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An international journal of news from the stellarator community
Editor: James A. Rome Issue 162 June 2018
E-Mail: James.Rome@stelnews.info Phone: +1 (865) 482-5643
On the Web at https://stelnews.info
Managing leading edges in the
Wendelstein 7-X divertor
1. The W7-X divertor
In Wendelstein 7-X (W7-X), the magnetic islands occurring
naturally at magnetic surfaces with rational values of
the rotational transform are used for divertor operation [1].
In such an island divertor, the islands are intersected by
target plates. Due to the low magnetic shear in W7-X,
rather long paths (in the range 150–400 m) along the magnetic
field within an island may be achieved before a field
line intersects a target plate in spite of the small direct distance
between the X points of the island chain and the target
plates [2].
The target plates are three-dimensionally shaped to ensure
that the thermal load does not exceed the design value of
10 MW/m2 locally, using a simple field line diffusion
model (as described, e. g., in Ref. [3]) with an assumed
diffusion coefficient of 1 m2/s, for any of nine reference
magnetic vacuum configurations. The resulting target
plates intersect the magnetic field at angles of 1°–3° in
those regions where the heat flux along the magnetic field
is largest, thus limiting the thermal load to the targets as
required.
Under the aforementioned assumptions, the maximal heat
load onto surfaces oriented perpendicular to the magnetic
field (such as leading edges) is expected to be of order
200 MW/m2. Therefore, the height of leading edges must
be minimized. The locations of the regions of highest thermal
load (strike lines) on the target plates in reality may
deviate from the modeling due to deviations of the magnetic
field from the underlying vacuum configurations
(finite-beta effects, additional driven plasma currents, use
of different magnetic configurations), but also due to particle
drifts which are not taken into account in the model.
Therefore, the design of the targets was chosen to minimize
leading edges between target elements everywhere
on the target surface by introducing steps and slight inclinations
at gaps, so that leading edges are shadowed by the
adjacent target element.
Fig.1. Incident angle of the magnetic field on a portion of
the target plates. The larger gaps of 5 ± 1.5 mm between
adjacent target elements are indicated by arrows. The
direction of incidence is not shown, but the poloidal “watershed”
with an angle of incidence of 0° is clearly visible
along most of the horizontal target plates. The direction of
incidence changes sign when crossing the poloidal watershed
(see Fig. 2).
In this issue . . .
Managing leading edges in the Wendelstein 7-X
divertor
To handle heat loads, the design of the divertor targets
was chosen to minimize leading edges between
target elements everywhere on the target surface by
introducing steps and slight inclinations at gaps, so
that leading edges are shadowed by the adjacent target
element. Computer models were used to realign
the divertor plates so that modeled evaporation rates
after the realignment did not exceed the level of
2×10-3 g/s................................................................. 1
Stellarator News -2- June 2018
Due to technical boundary conditions, there exist gaps of
up to 7 mm between target modules in some locations, and
there is a poloidal watershed in some areas of the targets
(see Fig. 1). The watershed is defined as the line where the
angle of incidence of the magnetic field on the target surface
changes sign, here between the torus outboard and
inboard sides of the target plates.
2. Requirements for accuracy during divertor
assembly
If there is a gap between target elements crossing the
watershed, the potential leading edge will change side
between the two target elements at the watershed, and the
target elements must be inclined relative to each other in
order to achieve shadowing on both sides of the watershed
(Fig. 2). However, the position of the watershed changes
by several centimeters between magnetic configurations,
such that a compromise for the design of the target elements
had to be found, which required a relative positioning
accuracy of 0.2 mm between adjacent target modules.
In order to gain experience in W7-X divertor operation
before the installation of the water-cooled high heat flux
(HHF) divertor, which will be capable of steady-state
high-power discharges, W7-X will first be operated with
an inertially cooled test divertor unit (TDU) with the same
surface shape as the HHF divertor. With this TDU, the
power input will be initially limited to 80 MJ per discharge
(e. g., 8 MW ECRH for 10 s). As (1) the alignment
of the divertor frames to the originally defined relative
accuracy of 0.2 mm requires a significant amount of
assembly time, (2) in contrast to later HHF divertor operation,
stationary target temperatures will not be reached
during TDU operation, and (3) only a limited number of
magnetic configurations is envisaged for the TDU operation
phase, we decided to reassess the potential impact of
leading edges.
Rather than avoiding leading edges with an effective
height above 0.2 mm everywhere on the target surface, we
chose to focus on the locations of the strike lines (according
to the field line diffusion model) and on magnetic configurations
selected for high-power operations in the TDU
phase. We would allow for larger tolerances in the assembly
of the targets but would measure the resulting step
height along the gaps between target elements. We would
estimate the expected carbon evaporation rate due to the
as-built steps for the specified magnetic configurations
and would realign the targets if the estimated carbon evaporation
was so large that radiation collapses would likely
limit the operation of W7-X.
3. Approach to assess the impact of leading
edges
The goal for positioning the TDUs was to avoid steps
between adjacent target elements that might cause such
high carbon evaporation rates due to plasma hitting leading
edges that operation in OP1.2 would be severely limited.
We use a carbon evaporation rate given in Ref. [4].
In a finite element method (FEM) model of a leading edge,
the local temperature is obtained by balancing the incident
power onto the target surface P with radiation, heat conduction
into the bulk graphite, and sublimation. The resulting
temperature field is used to calculate the local
evaporation rate per unit length of the leading edge, which
depends on the time from the start of the discharge t, on P,
and on the height of the leading edge hLE (taking into
account the gap width, the incident angle of the magnetic
field, and a possible step between the target surfaces on
both sides of the gap). By integrating this local evaporation
rate along a gap, where P and hLE will vary along the
gap, a time-dependent total evaporation rate for a specific
gap and a specific magnetic configuration is calculated.
The FEM model is used to create a table of local evaporation
rates for t in the range 0–40 s, P in the range 10–
230 MW/m2, and hLE in the range 0–3 mm. This table can
then be used to quickly compute, by appropriate interpolation,
the local evaporation rates for any given set of (t, P,
hLE) values along the gaps between divertor modules.
A simple estimate of the local power density to a target
surface is provided by the “field line diffusion” model:
Magnetic field line tracing is started at arbitrary points
slightly inside the last closed magnetic flux surface, and
after some “free path” length, a random step perpendicular
Fig. 2. Schematic view of the relative inclination of two target
elements with a gap that might cause leading edges to
be exposed to heat flux from the plasma along the magnetic
field lines. In this case, the direction of incidence of
the magnetic field reverses along the gap, and the two target
elements are inclined in such a way that the leading
edge is always shadowed by the adjacent target element.
Since the point of field inclination reversal shifts poloidally
for different magnetic configurations, and since the target
elements are positioned within a given accuracy, the shadowing
will not be perfect for all configurations.
Stellarator News -3- June 2018
to the magnetic field is added. This corresponds to test
particles moving parallel to the magnetic field and undergoing
a diffusion perpendicular to the magnetic field [3].
The parameters of this process are chosen to reflect the
ratio of parallel to perpendicular transport in the plasma
edge. In the end, all the tracing paths will intersect some
wall component. The density of hit points on the target
surface is then a measure for the power density in this
location for the underlying magnetic configuration. The
incident power density can also be computed by more
sophisticated models such as the EMC3/Eirene code [5, 6],
which is, however, significantly more computationally
expensive.
Both approaches were used, but most calculations were
performed with field line diffusion.
In order to obtain good spatial resolution along the gaps
between target elements, we subdivided the target surface
into pieces with a length of 5 mm along the gap and
40 mm perpendicular to the gap. To achieve good statistics
then requires the tracing of O(106) test particles.
In most areas, the magnetic field intersects the target surface
under a small angle . The incident power density
parallel to the magnetic field is therefore larger by a factor
of 1/sin . A leading-edge hit under an angle  is therefore
exposed to a power density larger by a factor sin /sin 
than the power density to the target surface, and  is typically
much larger than . We therefore obtain the power
density P to the leading edge from the power density to the
local target surface Psurf from field line diffusion or
EMC3/Eirene as
.
As examples, we show in Fig. 3 surface loads Psurf and
resulting loads P on leading edges at the gap between two
modules of the horizontal target for two different magnetic
configurations (“scraper element mimic configurations”
for a toroidal plasma current of 0 and of 43 kA, see Ref.
[7]).
The outlined procedure for calculating the projected power
density to the face of a leading edge might appear to be
unnecessarily complicated. Obviously, it would be more
straightforward to count the hits to the face of a leading
edge directly in the field line diffusion model. However,
since this face area is very small for the design case, an
even higher number of test particles would be required to
achieve good statistics. During the optimization procedure
(see Section 4), the step height between adjacent divertor
modules is varied, which would require a multiple repetition
of the field line diffusion calculation for different step
heights. In our implementation, the local power density
Psurf as a function of position along the gap remains independent
of the step height and depends only on the choices
of gap and magnetic configuration. The total evaporation
rate along the gap etot(t) can then quickly be evaluated
from the table of local evaporation rates, as outlined
above.
After installation of each TDU in the W7-X plasma vessel,
the step heights were measured with a dial gauge at 3–4
positions along the gap between any two adjacent target
modules. The deviation of the measured steps from the
design values was then interpolated to obtain the as-built
step height along the gap. An example is shown in Fig. 4.
F
P Psurf
sin
sin
= -----------
Fig. 3. Left: surface power loads Psurf to the target surfaces
adjacent to the gap between target modules 3h and 4h, as
calculated in a field line diffusion model, for two magnetic
configurations (black and green); right: resulting power
loads P|| to a leading edge. The abscissa coordinate is the
length along the gap, with the major radius of the inboard
end of the gap as an offset.
Fig. 4. Step height of target module 4h above target module
3h in divertor unit 39 design (black) versus as-built
(red). The three measurements along the gap are marked
by circles. According to the design, the edges of the two
target modules are tilted relative to each other, as shown in
Fig. 2.
Stellarator News -4- June 2018
The height of the leading edge can now be calculated for a
given magnetic configuration from the as-built step height
and from the magnetic field vector and the surface normal
vector for each point along the target edge.
Likewise, for each point along the target edge the incident
power density PLE to the leading edge can be calculated
for each magnetic configuration from P|| and from the
magnetic field vector and the normal vector on the face of
the leading edge.
Finally, the local evaporation rate along the gap and the
total evaporation rate for the entire gap etot(t) can be calculated
(see Fig. 5).
4. Optimizing the steps between divertor modules
In the framework of our model, we calculated evaporation
rates for the as-built state of all gaps between target modules
in all ten test divertor units of W7-X, as described in
Section 3. We did this for a number of magnetic configurations,
which were considered relevant for the TDU operation
phase.
Our criterion was to avoid a situation where the operation
of W7-X would be limited by strong carbon evaporation
from leading edges. With a plasma volume of roughly
30 m3 and an assumed plasma density of at least
2×1019 m3, we obtain an overall particle content of
6×1020. Therefore, the plasma should certainly not contain
more than 1020 carbon ions, corresponding to 2×10 g.
Considering the uncertainties of modeling as an order of
magnitude, a carbon evaporation rate of 10 g/s will be
regarded as a second critical threshold.
As mentioned in Section 2, precise alignment of most of
the target modules would require a significant amount of
assembly time. However, one type of target module can be
re-aligned within its divertor frame without major time
effort. This type of target module (TM5h1) happens to
have a critical gap toward TM4h, where a large carbon
evaporation rate is likely in case of misalignment. Our
compromise for the assembly strategy therefore was to
plan for a realignment of TM5h1, in all ten divertor units,
whereas we would monitor all other gaps between target
modules and take action only if our model would indicate
a particularly high evaporation rate at such a location.
Once the as-built step heights between the target modules
4h, 5h1 and 5h2 had been measured in a divertor unit, the
necessary shifts were calculated. TM5h1 is positioned into
the divertor frame by means of two fitted brackets and one
fitted bushing. For the new alignment, TM5h1 was
removed, new brackets and a new bushing were manufactured
to the newly calculated measures, the target module
was reinstalled, and the step heights were measured again.
A typical accuracy of 0.2 mm was thus achieved.
The realignment changes the step heights at the gaps
between TM4h and TM5h1, and between TM5h1 and
TM5h2. In order to find the optimum alignment of
TM5h1, the step heights at the inboard end and at the outboard
end of the gap are represented by a pair of values
(h1, h2), which are then varied. For each realization of (h1,
h2), the new total evaporation rates after 8 s at the two
gaps in question are calculated, and a weighted average of
the evaporation rates is taken for the magnetic configurations
considered relevant. This result is then minimized by
varying the parameters (h1, h2).
In fact, we perform a stochastic optimization; that is, we
optimize the evaporation rate for Gauss distributions of
shifts around each pair of nominal (h1, h2) values, with
standard deviations of 0.2 mm. By this procedure we avoid
choosing a narrow minimum which will most probably not
be realized due to the finite accuracy of the realignment.
In the most severe case, the sum of the weighted averages
of the evaporation rate from both gaps would be reduced
from 0.27 g/s (initial installation) to 6.7×104 g/s (realigned).
5. Results of test divertor assembly
The resulting carbon evaporation rates from our model for
each gap between target modules and for the steps
between certain target elements are summarized in Fig. 6.
The value for each gap is the weighted average of the
Fig. 5. Left: Local evaporation rate after 8 s, corresponding
to the surface power shown in Fig. 3. Of the two magnetic
configurations, only for the one shown in black in Fig. 3,
and only at the leading edge at target module 3h are the
temperatures high enough to yield significant carbon evaporation
after 8 s. Whereas in the CAD alignment (gray
curves),
the evaporation rate is still tolerable at this time of the discharge,
it becomes three orders of magnitude larger for the
as-built alignment (pink curves). Right: Time evolution of
the total evaporation rate etot, integrated along the entire
gap.
5200 5300 5400 5500 5600 5700
position along gap [mm]
10−12
10−10
10−8
10−6
10−4
evaporation at 8 s [g/s/mm]
0 10 20 30 40
t [s]
10−12
10−10
10−8
10−6
10−4
10−2
total evaporation [g/s]
Stellarator News -5- June 2018
results for different magnetic configurations, as considered
appropriate for the TDU operation phase of W7-X. In
many cases, only one or two magnetic configurations contribute
significantly; the values for such configurations
could be higher by a factor of ~10.
We note that the critical values 104 g/s and 2×103 g/s
(gray horizontal lines in Fig. 6) are already reached or
exceeded at several gaps. The limit of 2×103 g/s was only
exceeded at gap 6, which is between the target modules 4h
and 5h1, before the realignment described in section 4
(full symbols), and at gap no. 3 in one case (full red
square). The latter case concerned one target element with
inlays for the investigation of plasma-wall interactions,
which was also realigned. In all cases, the modeled evaporation
rates after the realignment (open symbols) did not
exceed the level of 2×103 g/s.
An extended version of this report is in preparation for
publication as a journal article.
This work has been carried out within the framework of
the EUROfusion Consortium and has received funding
from the EURATOM research and training programme
2014–2018 under grant agreement no. 633053. The views
and opinions expressed herein do not necessarily reflect
those of the European Commission.
References
[1] H. Renner, J. Boscary, V. Erckmann, H. Greuner, H.
Grote, J. Sapper, E. Speth, F. Wesner, M. Wanner, and
W7-X Team. “The capabilities of steady state operation
at the stellarator W7-X with emphasis on divertor design.”
Nucl. Fusion 40(6) 1083–1093, June 2000.
[2] Y. Feng, C. D. Beidler, J. Geiger, P. Helander, H.
Hölbe, H. Maassberg, Y. Turkin, D. Reiter, and W7-X
Team. “On the W7-X divertor performance under detached
conditions.” Nucl. Fusion 56(12) 126011, December
2016.
[3] S. A. Bozhenkov, J. Geiger, M. Grahl, J. Kißlinger,
A.Werner, and R. C.Wolf. “Service-oriented architecture
for scientific analysis at W7-X. An example of a
field line tracer.” Fusion Eng. and Des. 88(11) 2997–
3006, November 2013.
[4] V. Philipps, U. Samm, M. Z. Tokar', B. Unterberg, A.
Pospieszczyk, and B. Schweer. “Evidence of hot spot
formation on carbon limiters due to thermal electron
emission.” Nucl. Fusion 33(6) 953–961, June 1993.
[5] Y. Feng, F. Sardei, J. Kisslinger, P. Grigull, K. McCormick,
and D. Reiter. “3D edge modeling and island divertor
physics.” Contrib. Plasma Phys. 44(1–3) 57–69,
April 2004. Special Issue: Proceedings of 9th Workshop
on Plasma Edge Theory in Fusion Devices (PET-
9), September 3–5, 2003, University of California in
San Diego, USA.
[6] P. Sinha, H. Hölbe, T. S. Pedersen, S. Bozhenkov, and
W7-X Team. “Numerical studies of scrape-off layer
connection length in Wendelstein 7-X.” Nucl. Fusion
58(1) 016027, January 2018.
[7] H. Hölbe, T. Sunn Pedersen, J. Geiger, S. Bozhenkov,
R. König, Y. Feng, J. Lore, A. Lumsdaine, and Wendelstein
7-X Team. “Access to edge scenarios for testing a
scraper element in early operation phases of Wendelstein
7-X”. Nucl.Fusion 56(2) 026015, February 2016.
Michael Endler for the W7-X Team
Max-Planck-Institut für Plasmaphysik
D-17491 Greifswald, Germany
E-mail: michael.endler@ipp.mpg.de
Fig. 6. Carbon evaporation rates etot(t = 8 s) for 10 MW
heating power, weighted average for magnetic configurations
relevant for the TDU operation phase, as calculated in
our model, for the step heights measured after the integration
of the TDUs into W7-X. The gaps are numbered in
toroidal order for the horizontal target modules within one
divertor unit, and the colors indicate different machine modules.
Symbol code — circles: upper divertor units; squares:
lower divertor units; full symbols: before realignment; open
symbols: after realignment, or no realignment. Horizontal
gray lines: indication of the thresholds 104 g/s and
2×103g/s.
2 4 6 8 10
Gap number
10−12
10−10
10−8
10−6
10−4
10−2
100
Configuration averaged evap. rate [g/s]