π R. √ g is the volume element. Φ is the electric potential.
↔ π is the viscous stress tensor. The S m 's are the various momentum sources: the coriolis force, friction forces, thermal forces, ionization, recombination, charge exchange, anomalous current, and plasma-neutral interactions from EIRENE, respectively. The other variables have their usual meaning. Examining the sources shows that the friction force due to the deuterium on the lithium is generally the dominant source of momentum towards the divertor plate for the lithium while the thermal force pushes the lithium towards the main chamber. These sources are balanced by the gradients on the left side of this equation from which SOLPS solves the plasma state. Though the thermal and friction forces are quite similar in magnitude above the main ion flow stagnation point, below the main ion flow stagnation point the pressure gradient term is dominant and balances the mismatch between the friction and thermal forces. This result aligns with other work on impurity forces in the divertor [21].

Varying puffing location and recycling
A central comparison made here is the effect fuel puffing has on the plasma when located on the CFR side or the PFR side. The two locations used are shown in figure 3. There may be further variation of the effect of the D-puffs for upstream compared to downstream puffing, however the scope of this article has been limited to allow for more in depth analysis comparing these two specific locations. For clarity, 'puffing' shall be used to indicate deuterium injection into the simulations while evaporation shall be used to indicate lithium injection into the simulations. Deuterium is injected as D 2 gas with an energy of 0.03 eV. The deuterium is pumped at the targets which are given a deuterium recycling coefficient of R D = 0.95 in the baseline scenario, though this parameter is scanned in section 4.1. In the baseline scenario all non-target walls have a R D = 1.0. In sections 4.1 and 4.3 this assumption is examined for how it affects fuel puff efficacy. For all simulations presented here, the lithium recycling coefficient at all walls is set to zero (entirely absorbing). Lithium has a very high sticking coefficient on liquid lithium and cold PFC surfaces, both of which would be expected for NSTX-U. A commercial reactor will likely have hotter walls and non-zero lithium reflection will be a topic for future modeling efforts.

Target lithium evaporation
The lower outer divertor target (orange walls in figure 3) is given evaporation consistent with a 500 • C lithium surface, which amounts to 1.514×10 22 Li/s for the given surface area. This temperature was found to be reasonable for an advanced capillary porous systems using liquid lithium [8] when the heat flux was limited to 11 MW m −2 . In the present analysis the effect of thermal and physical sputtering are ignored, as the model for target evaporation here is very simple. At higher temperatures, evaporation is expected to be the dominant mechanism for lithium emission, however the temperature range at which sputtering becomes negligible is uncertain as the sputtering yield of lithium is not well established [22,23]. In the present analysis, any increase in lithium emission due to sputtering effectively corresponds to a smaller assumed target temperature. A reduced temperature should, in principle, be possible given changes to the design criterion of the advanced capillary porous system [8]. Accurate calculation of target evaporation is a point of future work where more advanced modeling shall be performed to give self-consistent target temperatures that include the effect of sputtering with the lithium emission, which has a non-negligible effect as seen in other work modeling lithium capillary porous systems [24,25].
Lithium evaporated from the target is rapidly ionized and redeposited in the more attached cases, causing only a small decrease in the target heat flux and a small amount of lithium upstream. For all simulations shown here the target is assumed to be held at this 500 • C evaporation rate, hence setting the side evaporators to zero lithium flux yields non-zero lithium concentration in figure 4. Lithium evaporation rates quoted are only the amount of lithium coming from the side evaporators (red walls in figure 3) as the target evaporation is kept at a constant value. In the more detached cases the large evaporator fluxes from the side evaporators dominate such that the added lithium from the target has a minimal effect.
Changing the PFCs alone changes recycling conditions in the simulations and reduces the heat flux down to 85.3 MW m −2 down from 91.2 MW m −2 . Just evaporating from the target within the box while turning the side evaporators off gave a heat flux of 73.8 MW m −2 . Thus the heat flux reduction from 91.2 MW m −2 to 73.8 MW m −2 is given just by the PFC geometry and assumption of lithium conditions at the target.

PFR puffing reduces upstream flow of lithium ions further than CFR puffing
The main effect of changing the puff location is shown in figures 3 and 5. There the upstream-directed flow is shown in blue while the downstream-directed lithium flow is shown in red. The region of downstream-directed deuterium and lithium flow moves away from the target as deuterium puffing is introduced, more so on the side where the deuterium puff is located. Importantly the flow of lithium upstream at the Xpoint is heavily reduced for both puffs. Without a puff, the total lithium flow upstream at the poloidal location of the X-point is 5.82×10 21 Li/s when 10 24 Li/s is evaporated from the side evaporators. With a 10 23 D 2 /s CFR puff that flow is reduced to 1.80×10 21 Li/s, while the same puff from the PFR reduces that flow to 1.07×10 21 Li/s. Differences in the upstream lithium ion concentration upstream and, more indirectly, the target heat flux are explained by this alteration of flow. The flow at the separatrix is more significantly reduced with a PFR puff due to the close access the PFR puff has to this region. Simultaneously, the near-SOL is the largest source of lithium from the target. This is expanded upon in section 3.2.

Discussion of forces acting on lithium
A large thermal force acts on the lithium ions at the separatrix since it has the highest temperature upstream and thus largest temperature gradient, which indicates the separatrix is the most critical region to be affected by the friction force. The deuterium puff's effect on the friction force and the interplay with the thermal force acting on the lithium has been noted in other publications on the lithium vapor box [7]. The friction force approximately balances the thermal force, with the caveat that the pressure gradient becomes significant in regions below the stagnation point [21,26]. The friction and thermal forces, as well as the pressure gradient term from section 2.2 are plotted in figure 6 and is seen to agree with this description. In the figure it is seen that the addition of the friction and thermal forces acting on the lithium is significantly more positive near the target when a D-puff is added. This heavily reduces the lithium concentration that makes it upstream. The friction force and thermal force are seen to be the dominant terms of momentum balance for the simulations presented here. Figure 5. The deuterium ion poloidal particle flux for three representative cases, (a) without deuterium puffing, (b) CFR deuterium puffing, and (c) PFR deuterium puffing. Red areas indicate target-directed deuterium flux while blue areas indicate upstream-directed deuterium flux. Deuterium puff location is shown via the green dot, lithium evaporators are shown in red and the target (which has a non-unity recycling coefficient for deuterium, R D ) is shown in orange. Note that the down stream directed deuterium flux significantly increases when the fuel puff is turned on, with a bias towards the side where the fuel puff is located.

PFR-side gas puffing reduces upstream lithium concentration more effectively
The effect of the gas puffs on the lithium concentration, n Li /n e , at the LCFS is shown in figure 4(b). Lower lithium content is the result of either puffing location, however the PFR located puff is clearly more effective at minimizing contamination. This is due to the changes in lithium flow mentioned in section 3.1. The difference between the two scenarios becomes more extreme as the puff intensity is increased. At the largest puffing intensity, 10 23 D 2 /s, the first lithium evaporation rate to reach sub-10 MW m −2 was 4×10 23 Li/s for the CFR puffing and 6×10 23 Li/s for the PFR puffing. The difference in lithium fraction between these two cases was over a factor of three, 0.041 and 0.013 respectively. Naturally, this difference is minimized at lower deuterium puffing intensities. At the lowest puffing rate tested, 10 22 D 2 /s, the lowest lithium evaporation rates to achieve sub-10MW m −2 had lithium concentrations of 0.100 for the PFR puff and 0.108 for the CFR puff. Thus this different behavior may not be significant if deuterium puffing rate is limited to lower values. However SOLPS simulations here show little sign of problems at the highest puff rates tested, see section 3.5.

CFR-side gas puffing reduces heat flux further
The heat flux to the lower outer target, including radiation and neutral contributions, is shown in figure 7. From the figure, it can be seen that CFR puffing succeeds at reducing the heat flux to acceptable levels at lower lithium evaporation rates than the PFR puffing, albeit at the cost of higher upstream contamination. However both puffing sides were able to get to sub-10 MW m −2 target heat flux with the correct combination of lithium evaporation and puffing.

Changes to upstream temperature
The upstream temperature was slightly lowered as the solutions became more detached. The upstream temperature is shown in figure 8. In the worst case, the temperature is reduced by about 10% from the case with no puffing or lithium efflux from the evaporators. The temperature is reduced more significantly for PFR puffing, a result of some radiation near the X-point which occurs due to deuterium neutral line radiation. However, the temperature reduced cases' density is significantly higher from the larger deuterium puffs such that the upstream plasma pressure is overall higher from the addition of neutrals to this system. The density increase at the outer midplane is shown in figure 9. Figure 9 shows that the density increase due to lithium evaporation can be significant, from 10% to 40% at the same rate of deuterium injection. With the lowest increases corresponding to the lowest upstream lithium concentration cases.
The radiation in these simulations is strongly localized, a natural result of a lithium vapor box divertor. The plasma and neutral line radiation is shown in figure 10. The radiation only reaches the X-point in the most extreme cases of PFR deuterium puffing, and this radiation is due to deuterium neutral line radiation, not the lithium based radiation. In the case with 10×10 23 Li/s and 10 23 D 2 /s CFR puffing, 87% of line radiation occurs in cells poloidally below the lower X-point or poloidally above the upper X-point (SOLPS divertor regions). The same case with PFR puffing has 95% of radiation in the divertors, while no puffing gets 87%. Thus regardless of puffing, the lithium vapor box succeeds in localizing the radiation to the divertor regions.
The radiation is mostly due to the lithium in these simulations. For example, the case with 10×10 23 Li/s and no puffing had 0.39 MW of radiation from D 0 , 1.05 MW from Li 0 , 0.21 MW from Li + and 0.86 MW from Li ++ . The radiation from the higher charge states of lithium drops off with deuterium puffing, as this contains the lithium plasma Figure 6. The friction, thermal, and pressure gradient force acting on the lithium at the separatrix, normalized to the lithium content at the separatrix. This is thus a plot of force per lithium ion. Note that positive is taken to be downstream-directed force. In agreement with other literature [21,26], the pressure gradient mostly compensates for any mismatch between the thermal and friction forces acting on the lithium below the main ion flow stagnation point. The location of the stagnation point is marked by the dashed vertical line. The stagnation point is significantly more upstream due to the PFR deuterium puff compared to the no D-puff and CFR D-puff scenario. The change in stagnation point location combined with the addition of thermal and friction forces being more positive leads to improved lithium containment.  to cooler regions below the second ionization point. When 10 23 D 2 /s of PFR puffing is added, the Li ++ line radiation decreases to 0.16 MW, CFR puffing decreases this number  to 0.24 MW, reflecting the reduced containment of lithium due to CFR rather than PFR D-puffing. The neutral line radiation increases for both species as a result of deuterium injection, though primarily the increase is due to neutral deuterium. 10 23 D 2 /s of PFR puffing causes the D 0 neutral line radiation to increase from 0.39 MW to 0.78 MW, while Li 0 increases from 1.05 MW to 1.35 MW. Similar increases are seen for the CFR puffing, which saw 0.81 MW and 1.24 MW for D 0 and Li 0 radiation respectively. Much of the remaining power flux to the lower outer divertor target is dissipated by increased energy transfer to the neutrals which spreads the heat flux across the first wall rather than the target.

Puffing from both sides
Naturally, after the previous analyses, puffing from both sides was also simulated with puffing evenly distributed between the PFR and CFR puffing locations. The results on the lithium concentration for the highest lithium evaporation rate are shown in figure 11. It is seen that, for the same total deuterium puffing rate, the simultaneous puffing has lithium concentration lower than the CFR puffed scenario but higher than the PFR puffed scenario. The PFR puff is thus the most efficient use of deuterium puffing of the three combinations of puffing locations examined. The reduction of lithium flow at the separatrix appears to be the most important parameter for determining upstream lithium concentration.

Sensitivity to deuterium recycling
The deuterium recycling coefficient at the various surfaces in experiment is very uncertain, with strong dependencies on the temperature and relative particle fluxes of the deuterium and lithium species [9,27,28]. Generally, it is expected that deuterium absorption above 400 • C-500 • C falls off rapidly due to thermal decomposition of lithium deuteride, though below this temperature deuterium absorption is high. As the walls in NSTX-U are not heated, this could indicate that non-unity recycling is to be expected. However, one could also reasonably assume that at deuterium fluxes significantly higher than lithium fluxes, the wall could be in a regime with surface saturation of the lithium coating and consequently have high recycling. For the cases shown here, the flux of lithium to walls outside of the box is drastically lower than the flux of deuterium to those same walls. For instance in the case with the worst lithium contamination, 10 × 10 23 Li/s with no deuterium puffing, the first wall segments above the box (those poloidally upstream of the box) received roughly forty times as much deuterium nuclei flux as lithium nuclei flux. This deuterium nuclei flux includes D 2 , atomic D made up roughly half the deuterium nuclei flux to this surface. The fraction of particle flux that is deuterium only increases as the walls get further from the lithium vapor box.
Thus, the assumption of unity deuterium recycling coefficient outside the box is not unwarranted. Regardless, the chosen recycling coefficients for each of the plasma facing surfaces were found to be important to the predictions presented in the previous sections. The total deuterium nuclei flux to the walls in these simulations can range from 3 × 10 23 D/s to 7 × 10 24 D/s depending on the puffing and evaporation rates. Thus the recycling coefficient can significantly change the deuterium flow in the simulation. Select deuterium recycling coefficients were changed to determine the sensitivity of section 3 results to the assumed recycling coefficients. Throughout all simulations presented, the lithium recycling coefficient at the wall is taken to be zero, i.e lithium is entirely absorbed by the walls [29].

Target recycling coefficient
The target is assumed to be evaporating at 500 • C which should retain very little deuterium [8,9,28]. However if the temperature was not well controlled and reduced below that value, lower recycling may occur. R target is lowered to 0.90 and raised to 0.99 from 0.95. The result on lithium concentration at the LCFS is shown in figures 4(a) and (c). There it is seen that the upstream lithium concentration predictions become higher at a lower recycling target. Without the puff, the lower target recycling has little effect, with the prediction staying around n Li /n e ≈ 0.12. Conversely, higher recycling cases do significantly better. For large puffs and high recycling cases, the lithium density is much less than a percent of the electron density. However it should be noted that at the highest puffing rate and R target = 0.99, the plasma is experiencing detachment primarily due to heat transfer to the deuterium neutrals. Without a puff, the increased recycling can have a significant effect on its own, reducing (n Li /n e ) LCFS to around 0.07. It is also seen that the CFR puff is significantly more dependent on recycling to reduce the lithium fraction. At high recycling, the difference between the CFR puffing and the PFR puffing becomes small while at low recycling the difference between the two locations becomes more significant. Though in all cases PFR puffing still achieves lower lithium contamination. Summarily, the assumed target recycling coefficient can significantly affect predictions of the deuterium puff's effect but does not strongly affect the predictions for the lithium concentration for cases without a deuterium puff. In cases without a puff the target plasma is 90%-99% lithium, making the deuterium recycling coefficient not as impactful. Cases with puffing, where target lithium density can be 40%-90% of the electron density, are more dependent on the target recycling coefficient. This is also seen via table 2 which summarizes the relevant numbers for the three R target values.
The beneficial effect of recycling is due to it is effect on the poloidal deuterium ion flow. The peak separatrix poloidal deuterium ion flow towards the target increases in the higher recycling cases, causing improved containment of the lithium via a stronger friction force for the given thermal force as discuss in section 3.2. Furthermore the flow stagnation point (the area coinciding with the deuterium ionization front where the flow switches from downstream directed to upstream directed as shown in figure 5) moves upstream when higher recycling occurs, which has been known to improve impurity containment to the divertor [21]. These effects are found to varying degree across the three sets of surfaces where recycling was altered. At fixed injection of both deuterium and lithium, there is a dramatic effect of changing the deuterium recycling coefficient on upstream lithium concentration. The effect of decreasing lithium injection rate alone is comparatively less pronounced.

Evaporator recycling coefficient
The evaporators (red walls in figure 3) are unlikely to pump deuterium via lithium deuteride formation given that they are assumed to be around 600 • C-700 • C where LiD would decompose very rapidly. However, co-deposition may be present, leading to some effective deuterium pumping. In the baseline recycling scenario (R Target = 0.95, everywhere else unity) the ratio of lithium to deuterium flux to the evaporators is 0.5-3.0. The evaporator recycling coefficient is made nonunity and the effect on the upstream lithium concentration is shown in figure 12.
It is seen that the evaporator recycling coefficient is highly consequential for the CFR puffing predictions. At R evap = 0.50, the CFR puffing barely has any effect on the upstream contamination at all, with (n Li /n e ) LCFS ≈ 0.10-0.12 regardless of CFR puffing rate. The reduced CFR puff effectiveness was also seen with the reduction in target recycling in figure 4. The CFR puff does not have good access to the separatrix, a primary source of lithium contamination as noted in section 3, thus it relies heavily on recycling to increase friction acting at the separatrix and reduce lithium concentration upstream. Should the lithium vapor box find itself in a low recycling regime, a CFR puff may be ineffective at reducing upstream lithium contamination.

Main chamber recycling coefficient
Not much lithium is expected to escape the main chamber, as noted at the beginning of this section. However additional lithium coatings may be of experimental interest given the benefit of low recycling regimes caused by low temperature lithium coatings [30]. Furthermore, between discharges the lithium vapor box will remain hot for some period thus leading to some lithium escaping the box region. The main chamber recycling coefficients are scanned to determine the effect of non-recycling lithium coatings. Lithium concentration as a result of this scan in shown in figure 13. Results are similar to the scan in evaporator recycling coefficients, showing the reliance of the CFR deuterium puff on recycling in order to achieve the reduction in upstream lithium concentration.

Discussion on acceptable lithium contamination
The SOLPS simulations presented here cannot answer the question of what is the acceptable quantity of upstream lithium concentration in a commercial reactor. To balance the fusion heating with the losses due to radiation, characterized as the burn-condition [31], the maximum lithium concentration is between n Li /n e = 0.01-0.20, depending on the value of ρ * = τ * He /τ E . ρ * of around four has been observed in ASDEX Upgrade plasmas [32], for which this analysis would indicate n Li /n e ≈ 0.10 would be acceptable. A more conservative ρ * of 10 would have the limit of lithium concentration be n Li /n e ≈ 0.05-0.07 depending on the temperature and density profiles [31]. Previous analysis [35] of lithium's dilution effect put similar limitations on lithium concentration in the main plasma. The maximum concentration where operation of a fusion reactor in a steady state is still possible was taken to be around 10%. This indicates that if the burncondition was the main limitation on lithium concentration, the SOLPS simulations presented here would indicate the lithium vapor box has a reasonable operation space for a burning plasma. However, the analysis of [31] ignores β and density limits, which affect fusion power production and transport that affects Q. β limits the plasma pressure while the Greenwald density limits the line averaged electron density [33,34]. The analysis of [35] provides a maximum allowed impurity concentration but does not give the reduced power density resulting from lower concentrations which have an associated cost for a fusion reactor. Here we give simple estimates of the reduction in fusion power density for the range of lithium concentrations predicted by SOLPS. Note that other SOLPS work on a reactor-scale device has predicted similar upstream concentrations of lithium for lithium vapor based detachment [23]. First, consider a scenario where operation is near the Greenwald density limit. Should there be lithium contamination, then the density of the main ion species must decrease compared to an uncontaminated plasma as a result in order to keep electron density at the same level. Take n DT = n T + n D and the uncontaminated density to be n 0 Taking a lithium concentration of n Li /n e = 0.01-0.10, as SOLPS analysis indicates is reasonable to expect, then n DT must decrease to 0.70-0.97n 0 DT . A caveat to this calculation is that it assumes that the lithium concentration at the separatrix is the same as the lithium concentration through out the core, which is not always the case. Since fusion power density p fus scales as n D n T , this results in the fusion power density decreasing to 0.49-0.94 of the uncontaminated value. Alternatively, suppose a device is limited by β ∼ n = n DT + n Li + n e and assume temperature to be constant at the optimal value. The main ion density reduction factor in order to keep the total density constant is calculated as Using n Li /n e = 0.01-0.10, then n DT would need to be 0.78-0.98n 0 DT in order for the plasma to remain at the same total density. p fus is thus reduced to 0.60-0.96 of the uncontaminated value. These estimations of reductions in p fus neglect the presence of helium, which would cause p fus to be lower still from a pure DT plasma, but reduces the additional effect of adding lithium.

Conclusions
Notably, throughout all the recycling coefficient scans, (n Li /n e ) LCFS is always kept between 0.11-0.125 when no puff is utilized. The prediction of upstream lithium contamination is thus fairly consistent when no deuterium puff is used, which then depends on other parameters such as upstream density and particle flux. The effect of the deuterium puff on lithium concentration is more variable under changes to recycling, ranging from (n Li /n e ) LCFS = 0.01-0.12 depending on puff strength and location. The PFR puff is systematically able to reduce the lithium fraction much more significantly than the CFR D-puff or puffing from both sides while limiting the target heat flux to <10 MW m −2 . This is seen to be due to a larger downstream friction force relative to the thermal force. In the baseline deuterium recycling scenario, R Target = 0.95 and unity elsewhere, the best scenario with acceptable heat flux with CFR puffing had an upstream lithium concentration of 0.039 while the PFR puffing was able to reduce the lithium concentration to 0.017. Varying recycling coefficients finds that PFR puffing could achieve an upstream concentration between 0.01-0.07 with no limit on the deuterium puffing rate. CFR puffing could achieve an upstream concentration between 0.01-0.11 and is more dependent on PFC recycling coefficient. Because of this, the PFR puff is more successful at reducing lithium concentrations for a given target heat flux.
It is expected that the vapor box contains lithium well enough to allow for the 'burn-condition' to be satisfied, as even moderately puffed scenarios can bring the lithium concentration to the 0.05-0.10 range. It is also calculated here that a vapor box could reduce the fusion power density by a factor of 4%-50% from a plasma uncontaminated with lithium, with more limited reductions occurring for a β-limited scenario. Furthermore, for the parameters studied here, there is always a lithium evaporation rate capable of reaching acceptable target heat fluxes while maintaining only marginally reduced upstream temperatures. The results shown are promising in this regard, indicating that, while the previous analysis [6] had a more relaxed set of transport coefficients, this ultimately did not change the fundamental result that a lithium vapor box is capable of detaching a plasma with reactor relevant heat fluxes, at the expense of (n Li /n e ) LCSF ∼ 0.04-0.05.

Future work
Future work includes further optimization of the lithium vapor box, including evaporation location, which until now has assumed to be split equally between HFS and LFS of the box and never primarily from the target. Evaporating and puffing from the PFR side is a promising direction to pursue. Exploring target evaporation rates that are self-consistent with the SOLPS heat flux and target temperatures requires coupling with divertor temperature calculations. The effect of drifts is also consequential for the inner divertor and must be included.