3D effects on hydrogen transport in ITER-like monoblocks

The influence of recombination on the poloidal gaps of ITER-like monoblocks on hydrogen transport simulations is investigated. A 3D FESTIM model is first built and transient simulations up to 1×107  s of continuous exposure are run with or without instantaneous recombination on the poloidal gaps. In the case of instantaneous recombination, the poloidal gaps act as a strong sink for hydrogen leading to a decrease in the monoblock inventory. The total desorption flux on the poloidal gap is greater than on the toroidal gap but remains orders of magnitude lower than the retro-desorbed flux at the top surface. For a monoblock thickness of 4 mm, the relative difference in the hydrogen inventory per unit thickness between the two cases is 500%. As the thickness of the monoblock increases, this difference decreases (55% at 14 mm). The monoblock’s response to baking is then studied at different baking temperatures. At 600 K, almost all the hydrogen content in the monoblock is removed after 15 days of baking. Assuming a non-instantaneous recombination on the tungsten surfaces would not have a major impact on the monoblock desorption for baking temperatures above 600 K.


Introduction
Plasma facing materials in fusion reactors such as ITER and DEMO will be submitted to extreme conditions [1].The tungsten divertor of ITER-made of monoblocks-demonstrated that it met the requirements for high heat flux [2].Even though some studies proposed alternative designs for DEMO 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.monoblocks [3][4][5], the ITER-like monoblock concept has been selected in the baseline of DEMO [6].
Bombardment of high energy hydrogen particles (deuterium and tritium) on the tungsten divertors will lead to a build-up of hydrogen inventory, which can induced embrittlement [7] and therefore reduce the component lifetime.Because tritium can be trapped, permeate through materials and contaminate the coolants, it also a radioactive hazard.The control and assessment of this tritium inventory is therefore crucial [8] and it will be optimised to remain as low as reasonably achievable.That is why a maximum inventory of 1 kg of tritium in the vacuum vessel at any time is set as a design constraint [9].Baking of the plasma-facingcomponents could reduce the tritium inventory in the divertor [10].For the ITER baking phase, the divertor's monoblocks will be kept at a homogeneous temperature up to one month, by heating from the cooling tube.
Hydrogen transport in ITER relevant materials has already been modelled numerically, based on the McNabb and Foster equations [11], with 1D simulations for large sets of radiations conditions [12][13][14][15][16]. 2D simulations for large sets of radiations conditions [17,18] were also done assuming no effect of the block axial thickness due to the large size defined for ITER design (12 mm).Since the conceptual design for the DEMO monoblocks can still change, the aim for this study is to explore the impact of the block axial thickness on the retention and permeation during plasma operations.Desorption from both the toroidal and poloidal gaps are also studied during the baking phase.

Methodology
Figures 1 and 2 shows the ITER-like monoblock geometry used for this study.The toroidal gap between each monoblock is assumed to be 1 mm.To mimic the CuCrZr pipe continuity, a 0.5 mm extrusion of the pipe on each side of the W armour and Cu joint is done on toroidal axe.The poloidal surfaces Γ poloidal include W and Cu materials.
The model is based on previous works on hydrogen transport (including fickian diffusion and transient trapping) coupled with transient heat transfer [22,23].The thermophoresis (Soret) and mechanical field effects are not included in this study.The spatio temporal evolution of the mobile and trapped hydrogen concentrations c m (m −3 ) c t,i (m −3 ), respectively, are described by the following reaction-diffusion system: In equation ( 1) the first term on the right hand side corresponds to the Fick's law, where is the diffusion coefficient of hydrogen in the considered material in m 2 s −1 , T the temperature in K and k B = 8.6 × 10 −5 eV K −1 the Boltzmann constant.In equation ( 2), the trapping of mobile particles depends on the number of empty trapping sites n i − c t,i , the amount of mobile particles c m and the rate The second term describes the detrapping process characterised by the rate p i = p 0,i • exp( Two intrinsic traps were set in W, one trap in the Cu inter-layer and one trap in the CuCrZr cooling pipe (see table 1).The materials properties used in the simulations are described in table 2. A chemical potential continuity at interfaces is ensured by the continuity of the ratio c m /S where S = S 0 exp (−E S /k B T) is the solubility.This interface condition also ensures the conservation of particle flux [22].
The heat equation is solved in stationary form: where λ is the thermal conductivity expressed in W m −1 K −1 depends on material and temperature (table 2).
A heat flux is imposed on the plasma-facing surface and a convective flux on the cooling surface (see equation ( 4)): where n is the outward normal vector, is the convective heat exchange coefficient and T coolant = 323 K is the coolant temperature.A zero flux condition is imposed on the other surfaces.
A non-homogeneous mobile concentration is assumed at the plasma exposed surface to simulate an implanted source of particles (equation (5a)) [24,25].Depending on the simulation case, a zero concentration or a zero flux is imposed on the poloidal surfaces, respectively when an instantaneous recombination or an non-instantaneous recombination is assumed on the gaps (equation (5b)).A recombination flux is assumed on the cooling surface (equation (5c)).The other external surfaces will either be insulated (equation (5d)) or an instantaneous recombination (equation (5e)) will be assumed. where The heat flux is mainly caused by incoming particles at the surface and so depends on the incident particle flux and energy, which vary along the divertor.The set of heat flux φ heat = 10 MW m −2 and particle flux φ imp = 1.6 × 10 22 H m −2 s −1 is therefore not necessarily representative of all monoblocks in ITER and corresponds to the hottest point (strike point) on the divertor [18].This particular hot-point was selected for this study as it is hot enough for hydrogen to diffusion in the bulk.Indeed, for fluxes below 1 MW −2 , monoblocks remain around coolant temperature (due to the high thermal conductivity of materials).Lower temperatures meaning a smaller diffusivity and higher trapping effects, hydrogen therefore remains closer to the exposed surface [27].In this case, the aspect ratio of the problem being much higher, edge effects become negligible and the problem is effectively 1D.The open-source FESTIM code (v0.10.2) [28] was used run the model.All the scripts and datasets to reproduce the results are available at: https://github.com/RemDelaporteMathurin/3d_monoblocks [29].Since the model has two symmetry planes, only a quarter of the monoblock is modelled as illustrated on figure 2.

Results
First, thermal and H transport behaviours during plasma exposure are presented for a standard case (e = 4 mm) with and without desorption on the gaps.Next, the influence of the monoblock's thickness (e varying from 4 mm to 14 mm) is shown.The third part is dedicated to the desorption occurring during baking phase.Last, non-instantaneous recombination during baking is discussed.

Standard case
The temperature field obtained during plasma exposure is shown on figure 3. The top surface temperature of approximately 1200 K.There is no temperature gradient along the poloidal axe because a zero flux condition is imposed on the poloidal surface.The temperature field was therefore similar to a 2D case.
As expected, a higher retention (mobile and trapped hydrogen) was observed in the case without desorption (see figure 4).This is explained by the surface losses (see figure 5).
The total H inventory in the monoblock was also between one and three orders of magnitude lower in the case with desorption (see figure 6).This difference increased with the exposure time.Moreover, the steady state was reached way earlier for the case with desorption whereas the inventory kept increasing after 1 × 10 6 s for the insulated case.This means that not taking desorption from the gaps into account in 2D simulations is a conservative assumption in terms of H inventory.The 2D simulations performed in previous studies [17,18,22,23,28] then overestimate the monoblock H inventory (up to a factor 10 after 1 × 10 6 s) .This conclusion is valid as long as the tritium can desorb from poloidal surfaces i.e. the gaps are in vacuum or pressure free.As studied in [30], some co-deposited materials on the gaps can change the tritium inventory.
These 3D simulations are however essential to estimate the outgassing fluxes from the poloidal gaps.The particle flux at the poloidal gap is six times higher than the flux at the toroidal gap (see figure 7).The permeation flux to the coolant is five to six orders of magnitude lower than the fluxes at the gaps.The particle fluxes at the gaps (poloidal and toroidal) were approximately 1 × 10 12 H s −1 whereas the flux towards the cooling channel was below 1 × 10 7 H s −1 .The values of the outgassing fluxes from both the gaps are orders of magnitude lower than that of the retrodesorbed flux (i.e. the flux of implanted particles that diffuse back to the exposed surface, equal to the implanted flux φ imp assuming a recycling coefficient equal to 1 [25]).This means 3D edge effects will not affect previous results regarding the outgassing to the vessel.They will however impact the value of the contamination flux towards the coolant as assuming an instantaneous recombination on the gaps will lead to way less particles reaching the cooling surface and therefore a lower flux.The flux towards the coolant reaches a maximum close to 1 × 10 5 s, and next decreases due to the desorption from the top surface of extrusion of the CuCrZr pipe between monoblocks (Γ top_pipe shown on figure 1).

Influence of the monoblock thickness
Several simulations were run with monoblock thicknesses varying from 4 mm to 14 mm.
As the thickness increases, the inventory per unit thickness increases for the case with instantaneous recombination on the poloidal gap (see figure 8).It remains constant for the case without recombination and without the CuCrZr extrusion.This case corresponds to a pure 2D case and the inventory is independent of the thickness.The decrease observed at low thicknesses is due to the impact of the CuCrZr pipe between monoblocks Γ top_pipe as shown on figure 1.The relative difference between the cases with or without recombination on the poloidal gap decreases as the thickness increases.After 1 × 10 5 s of exposure, for a thickness of 4 mm the relative difference is 500% and drops at 55% for 14 mm.This result was expected as the edge effects become negligible at large thicknesses.
The permeation flux towards the cooling channel per unit thickness globally increases with the monoblock thickness (see figure 9).It is always higher in the case without recombination on the gaps.Similarly to the inventory, the relative  difference between the cases with or without recombination on the poloidal gap decreases with the thickness.

Baking
Baking was simulated for 30 days by applying an homogeneous temperature on the monoblock after application of a 10 min temperature ramp between 343 K and the baking set temperature.Instantaneous recombination was assumed on all surfaces (including the cooling surface).Only the initial trapped concentrations were taken from the steady state results of the standard case with desorption on the gaps (see figure 10).The steady state inventory was approximately 1.5 × 10 14 H, which corresponds to 2.5 × 10 −4 µg.Several temperatures were tested from 500 K to 673 K.
As expected, the monoblock hydrogen inventory decreased for all baking temperatures (see figure 11).The inventory decreased faster at higher baking temperatures.For instance, at 600 K, 5 days of baking are required to decrease the relative inventory to 20%, more than 15 days are required at 550 K.At the highest baking temperature (673 K), the relative inventory is lower than 1% after only 5 days of baking.At the lowest temperature (500 K), after 30 days of baking, more than 60% of the initial inventory still remains in the monoblock.
As for the inventory, the higher the baking temperature, the faster the desorption flux decreases.The desorption from the poloidal and toroidal gaps represents 80% of the total desorption flux (see figure 12).The desorption to the coolant represents ≈10% of the total desorption.Although this value may appear high, it should be noted that it was obtained from a monoblock at steady state.In monoblocks where most of the hydrogen is trapped near the plasma facing surface, the desorption to the coolant is expected to be smaller.

Influence of non-instantaneous recombination on baking
In the previous simulations, hydrogen was assumed to recombine instantaneously on the surfaces (i.e. the concentration of mobile particle is zero on surfaces).However, a noninstantaneous molecular recombination flux can be applied to surfaces:  where K r is the recombination coefficient in expressed in m 4 s −1 and c m is the concentration of mobile particles.
A recombination flux was set on the tungsten surfaces using Anderl's recombination coefficient [31]: At high temperature (673 K), there was little difference between instantaneous and non-instantaneous recombination (see figure 13).As expected, at lower temperatures, the noninstantaneous recombination inhibited desorption from the monoblock.
Studies proposed a revised recombination coefficient that would better reproduce experiments [32].However, this other recombination coefficient has a much higher value.Using this recombination coefficient instead would therefore result in a smaller impact on the desorption.

Conclusion
3D edge effects of hydrogen transport in ITER-like monoblocks were studied with the FESTIM code.It was shown that neglecting the desorption from the poloidal gaps leads to an overestimation of the tritium inventory in monoblocks up to a factor four after 10 5 s of plasma exposure for a thickness of 4 mm.This proves that the 2D assumption is conservative in terms of inventory.Moreover, as the thickness of the monoblock increases, the 2D assumption (i.e.neglecting desorption on the poloidal gaps) becomes more valid.For a thickness of 14 mm, the overestimation is reduced to 50%.The 2D assumption is also valid when recombination on the gaps is negligible.
However, these 3D edge effects cannot be neglected when studying baking since the desorption from the poloidal gaps dominates the desorption during baking phases due to the higher surface area.Non-instantaneous recombination on the tungsten surfaces can have an impact on baking at the lowest temperature (below 600 K).At 538 K, the relative inventory after 30 days of baking increases from 20% to 70% with non-instantaneous recombination on the tungsten surfaces.Uncertainties on the recombination coefficients in tungsten and CuCrZr (in contact with water) should therefore be lifted.

Figure 1 .
Figure 1.Cross sections of the DEMO monoblock used for this study showing W armour , Cu interlayer , CuCrZr alloy cooling pipe .

Figure 2 .
Figure 2. Tetrahedral mesh for e = 6 mm made with SALOME corresponding to a quarter of a monoblock (154 130 cells).Units in mm.

Figure 4 .
Figure 4. Retention fields of the monoblock with (left) or without (right) recombination on the poloidal gaps (standard case, e = 4 mm).Note the colour bars are different.

Figure 5 .
Figure 5. Mobile hydrogen concentration field and stream lines at t = 1 × 10 6 s for the instantaneous recombination case.The streamlines are computed from the gradient of the mobile concentration field.

Figure 6 .
Figure 6.Temporal evolution of the monoblock inventory (standard case).

Figure 7 .
Figure 7. Temporal evolution of outgassing fluxes for the standard case with desorption from the poloidal gaps.Blue lines correspond to the fluxes towards the vacuum vessel, the orange line is the flux towards the coolant.

Figure 8 .
Figure 8. Evolution of the inventory with or without recombination at the poloidal gap for several monoblock thicknesses at t = 1 × 10 5 s.The dashed line represents the case without recombination and without the CuCrZr pipe extrusion.

Figure 9 .
Figure 9. Evolution of the permeation flux to the coolant with or without recombination at the poloidal gap for several monoblock thicknesses at t = 1 × 10 5 s.

Figure 10 .
Figure 10.Steady state retention field in the standard case (for e = 4 mm).

Figure 11 .
Figure 11.Temporal evolution of the monoblock inventory at several baking temperatures.

Figure 12 .
Figure 12.Contribution of the monoblock's surfaces to the total desorption at a baking temperature of 623 K.

Figure 13 .
Figure 13.Influence of non-instantaneous recombination on the evolution of the relative inventory for several baking temperatures.

Table 1 .
Traps properties used in the 3D monoblocks simulations.