A Compact NonPlanar Coil Design for the SFLM Hybrid
Authors
 First online:
DOI: 10.1007/s108940119479z
Abstract
A nonplanar single layer semiconductor coil set for a version of the Straight Field Line Mirror Hybrid concept with reduced magnetic field has been computed. The coil set consists of 30 coils that are somewhat similar to baseball coils with skewed sides. The coil set has been modeled with filamentary current distributions and basic scaling assumptions have been made regarding the coil widths. This coil set is expected to be considerably cheaper than a previous computed coil set. The coils can probably be produced with technologies known today.
Keywords
Superconducting coils Hybrid reactor Fusion Fission Mirror machine TransmutationIntroduction
A design of superconducting coils [1] for the Straight Field Line Mirror (SFLM) hybrid concept [2] has been made in an earlier work with semiplanar coils for a midplane vacuum magnetic field B _{0} of 2 T and a mirror ratio of four. This paper presents a design of a nonplanar superconducting coil system for the SFLM hybrid concept where B _{0} has been reduced from 2 to 1.25 T and the maximum midplane β has been increased from about 0.2 to about 0.5. The main reason for the new design is to reduce the cost and size of the coils. With the new coil design, a single layer of coils will be sufficient to produce the magnetic field while the old coil system had two layers. Since the magnetic field strength is now reduced, the sizes of the coils are significantly reduced and most coils can be made of NbTi instead of Nb_{3}Sn or Nb_{3}Al, which will reduce the cost of the coils substantially. Manufacturing of nonplanar coils has been made for the Wendelstein 7× stellarator [3], and is considered to be a known technology.
The fusionfission reactor (hybrid reactor) is a combination of a fusion reactor and a fast fission reactor, where the fusion reactor is used as a neutron source for the surrounding subcritical fission core. The fusion neutrons trigger fission reactions in the fission core which produces fission neutrons, and on average a cascade of fission neutrons and subsequent fission reactions will be created for each fusion neutron. Thereby, the fission core will act as a strong energy multiplier for the fusion reactor, and a fission to fusion power ratio of about 100–150 is possible with a fusion mirror machine driver [4] in steadystate mode. A subcritical fission core is a fission core with an effective neutron multiplication factor K _{ eff } of less than unity, typically 0.9–0.97. Such a core is not selfsufficient in neutrons, and must have an external neutron source to produce energy. Systems with subcritical cores and external neutron sources are called driven systems. The aims of hybrid reactors are energy production, breeding of fertile fissile material from U238 or Th232 or transmutation (by fission) of TRansUranic (TRU) elements in radioactive waste from fission plants. Competitors to hybrid reactors are ADS (AcceleratorDriven Systems) and fast fission reactors, but even LWRs (Light Water Reactors) can transmute TRU to some extent (plutonium).
The reason to study mirror machines for hybrids is the steadystate option and the simple geometry, which allows access to the plasma from the mirror ends and reduces fusion neutron leakage through holes in the fission mantle. The concern for insufficient plasma confinement is a less serious concern for a mirror hybrid than for a pure fusion device.

Stacey et al. [11] at Georgia Tech who have studied several tokamakbased concepts with downscaled ITER parameters.

Wu et al. [12] in China who studies tokamakbased hybrids (several FDS concepts) and are putting large resources into hybrid studies.

Kotschenreuther, Mahajan et al. [13] of Institute for Fusion Studies who studies spherical tokamakbased hybrids.

Gryaznewich et al. at Culham Laboratory is examining the possibilities to use spherical tokamaks as neutron sources.

Researchers at Budker Institute who studies a mirrorbased hybrid scenario using the axisymmetric Gas Dynamic Trap (GDT) as a driver [14].

Moiseenko et al. in Kharkiv, Ukraine is working on fusionfission and neutron sources [15].

Moir et al. [16] at Lawrence Livermore who recently presented an axisymmetric mirrorbased concept.

Taczanowski et al. [9] has some activities in fusionfission.

Yapici et al. [17] are working with fusionfission using catalyzed fusion as a driver.

Manheimer [10] is advocating fusionfission.

An Uppsala University group (the authors) are evaluating quadrupolar single cell mirror fusionfission hybrids [2].
In section “Geometry”, the geometry of the device is described. Section “The Vacuum Magnetic Field” decribes the properties of the vacuum magnetic field. Section “Coil Parameterization” describes the coil type used. Section “Coil Optimization” describes the coil optimization procedure. The results and discussion is in section “Results and Discussion” and section “Conclusion” concludes the paper.
Geometry
In this section, the geometric constraints for the coil system are briefly described. The geometry of the vacuum chamber, fission mantle, shielding etc. in the confinement region is changed from the previous coil design described in Ref. [1] in two ways. Outside the confinement region, the vacuum chamber radius is expanded to 1 m close to the confinement region and expands more towards the ends. This is since the magnetic field strength drops off in this region before the flux tube has been recirculated, giving a somewhat larger flux tube outer radius. Also, the space allocated in Ref. [1] for neutron shielding was too thin on the old design. A redesign of the fission mantle has been made, and at this stage the fission mantle with shielding has an outer radius of 199.8 cm [18]. Thereby, the inner coil radius of 210 cm which was used in the old coil system can be used here as well which provides some extra space for expanding the neutron shield if this should prove necessary.
A simple model for the coil sizes are used, where the cross sectional area of the winding packs (including the jacket) are assumed to be proportional to the current. The current density of 2,280 kA/(29.5 × 29.5 cm^{2}) = 2.6 kA/cm^{2} is approximately taken from the design of the toroidal field superconducting coils of JT60SA, which uses NbTi coils and have a magnetic field modulus similar to that of this device [19]. The coil structure material that surrounds the winding pack will also occupy some space. This is roughly accounted for by adding 10% of the coil width at each side, expanding the coil sides with a factor 1.2. This is roughly in accordance with Ref. [19]. For simplicity, the coil cross sections are made quadratic.
The Vacuum Magnetic Field
Coil Parameterization
Coil Optimization
By the obvious demand that r _{ i } should be kept as small as possible to save current, r _{ i } is set to 2.10 m for all coils except the circular cusp coils at the magnetic expanders which have r _{ i } = 3 m due to the large plasma radius at the magnetic expander. Also, the parameter a is preset for all coils (to 1 m for all coils). Thereby, the varying parameters for each coil pair are z, L and I. There is also a constraint that the coils should not intersect. The coils are formed so that they can be inserted into each other like a pile of drinking glasses. In practice, this limits the variation of the L parameter along the coil set and sets restrictions on the optimization process. The problem now looks like a straightforward optimization problem to solve with a local optimizer to find the parameters for the coils in a similar manner as has been done in [1]. It proved, however, to be sufficient to optimize the coils by hand, which was quite easily done since the parameters are so well separated in the sense that one parameter more or less controls each function \( \tilde{B}(z) \) and g(z). The hand optimization gave more practical solutions than the attempted numerical optimizations that were investigated.
Results and Discussion
The 3D coil parameters on the z > 0 side of the midplane
Coil name 
z(m) 
I(kA) 
L(m) 
r(m) 
a(m) 
Coil width (m) 

C01 
0.4 
787 
0.915 
2.1 
1 
0.209 
C02 
1.25 
700 
1.085 
2.1 
1 
0.197 
C03 
2 
700 
0.905 
2.1 
1 
0.197 
C04 
2.75 
710 
1.05 
2.1 
1 
0.198 
C05 
3.5 
762 
1.035 
2.1 
1 
0.205 
C06 
4.25 
720 
1.06 
2.1 
1 
0.200 
C07 
5 
680 
1.20 
2.1 
1 
0.194 
C08 
5.75 
750 
1.68 
2.1 
1 
0.204 
C09 
6.5 
1,100 
0.88 
2.1 
1 
0.247 
C10 
7.7 
1,640 
2 
2.1 
1 
0.301 
C11 
9.5 
2,750 
2.33 
2.1 
1 
0.390 
C12 
11 
5,600 
2.32 
2.1 
1 
0.557 
C13 
12.45 
11,150 
0.71 
2.1 
1 
0.786 
C14 (recirculation) 
15.05 
11,200 
−2.98 
2.1 
1 
0.788 
C15 (cusp) 
17.4 
−5,070 
0 
3 
1 
0.530 
There is no need to round off sharp angles in the new coil set, as there were in the old coil set. It is not expected that the moderate curvature of these coils will cause problems for the superconductors. The 3D coils are standalone which makes them easier to handle and transport. The quadrupolar coils of the old coil system are coupled. Also, the maximum magnetic field is expected to be low for the coils, and most coils can be made of NbTi which is considerably cheaper than Nb_{3}Sn. Some coils at the mirror peak may need to be made of Nb_{3}Sn or Nb_{3}Al, which may increase their size somewhat. It is expected that they can be fitted in in that case. The strain in the coils has not been examined in detail yet.
The lower magnetic field will increase the required maximum β from about 20% to about 50% to achieve the maximum fusion neutron production of n _{ n } = 7.1 × 10^{18} neutrons/s for the 1.5 GWth option in Ref. [4]. The desired neutron production and thus the β would vary with the fission fuel burnup during a fuel cycle if the total output power should remain constant. Mirrors are known to be able to produce higher β values (GDT about 60% [23], and 2XIIB more than unity [24]), but the high β will deteriorate the MHD stability of the plasma [25] and a more detailed examination is needed to fully address the finite β effects. There are also other stabilizing effects, such as finite Larmor radius effects [26] and the extra stability gained from the magnetic expanders [23]. Ballooning modes are not addressed in this work, partly since the authors of Ref. [26] do not believe that they will be βlimiting in (tandem) mirrors.
The design of the magnetic expander is not carried out in detail. There is a strong cancellation in the magnetic field at the magnetic expander, and it is probably necessary to have in situ adjusted correction coils at the expander to distribute the plasma load evenly on the plasma receiving “divertor plates”.
In this design, neoclassical transport is partly addressed through the use of the omnigenious SFLM field in the central part of the confinement region. There will, however, be drifts in the other parts of the confinement region, and small drifts from the field inaccuracies caused by the coil system.
There is a possibility to optimize the coils with a numerical optimization method, for example similar to the one used in Ref. [1]. It is probably possible to find a coil set with that reproduces the ideal field with somewhat better accuracy. The current solution is, however, considered satisfactory, and the small deviances from the ideal field are not considered important. The 2% “ripple” seen in Fig. 6a is not a ripple in the ordinary sense since the magnetic field is increasing in this region and there are no regions where particles will be locally trapped. The “ripple” is merely a small modification of the magnetic field. If there should arise a need to reduce the deviations, they could also be considerably reduced by the use of ferromagnetic inserts [27].
Conclusions
A 3D single layer coil set for a lowfield version of the SFLM Hybrid has been computed that reproduces the desired magnetic field with satisfying accuracy, and such a coil set can probably be realized with known technology. The set consists of 30 coils of a type which we have named “fishbone coil” due to its resemblance of a fish skeleton when several coils are added. The coils are somewhat similar to a baseball coils with skewed sides. The coil set would be considerably cheaper than the earlier design described in Ref. [1]. The largest coil outer radius is reduced from 4.35 m in the old coil set to 2.89 m in the new design.
Acknowledgments
Prof. Mats Leijon is acknowledged for support. Johan Abrahamsson is acknowledged for assistance with 3D modelling in SolidWorks.
Open Access
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