Optimization of lithium vapor box divertor evaporator location on NSTX-U using SOLPS-ITER

Commercial fusion reactors will be faced with extremely high divertor target heat fluxes that will require mitigation. Simulations of detachment in an NSTX-U scenario projected to have 92 MW m−2 unmitigated peak target heat flux are presented, which reaches sub-10 MW m−2 target heat flux using a highly dissipating lithium vapor box divertor design. The lithium vapor box is a detached divertor design which employs lithium vapor evaporation and condensation to contain lithium below the X-point. Previous SOLPS modeling has indicated a lithium vapor box can reduce the heat flux down to 10 MW m−2 via simultaneous evaporation from the Private Flux Region (PFR) and the Common Flux Region (CFR) sides of the vapor box. It is found here that PFR evaporation has improved access to the separatrix leading to significantly more efficient power dissipation than CFR evaporation. Simulations of target evaporation with an evaporation distribution that is self-consistent with the temperature of a Capillary Porous System with Fast flowing liquid lithium could reach nLi / ne∼ 0.025–0.030 at the Last Closed Flux Surface (LCFS) depending on the liquid metal flow speeds and lithium sputtering yield, while PFR-side evaporation can reach acceptable heat fluxes with nLi / ne∼ 0.038 at the LCFS. However, PFR evaporator performance can be improved if the target is allowed to be hot enough such that it reflects lithium, reaching nLi / ne∼ 0.028 and reducing required lithium evaporation. Ultimately PFR evaporation and target evaporation are found to have similar ability to produce acceptable heat flux solutions with minimal upstream concentration.


Introduction
Plasma facing components (PFCs) of future fusion devices, and more specifically the divertor, may reach greater than Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 100 MW m −2 [1,2].However, PFCs today can only recieve ~20 MW m −2 transiently before damage becomes unavoidable, with 10 MW m −2 often considered to be the safe operation limit on PFCs in steady state [2].This limit is reduced even further to 5 MW m −2 in the presence of fusion neutrons [3].Detachment, where the power is dissipated before reaching the PFCs, is thus a critical area of research to be developed for future reactors.The power dissipation should occur outside the Last Closed Flux Surface (LCFS), or risk reduction in energy confinement time of the main plasma [4].Impurities, necessary to induce the power dissipation in detachment, must have limited concentration in the main plasma or risk reductions in fusion power production, either via upstream radiation or fuel dilution [5,7].
The lithium vapor box is a detached divertor design which would rely on lithium evaporation to dissipate the power impinging on the divertor of a fusion device.The 'box' structure of the lithium vapor box limits the ionization location to be near the target, while the flow of the main ions work to prevent lithium from reaching the main plasma via the friction force [8,9].Past work using SOLPS-ITER [10] has shown that (n Li /n e ) LCFS ≲ 0.05 was possible in NSTX-U with moderate fuel puffing, indicating viability for a burning plasma scenario, should a reactor have similar results [5][6][7].(n Li /n e ) LCFS ≲ 0.05 was possible across changes in power, density, recycling scenarios and transport coefficients.Though the deuterium recycling assumed changed the fuel puffing required.
This article further optimizes the lithium vapor box via SOLPS by determining the optimal lithium evaporator location.Past work has evaporated equal amounts of lithium from the Private Flux Region (PFR) and the Common Flux Region (CFR) of the vapor box.However in principle there are three distinct evaporator locations: the PFR, the CFR and the target.Target evaporation requires the incoming plasma flux to calculate the temperature distribution of the liquid lithium surface, as well a robust target design which is capable of controlling the target surface temperature.Here the benefits of each evaporation location are compared.

Simulation set-up
In alignment with previous analyses [5,6,8], a simulated NSTX-U magnetic equilibrium is employed as an example for a lithium vapor box device.The lower outer divertor PFCs are altered from NSTX-U to create a contained lithium evaporation region.The evaporation location within these PFCs is varied in this article.The locations used are shown schematically in figure 1, results of the variation will be discussed in section 3. The cross-field transport coefficients used here are the same as a previous analysis of the lithium vapor box using SOLPS-ITER [5].The power boundary condition at the Core-Edge Interface (CEI) is 7 MW for all simulations presented in this article, indicating 10 MW of input power with 30% radiated power within the normalized poloidal flux function ψ n ∼ 0.85 surface, where the CEI is located.The CEI density boundary condition is 1.6 × 10 20 m −3 , resulting in a separatrix outer midplane density of 0.27 n Greenwald (n Greenwald ∼ I p /πa 2 ) with no lithium evaporation or deuterium puffing, with n Greenwald = 1.98 × 10 20 m −3 under this simulation's assumptions.
The deuterium recycling coefficients are chosen to be R D = 0.95 at the targets and unity elsewhere.Past analyses have shown that non-unity deuterium recycling affects predictions for upstream lithium concentration [5], however for the sake of simplicity this is kept constant for all simulations in this article.Generally it has been found that similar solutions can be created with higher D 2 gas puffing, if recycling is reduced.D 2 gas puffing from the private flux region has also been shown to significantly reduce lithium concentration upstream [5,6], here it is kept at a rate of 6 × 10 22 D 2 /s except when noted otherwise.The location of the gas puff is noted as a green circle on the wall in figure 1. Lithium is initially chosen to be absorbed at every material surface in the simulation, however this is changed later in sections 5 and 6.Changes to lithium recycling is described at the beginning of those sections.Drifts are not included for the cases presented in this analysis.

Inclusion of neutral-neutral collisions
An improvement over past analyses is the inclusion of neutralneutral collisions using the BGK collision operator available in EIRENE [11].In order to include this effect in the neutral transport, the rate coefficient < σv > for neutral-neutral collisions must be specified.The cross sections used to estimate the rate coefficient were also used in modeling of a lithium vapor box which used the DSMC code SPARTA [12,13].Experimental work using a lithium test stand determined these cross sections to be accurate within 15% in predicting lithium vapor mass flow [14,15].The cross sections were integrated over the bi-Maxwellian velocity distribution of the two colliding particles, resulting in different rate coefficients for each set of neutral collisions.SOLPS allows for reaction rates of the form ln < σv >= ∑ 8 n=0 b n (ln T) n where the b n 's are user input.The b n that emulate the reaction rate derived from SPARTA is given for each neutral interaction in table 1. b 1 is the same for each set of interactions, indicative of the very similar temperature dependence used for all species.Given the temperature dependence.To avoid over-fitting the SPARTA reaction rate, only n = 0 and n = 1 coefficients were used in the SOLPS reaction rate user input.

Self-Consistent Calculation of Target Temperature
In order to model the emission of lithium at the target, it is necessary to calculate the temperature as well as the liquid lithium sputtering yield.For the temperature, a Capillary Porous System with Fast flowing liquid lithium (CPSF) [16] is assumed to be placed at the horizontal wall of the lower outer target.This model is only used in section 6.Sections 3-5 scan the total lithium emission from the evaporator as it assumed to be externally controllable in the vapor box since the evaporator is away from the plasma particle and power flux.Similar flowing liquid metal temperature calculations to those presented in section 6 have been done in other literature [17] by coupling to a flowing liquid metal MHD code.There it was used in tandem with neon gas injection, making lithium evaporation less significant to the overall plasma solution.However, for the work presented here, the target plasma is strongly dependent on the CPSF solution which necessitates an analytical model to be employed when iterating with SOLPS, since many iterations of the calculation are necessary.The CPSF system uses porous material to create a stable uniform interface between liquid lithium and plasma, with simultaneous fast radial flow of liquid lithium beneath the porous wall.Currents are driven such that the flow of liquid lithium is created via the J × B interaction with the toroidal magnetic field.The analytical heat transfer model used for this system neglects diffusion along the interface, resulting in parabolic equations.For the tabulated heat flux distribution obtained from SOLPS, this analytical model results in an implicit algebraic equation for the interface surface temperature T surf at each radial cell i, which can be solved using traditional methods.Note that the flow is assumed to be along the radial direction.The implicit character of this dependence originates from the non-linear dependence of the evaporation and sputtering flux on surface temperature.Radial cell numbering starts from 1 at the beginning of the CPSF flow region and increases in the direction of the liquid lithium flow in the CPSF.The CPSF temperature model assumes all heat impinging on the target is removed either via the flow of liquid lithium beneath the capillary porous structure or via the evaporative cooling effect of emitted lithium.
The derivation of the temperature model is detailed in a separate publication [18] and is determined by solving the heat equation for the liquid lithium channel and above porous surface.The temperature is found by finding the temperature that satisfies the equation is the heat flux due to evaporation cooling and condensation heating on the interface, caused by the emission and absorption of lithium on the interface Γ Li = Γ Li Influx − Γ Li Emission .q Li is then taken to be 145920[J/mole] NA Γ Li .This emission includes sputtering, which is discussed in section 2.3.2.d w [m] and k w [W/(m•K)] are the thickness and thermal conductivity of the porous medium separating the flowing lithium from the plasma, α w [W/m 2 •K] is the heat transfer coefficient between the porous wall and the liquid metal lithium, given by where Ha is the Hartmann number, which is calculated to be 39.8 for the cases here.s [m] is the thickness of the flowing slab of liquid lithium beneath the porous layer.C w = (dw+t)s is an adjustment coefficient dependent on the lithium channel side wall thickness, t [m], derived in [18].z i [m] is the radial location of the cell centers in SOLPS, with z i±1/2 [m] being the radial location of the cell edges.Pé is the peclet number, ρc p sv/k Li , where ρ is the liquid lithium density, c p is the heat capacity of lithium, v is the flow speed of the lithium, and k Li is the thermal conducivity of lithium.For this analysis, we use s = 0.5 mm, C w = 1, d w = 1 mm, and k w = 100 W/(m•K).Equivalent channel depth s and coefficient C w are adjusted to include the effect of the sidewalls of the lithium channels, the parameters s = 0.5 mm and C w = 1 represents a 1 mm wide lithium channel with 1 mm sidewalls of thermal conductivity 255 W/(m•K).Detailed dependencies of s and C w on the Li channel side wall parameters are presented in [18].

Evaporation and sputtering models.
The sputtering yield of deuterium ions on a lithium surface is uncertain but has been predicted to play a significant role in the total lithium emission even for high temperature lithium surfaces [19][20][21].In order to examine the effect of sputtering, two different models are employed which are on the low [19] and high [20] ends of the range of predicted lithium sputtering yield.These models derive their values from different ion bombardment experiments of liquid lithium, with varying ion fluence, angle of incidence, liquid lithium surface temperature and deuterium saturation levels.Both models follow the relation where the adatom flux is the flux due to atoms which require less energy to be liberated due to their weakly bound state from a previous interaction with an energetic particle.f neut is a factor to account for the prompt redeposition of lithium particles after the sputtering process.The 'low sputtering' cases used the following parameters for the adatom and physical sputtering: A = 10

Comparison of PFR and CFR evaporation locations in solps
The evaporation location was altered in SOLPS in order to determine the benefits of each location.In this section constant temperature walls with a uniform evaporation distribution is assumed.The CPSF calculation is not employed for sections 3-5.PFR, CFR, and balanced (equal evaporation from CFR and PFR) evaporation scenarios were considered.The resulting radiation distributions are shown in figure 1, which shows the total radiation (neutral and plasma, for both deuterium and lithium species).It is seen that the radiation distribution of the plasma is altered by this choice, with the primary region of radiation being biased towards  the side where the evaporation is located.Importantly the PFR evaporation scenario induces radiation at the separatrix, denoted by the dashed red line of figure 1.The integrated radiation is not significantly different between these three cases, as noted in table 2. However, the different radiation distributions result in different peak lower outer target heat fluxes.At 14 × 10 23 Li/s, the PFR biased evaporation had 9.4 MW m −2 while the balanced evaporation had 14 MW m −2 and the CFR biased evaporation had 26 MW m −2 .As a result of the radiation distribution, PFR evaporation reduces the lower outer target heat flux to an acceptable heat flux (⩽10 MW m −2 ) for the lowest gross lithium evaporation rate of the three evaporation scenarios tested.As shown in figure 2, the PFR evaporation location reduces the heat flux to acceptable levels at 14 × 10 23 Li/s of gross lithium evaporation while the balanced evaporation requires 24 × 10 23 Li/s.Note that the gross lithium emitted from both evaporators is quoted, so 24 × 10 23 Li/s of balanced evaporation had 12 × 10 23 Li/s emitted from the PFR.
For the range of lithium evaporation rates tested, CFR evaporation did not succeed at reducing the target heat flux to an acceptable level.The CFR evaporators lack of success is due to the lack of penetration of the lithium to the strikepoint.The integrated lithium ionization rate for 14 × 10 23 Li/s gross emission was 1.7 × 10 23 Li/s for both PFR and CFR evaporator locations, indicating both had a significant loss of lithium to the walls and a significant difference in the power dissipation provided by different lithium ionization locations.The off-target walls of the box (red and orange walls that do not intersect with the plasma grid, figure 1(b) absorbed ~12 × 10 23 Li/s in both cases, indicating a similar particle balance between the two evaporation locations.Thus, an important conclusion of this study is that lithium evaporator access to the separatrix field lines, which is greater for PFR evaporation, is important to achieve acceptable heat flux at a lower gross lithium evaporation rate.
Figure 2 shows that, in order to reach a lower outer divertor peak heat flux of ⩽10 MW m −2 , the balanced evaporation scenario would reach a lithium concentration at the LCFS of (n Li /n e ) LCFS = 0.043 while evaporating from the PFR could reach a similar heat flux level at (n Li /n e ) LCFS = 0.038.This difference is due to the CFR evaporated lithium not providing as much energy per ionization dissipated.This results in the PFR-only evaporation reaching a 12% lower lithium concentration solution that is not attainable with the other evaporation scenarios tested.A previously published analysis of the vapor box [5], using the same parameters as these simulations but without neutral-neutral collisions, determined the minimum lithium concentration required for ⩽10 MW m −2 at the target to be 0.059 for 3 × 10 22 D 2 /s and 0.013 for 1 × 10 23 D 2 /s.This previous analysis used balanced evaporation.These concentrations resulted at 10 × 10 23 Li/s and 6 × 10 23 Li/s respectively.Running 10 × 10 23 Li/s with the same parameters as the previous analysis at 6 × 10 22 D 2 /s resulted in lithium concentration at the LCFS of 0.033 (without neutral-neutral collisions).Thus similar lithium concentrations are predicted to be required with or without the neutral-neutral collisions, however a significant change to the gross lithium evaporation rate occurs if neutral-neutral collisions are included.

The lithium vapor cave
The results of section 3 show that PFR lithium evaporation is preferable to CFR lithium evaporation.Figure 3 shows that in this case both the lithium ion and neutral densities are confined to the private flux and near-separatrix regions.Since the purpose of the baffles is primarily to contain neutral lithium, this suggests that the CFR baffles are not necessary.To test this, a new PFC configuration was created which removes the CFR baffles.
This configuration, no longer a box since half of the 'box' has been removed, will be called the lithium vapor 'cave' for the remainder of this article.The ionization distribution of the cave is compared with the box in a case with similar lower outer divertor heat flux in figure 4. The ionization profile is seen to be similar, further supporting the idea that the CFR baffles are not necessary.Further examination shows the cave is first able to reach ⩽10 MW m −2 heat flux at 18 × 10 23 Li/s gross lithium evaporation rate with (n Li /n e ) LCFS = 0.0406.Thus, while sacrificing some gross evaporation efficiency and a marginal increase in the upstream lithium concentration, a significant reduction in the PFC complexity has been achieved.The increases to the upstream concentration and evaporator efficiency are due to reduced divertor closure, which ultimately reduces the deuterium recycling which acts to contain the lithium.Further optimization of the vapor cave design is the subject of section 5.

Reflecting lithium surfaces in the vapor cave
The amount of gross lithium evaporation rate required to achieve the required reduction in the peak heat flux in the vapor cave is significant, corresponding to approximately 21 g s −1 .To reduce this requirement, selected surfaces have been made to reflect, rather than absorb, lithium.Adding reflecting surfaces increases the amount of ionization per emitted lithium atom.Reflection is a realistic boundary condition if the PFC satisfies the relation Li is the rate of lithium impinging on that surface.Thus as long as the temperature of the PFC is sufficiently high, the surface can be made to reflect lithium.The necessary temperature could be achieved by a combination of external heating of the walls and the radiative heating by the plasma.Two locations are considered for reflecting surfaces, the cave walls (green walls in figure 4(b) and the lower outer target walls (orange walls in figure 4(b).The lower outer target will be hot enough to prevent lithium build up unless it is externally cooled.
The effect of changing the lithium recycling coefficient at each of these surfaces is shown in figure 5. Changing the cave walls to reflect lithium but keeping the target walls absorbing (no lithium emission) can reduce the required gross lithium evaporation rate to 8 × 10 23 Li/s, representing a greater than 50% decrease to the required gross evaporation rate.This occurs with a (n Li /n e ) LCFS = 0.040, similar to the case with absorbing cave walls.The required gross evaporation rate further decreases as the target is made to reflect more and more lithium, reaching 1.5 × 10 23 Li/s in the case of R Tar Li = 0.95.Numerical convergence was difficult at R Tar Li = 1.0, preventing meaningful results from being found at this recycling level.However, at R Tar Li = 0.95 the vapor cave achieves target heat flux ⩽10 MW m −2 with (n Li /n e ) LCFS = 0.028, a 30% decrease to the upstream lithium concentration compared with the absorbing target solution.Furthermore, due to re-absorption of lithium at the evaporator, the net lithium lost from the evaporator is only 1.7 × 10 22 Li/s (~200 mg s −1 ).This is to be compared to the absorbing target, reflecting cave solution which lost a net 2.2 g s −1 and the absorbing target, absorbing cave solution which lost a net 15 g s −1 .Thus the lithium needs of the system was reduced by orders of magnitude with high reflectivity surfaces.The evaporator will thus need experimental flexibility to supply different lithium loss rates, depending on the lithium reflectivity found in experiment.
The reduction in required net evaporation is due to the change in the neutral lithium flow.The reflection of lithium at the cave walls directs the flow of lithium towards the plasma.Hence the minimal difference in the upstream lithium concentration when just the cave walls were made reflecting, since the actual total lithium ionization source rate changed very little with approximately 9.1 × 10 22 ionizations/s for both 18 × 10 23 Li/s gross evaporation with an absorbing cave and 8 × 10 23 Li/s gross evaporation with a reflecting cave.
The reflection of lithium at the target raised the ionization source rate to 2.1 × 10 23 ionizations/s for the same degree of detachment.In the reflecting target simulations, more of the ionization was happening in the cells near the target, where the ionized lithium would be swiftly pushed back into the target by the friction force of the main ion thus leading to less energy dissipated, lithium concentration upstream, and net evaporation required per ionization.Furthermore, in the reflecting target cases the lithium neutrals also form a 2D improving the fraction of evaporated lithium that is ionized.Recycling at the target walls (magenta) causes a recirculation of lithium, which further increases the fraction of evaporated lithium that is ionized.The target recycling also creates a more favorable ionization pattern that reduces upstream lithium, shown in figure 5.
poloidal vortex within the cave, as shown in figure 6(c), rising from the target towards the evaporator.Once re-ionized, the lithium is then pushed downwards back into the target where they are recycled once again, thus leading to multiple ionizations per evaporated lithium atom.This also causes significant re-absorption at the evaporator since most lithium impinges on the target, cave, or evaporator PFCs making it 'trapped' until it reaches the evaporator.

Target evaporation via CPSF
With high lithium reflection at the target shown to have considerable benefits both to upstream concentration and net lithium lost from the evaporator, the natural next step is to consider lithium evaporation from the target.As described in section 2.3, the CPSF concept [16] will be used as the design for target evaporation in this analysis.The CPSF employs radial flow of lithium beneath the capillary porous system, assumed to be along the divertor floor in this case, which in principle can flow radially outward (toward the CFR) or radially inward (toward the PFR).This flow direction is varied to determine the effect on the solutions.
In this section, SOLPS was run such that every few time steps the target temperature, and therefore the lithium evaporation rate given to SOLPS, is recalculated based on the equation described in section 2.3.This method is applied to solutions both with the vapor cave PFC structure introduced in section 4 and the unmodified NSTX-U geometry.The CPSF lithium emission calculation is only applied to the orange walls, shown in figure 4(b), where the CPSF is assumed to be placed.In the cases with the vapor cave, the side evaporation is also turned on to see if there is an optimal combination of PFR and target evaporation.Here, side evaporation refers to the evaporation from the red wall shown in figures 4(b) and 6.In all the cases in this section, the temperature of the CPSF is kept below 770 • C, despite the heat flux being up to ~20 MW m −2 in some cases.Lithium concentration increases with more side evaporation, which corresponds to lower CPSF temperature.

Combining the vapor cave with the CPSF
The gross lithium evaporation rate from the side evaporator is altered and the resulting lithium concentrations are shown in figure 7. The flow speed is kept at 3.45 m s −1 with the parameters mentioned in section 2.3 and a deuterium puff of 6 × 10 22 D 2 /s is present in the PFR, as in sections 3-5.At this high flow speed, regardless of flow direction or sputtering model, the lowest upstream lithium concentrationsolution is found by turning off the side evaporator.This then allows the lithium to be further below the stagnation point of the deuterium, leading to greater lithium containment.Integrating the lithium ionization source rate for two cases (no side evaporation and 8 × 10 23 Li/s side evaporation, both with flow towards the PFR and using the high sputtering model) shows the change in ionization pattern.The ionization upstream from the LFS deuterium flow reversal point is 2.0 × 10 20 Li/s without side evaporation vs 3.5 × 10 20 Li/s with side evaporation.Furthermore, the heat flux to the target is higher in cases with less side-evaporation, with the highest found being 0 Li/s from the side evaporator, a low sputtering model, and flow radially outward which saw a peak lower outer divertor heat flux of 18 MW m −2 .This occurs with a maximum target temperature of 714 • C. Thus, relying on the CPSF to carry the heat away at this flow speed also makes the task of reducing the heat flux more manageable, with less lithium needed to radiate power.With the same parameters, but 8 × 10 23 Li/s from the side evaporator, the heat flux to the target is 10 MW m −2 and the maximimum temperature is 598 • C. The side evaporator succeeds in reducing the incident heat flux on the CPSF, though with these plasma and CPSF parameters it is an excessive amount of lithium injection.
Flowing lithium from the CFR to the PFR (radially inward) is found to be marginally preferable.This difference is found to be small, though this may be due to a higher chosen flow speed, v Li = 3.45 m s −1 , which tends to minimize differences in flow direction.The preference for radially inward flow in the cave geometry may be due to the fact that this causes lithium emission to be located near the baffling which can capture the neutral lithium better due to the higher degree of closure, preventing uncontrolled ionization.This is quantified by the net lithium emission from the target in the high sputtering model and no side evaporation cases which is 5.6 × 10 22 Li/s in the case with flow towards the PFR, and 7.6 × 10 22 Li/s when the flow is towards the CFR.Thus more lithium is lost when emission is concentrated on the CFR side.The radial distribution of the lithium emission is shown in figure 8, which shows the spatial asymmetry of the lithium emission for the cases with high sputtering and no side evaporation.Overall, at this flow speed, the cases with no side-evaporation were found to have the smallest upstream lithium concentration.These cases had a peak surface temperature of 740 • C for the low sputtering model and 700 • C for the high sputtering model.These cases found upstream concentrations of 0.025 or 0.030 depending on the sputtering model, similar to the results with side evaporation and no CPSF shown in figure 5 and discussed in section 5. Lithium concentration in a case with a CPSF in the unmodified NSTX-U geometry.Lithium concentration can be kept low if the flow of lithium can be increased high enough resulting in lower target temperatures, regardless of flow direction, though results improve if lower sputtering is assumed.

Using the CPSF with the current NSTX-U PFCs
The CPSF coupling was also applied to simulations that used the current NSTX-U PFC geometry, shown in figure 10.The lithium concentration across a variety of lithium flow speeds is shown in figure 9 with flow towards the PFR (radially inward) and flow towards the CFR (radially outward).All cases in figure 9 have the same deuterium puffing rate in the PFR as the other cases presented here, 6 × 10 22 D 2 /s, which improves upstream lithium concentration [5].In figure 9, it can be seen that high lithium flow rates reduce the resulting upstream lithium concentration, however above ~4 m s −1 the improvement to upstream concentration is minimal when a larger sputtering rate is assumed.This is due to the fact that the high lithium flow rate creates a much lower temperature solution, which causes the lithium emission from the target to primarily be due to sputtering.Increasing the flow speed further can lower the temperature but evaporation eventually becomes insignificant compared to sputtering.The upstream concentration at 8 m s −1 was found to be 0.025 − 0.040, depending on the sputtering model and flow direction used.
In the cases without the cave PFC structure, the preferred lithium flow direction is opposite of the cases with the cave.The unmodified NSTX-U geometry has lower upstream lithium concentrations when lithium flows radially outward.This is because, without the PFC structure, lithium emission in the PFR is less contained due to the lower deuterium density and flow in that region.Lithium emitted in the SOL encounters the main ion flow of deuterium which pushes on the lithium via the friction force and leading to better lithium containment.With no PFC structure to contain the PFR emission, the net lithium emission from the target is less favorable.In the high sputtering model cases with flow speed v Li = 3.45 m s −1 , 7.5 × 10 22 Li/s is lost with flow towards the PFR, while 6.8 × 10 22 Li/s is lost when the flow is towards the CFR.The difference is minimized at higher flow speeds since the temperature at the edges of the CPSF becomes smaller the higher the flow of the lithium is.
The net lithium loss from the CPSF is also correlated to the ionization above the deuterium flow stagnation point, shown in figure 10.Lithium ionized above the stagnation point is significantly less contained than lithium ionized closer to the target.For example, the cases shown in figures 10(a) and (b) (high sputtering, v Li = 3.45 m s −1 ) have lithium ionization rates above the flow reversal boundary of 2.1 × 10 21 Li/s and 4.4 × 10 20 Li/s for flow towards the PFR and flow towards the CFR respectively.Thus both by metric of net lithium lost from the CPSF and lithium ionized above the deuterium flow reversal boundary, flow towards the PFR has worse lithium containment than flow towards the CFR in the unmodified NSTX-U geometry.
Solutions with flow speed ⩾3.45 m s −1 have similar upstream lithium concentrations as those created by the vapor cave without a CPSF, which saw a minimum concentration of 0.028, although the net lithium lost from the evaporator can be significantly higher.Thus the lithium vapor cave and a CPSF are found to have similar upstream concentration levels.Many aspects of both the CPSF and vapor cave require further testing to determine their practicality for a fusion device.The CPSF requires fast flow within 1 mm square channels, which is difficult to produce.The CPSF design also needs to show resilience to disruptions, which may damage the 1 mm scale structures that the CPSF requires.Meanwhile, the vapor cave requires strong control of the evaporator surface temperature, which may be difficult given the impinging radiation from the plasma during the shot unless the evaporator is moved out of the lineof-sight of the plasma.The vapor cave also requires a redesign of the PFR structure which will limit experimental flexibility.Both designs must also show translation of good performance to a reactor-scale device.Ultimately, further analysis must be done to determine the practical aspects of either design, which is beyond the scope of the work done here.

Conclusion
In this article the lithium vapor box PFC design has been simplified using SOLPS.Evaporating solely from the private flux region was seen to provide strong radiation near the separatrix while evaporation solely from the common flux region failed to achieve q max tar ⩽10 MW m −2 .The private flux region evaporator in the vapor box PFC configuration was able to reach this heat flux at the cost of an upstream lithium concentration of 0.038.The design of the lithium vapor box was then reduced to a lithium vapor 'cave', where the common flux region baffles are removed, which was able to achieve low heat fluxes at an upstream lithium concentration of 0.041.The vapor cave was able to achieve increased evaporator efficiency by making the cave walls reflect lithium, which could be achieved with sufficiently high temperature surfaces.Reflecting cave walls reduced the required net evaporation rate to 2.2 g s −1 compared to 15 g s −1 with completely absorbing walls.Reflecting lithium at the target further improved the required net lithium evaporation rate, down to 0.20 g s −1 .Reflecting lithium at the target also reduced the minimum required upstream lithium concentration to 0.028 due to the ionization of lithium being further below the deuterium stagnation point, resulting in better lithium containment.A flowing lithium target evaporator, assumed to be the CPSF design, with or without a cave has been shown to give similar upstream lithium concentrations when flow speeds are ⩾3.45m s −1 , though results are dependent on the sputtering model of lithium assumed.Future research is required to examine and improve the practicality of both designs, as well as determine their applicability to a reactor-scale device.

Figure 1 .
Figure 1.Radiation distribution comparisons for CFR vs PFR lithium evaporation locations.Radiation values shown here include neutral and line radiation from both D and Li species.PFR evaporation is seen to induce radiation at the separatrix, where the highest heat flux is located.All cases shown in this figure have a gas injection rate of 6 × 10 22 D 2 /s from the PFR.

Figure 2 .
Figure 2. Lithium concentrations across different evaporator locations.Cases with and without a deuterium puff are compared.The PFR evaporator location succeeds in reducing the heat flux for less lithium evaporation than the other evaporator locations.

Figure 3 .
Figure 3. (a) Neutral lithium density and (b) total lithium ion density for a case with PFR lithium evaporation and q max tar <10 MW m −2 .The lithium is entirely located on the PFR side, indicating the CFR baffle is unnecessary.

Figure 4 .
Figure 4. Ionization source rate for (a) the vapor box and (b) the vapor cave configurations.The negative (blue) regions indicate greater recombination than ionization while the positive (red) regions indicate greater ionization.The walls colored green and orange in figure (b) indicate walls which will change lithium recycling coefficient in section 5.

Figure 5 .
Figure 5. Lithium concentration for the vapor cave configuration with different lithium recycling coefficients given to the green and orange walls of figure 4(b).All cases shown here only emit lithium from the evaporator (red wall in figure 4(b)).

Figure 6 .
Figure 6.Neutral lithium flow vectors in three different recycling configurations.Flow vector length is normalized, flow magnitude is indicated by the color, direction by the arrow direction.Recycling at the cave (green) walls collimates the flow as shown in figure (b),improving the fraction of evaporated lithium that is ionized.Recycling at the target walls (magenta) causes a recirculation of lithium, which further increases the fraction of evaporated lithium that is ionized.The target recycling also creates a more favorable ionization pattern that reduces upstream lithium, shown in figure5.

Figure 7 .
Figure 7. Lithium concentration for cases with the vapor cave configuration and recycling cave walls (green walls in figure 4(b)) and a CPSF at the lower outer divertor target with v Li = 3.45 m s −1 .Lithium concentration increases with more side evaporation, which corresponds to lower CPSF temperature.

Figure 8 .
Figure 8.(a) Target temperature and (b) lithium emission distribution for cases with flow towards the PFR (red) and towards the CFR (blue), both with the high sputtering model assumed.Both of these cases have the cave PFC configuration and have no lithium emission from the side evaporator.

Figure 9 .
Figure 9. Lithium concentration in a case with a CPSF in the unmodified NSTX-U geometry.Lithium concentration can be kept low if the flow of lithium can be increased high enough resulting in lower target temperatures, regardless of flow direction, though results improve if lower sputtering is assumed.

Figure 10 .
Figure 10.Lithium ionization source distribution with the unmodified NSTX-U geometry, with a CPSF assumed to be constructed at the orange wall, flow direction indicated on the plot.Red indicates region of ionizaton, while blue is regions of recombination.The green line indicates the deuterium flow reversal location, green dot indicates the deuterium puff location.

Table 1 .
Values used to describe the rate coefficients of neutral-neutral collisions.

Table 2 .
Integrated radiation by species for three evaporation scenarios, in MW.Total lithium emission is 14 × 10 23 Li/s for each case.The primary lithium ion radiation source is line radiation.