Field and Coil Design for a Quadrupolar Mirror Hybrid Reactor
Authors
- First Online:
DOI: 10.1007/s10894-010-9353-4
- Cite this article as:
- Hagnestål, A., Ågren, O. & Moiseenko, V.E. J Fusion Energ (2011) 30: 144. doi:10.1007/s10894-010-9353-4
Abstract
A vacuum magnetic field from a superconducting coil set for a single cell minimum B fusion-fission mirror machine reactor is computed. The magnetic field is first optimized for MHD flute stability, ellipticity and field smoothness in a long-thin approximation. Recirculation regions and magnetic expanders are added to the mirror machine without an optimizing procedure. The optimized field is thereafter reproduced by a set of circular and quadrupolar coils. The coils are modelled using filamentary line current distributions. Basic scaling assumptions are implemented for the coil design, with a maximum allowed current density of 1.5 kA/cm^{2}. The coils are optimized using a local optimization method and the resulting field is checked for MHD flute stability and maximum ellipticity.
Keywords
Fusion-fission reactor Hybrid reactor Magnetic coils Mirror machineIntroduction
A fusion-fission reactor, considered by Bethe [1], Taczanowski [2], Manheimer [3] and others, is a combination of a fusion reactor and a fast fission reactor aimed for energy production, breeding of new fissile material or transmutation of radioactive waste from fission plants. The fusion device within the hybrid reactor is a neutron source and a fission reactor core surrounds the fusion device. The fusion-fission reactor is a driven system, which means that the fission reactor core has an efficient neutron multiplication factor k _{ eff } less than unity [4], typically 0.9–0.97, whereas a non-driven fission reactor operates at k _{ eff } = 1. In a fusion-driven system of the type considered here, the power output is mainly from the fission part, where the fusion part only contributes with about 1% of the total energy production [4–6]. For such a power producing device, this could reduce the requirements of the fusion Q factor by up to a factor of 100 or even more [6]. This suggests that even a low Q fusion device, with Q as low as 0.15, could be adequate for power production. Thereby, realization of a hybrid reactor is far less demanding than that of a pure fusion reactor, since the plasma confinement requirements are dramatically reduced. Other beneficial features are that the wall loads from plasma loss and neutron bombardment, which seriously limit the possibilities to realize a commercial fusion reactor [7], could be reduced by orders for a hybrid reactor.
Strong energy amplification by fission enables the use of several kinds of fusion devices as neutron sources, and that idea is being pursued by several groups. To mention some of them, the FDS Team [8] in China and Stacey et al. [9] in USA have considered tokamak hybrids with downscaled ITER parameters. Bethe [1] primarily considered mirror machines and breeding of fuel. The interest has gradually switched to the possibilities for transmutation and power production, where Taczanowski [2] has made early studies on tandem mirror hybrid aimed for power production and transmutation (incineration) of radioactive isotopes. Noack et al. has considered minor actinide burning based on the axisymmetric Gas Dynamic Trap (GDT) [4]. Demir et al. have performed studies on fission mantle concepts for catalyzed fusion [10].
Mirror machines have several beneficial properties for a fusion-fission device [1, 2], in particular a steady state operation. In addition, possibilities for large end “divertor” plates (provided by expanders beyond the confinement region), geometric simplicity, compactness, a fission mantle geometry that allows practically all the fusion neutrons to enter the fission mantle and radio frequency heating with antennas located outside the confinement region are other useful features for a fusion-fission scenario [6, 11]. For a mirror fusion reactor, an electron temperature T _{ e } in the order of 10 keV is a typical requirement for net power production, and this has been a major roadblock for the development of mirror fusion reactors. However, in a fusion-fission system with a high energy multiplication Q _{ r }, i.e. the ratio of fission to fusion power, the requirement on T _{ e } is reduced substantially [2, 5, 6]. We here extend previous studies on the Straight Field Line Mirror (SFLM) concept with magnetic coils and magnetic expander regions. The configuration, outlined in Ref. [6], has the possibility to reach Q _{ r } ≈ 147 with maintained reactor safety margins, and efficient power production is then expected with T _{ e } around 500 eV which seems reachable for up scaled mirror machines. A possible scenario for increasing the electron temperature by magnetic expanders beyond the confinement region is also outlined in Ref. [6]. The method relies on plasma depletion in the expanders, without violating an MHD (magnetohydrodynamic) stability condition and a gyro-resonant loss cone instability. In addition, the proposed device is aimed to keep plasma loads on “divertor” plates in the expanders tolerable, it would be possible to operate the device in a wide range of plasma β and the neutron source shutdown could be arranged in milliseconds (in a tokamak, shutdown is longer than 15 s as restricted by induction currents). Monte Carlo calculations for this SFLM hybrid concept have been carried out in Ref. [11] where fission mantle properties were set. The present paper is based on the “near-term” option [11] described therein, which corresponds to a fixed thermal power production of 1.5 GW in steady state. The fusion power increases slowly from about 10–20 MW during the fuel cycle (defined as 311 days of steady state 1.5 GW power production), as a result of the slow decrease in Q _{ r } due to fission fuel burning during operation (as an alternative, operation at a fixed keff and a fixed fusion power of 10 MW could be possible with control rods [6]). The fission mantle will be loaded with fresh fuel once a year. Calculations predict that the first wall will have a tolerable heat load and a lifetime of about 30 years power production (311 days a year) before a 200 DPA (displacement per atom) limit is reached [11]. Power load on end tanks are tolerable with Q = 0.15 and 4 m wide expander tank radii. Also, safety cases concerning local boiling of coolant and void effects were examined [11].
In most aspects, a driven system would be safer than a critical fast reactor, but the hollow geometry with the vacuum region carries within it a risk for fuel relocation (core collapse) where the reactor may become supercritical, as pointed out in Ref. [2]. These kinds of reactor accident scenarios have not been studied in Ref. [11].
The desired properties listed above would require appropriate device geometry, without holes for diagnostics and antenna power feeding in the fission mantle located in between the vacuum chamber and the superconducting magnetic coils. The aim is to show that a suitable coil design is possible within geometric and other constraints. The version outlined here is aimed for 1.5 GW thermal power (about 500 MW continuous electric power) with a confinement length of 25 m and a 40 cm midplane plasma radius [6, 11].
The design of magnetic field in a long-thin approximation for a fusion-fission hybrid reactor is carried out. Focus is on achieving MHD stability with the flux tube ellipticity within a tolerable range and keeping the gradients of the fields sufficiently low. Superconducting coils represented by filamentary line currents to reproduce the optimized field are generated. Early work on coil design has been made by Riordan et al. [12], who optimized ellipticity for MHD stable multiple mirrors. D’ippolito et al. [13, 14] made an optimization approach for tandem mirrors where omnigenuity and neoclassical transport were addressed.
Various coil systems have been developed for mirror machines, including MARS at Lawrence Livermore [15] and GAMMA 10 at Tsukuba [16, 17], where Gamma 10 uses a combination of baseball coils, racetrack coils and circular coils and MARS uses Yin-Yang coils and circular coils. Large superconducting coils have been developed in the Large Coil Task (LCT) [18], and will be manufactured for the ITER tokamak [19]. The machines with most challenging coil designs are stellarators [20].
In “Geometry of the Studied Test Device”, the reactor geometry is defined. The magnetic field is optimized in “The Vacuum Magnetic Field Properties” and appropriate coils for the optimized field are computed in “Calculation for Coil System”. The results are discussed in the fifth section and the last section concludes the paper.
Geometry of the Studied Test Device
The machine under consideration is a single cell minimum-B mirror with a mirror ratio of four. The magnetic field modulus is 2 T at the center of the mirror. The plasma radius is about 40 cm at the midplane, and the radius of the vacuum chamber is 90 cm, possibly with some small elliptic deformation around the maximum ellipticity regions to fit in the plasma edge. The length of the confinement region is set to 25 m, and beyond this 6.25 m long magnetic expanders with recirculation regions are added at each side, giving a total length of 37.5 m. The axial scale length z _{end} = 12.5 m is the distance from the midplane to the end of the confinement region. The 26 m long fission mantle is symmetrically placed along the magnetic axis, and the outer radius of this mantle is 1.95 m [11]. However, the neutron shielding for the coils is not present in Ref. [11], and must be added. In a work by Yapici et al. [10], 5 cm of Boron carbide (B_{4}C) is added for neutron shielding. Since no shielding calculations have yet been performed on the device considered here, some margin is applied and the total outer radius of the fission mantle is set to 2.10 m. This also gives space for some extra structure material if this is needed.
Beyond the fission mantle along the z direction, space is needed for power feed to ion cyclotron resonance heating and outflow of coolant from the fission mantle. Thus, no large coils are allowed in the area between \( \left| z \right| = 14.7 \) and \( \left| z \right| = 16\,{\text{m,}} \) although quadrupolar coils parallel with the z axis will be accepted. In the nearly neutron free transition region to the expander, between \( \left| z \right| = 16 \) and \( \left| z \right| = 17.5\,{\text{m,}} \) there is only a thin neutron shield and the vacuum vessel between the coils and the plasma. Beyond the transition region, the magnetic expander with divertor plates is located, and here cusp coils are placed.
The size of the machine is determined from several aspects. Strong magnetic field gradients are harder to produce with a thick fission mantle, which makes the coil system for a long-thin machine easier to build. The plasma radius needs to be wide enough to give sufficient plasma volume for power production and to confine alpha particles, and is set to 40 cm. A long-thin plasma column is achieved with a 25 m long confinement region. A fusion power of 10 MW which corresponds to a neutron production of 3.6 × 10^{18} neutrons per second could give almost 1.5 GW thermal power output with Q _{ r } ≈ 150. The 1.2 m thick fission mantle provides enough space for fission materials, protection of first wall from fission neutrons, neutron reflectors, tritium reproduction and neutron shielding of magnetic coils. An available empty space could be used for control rods or adding of fission fuel if required [11]. With ion cyclotron heating antennas located outside the confinement region, it is possible to avoid holes in the fission mantle, whereby almost all fusion neutrons enters the fission mantle and a very high Q _{ r } (Q _{ r } ≈ 150) is possible with reactor safety margins [11]. Plasma is accessible only through the mirror ends.
The Vacuum Magnetic Field Properties
- 1.
The field is designed to avoid gross MHD instabilities. The average minimum B criterion was derived by Rosenbluth et al. [22] and is equivalent to a favorable average curvature of the field lines. Later the criterion was experimentally verified by Ioffe [23], demonstrating a striking stabilization of gross MHD modes and improvement in plasma confinement by a sufficiently strong quadrupolar field. The flute stability criterion [24] which in the low β limit can be written as
- 2.
The magnetic flux tube ellipticity should be minimized. A high ellipticity, caused by the stabilizing quadrupolar field, could give impractically thin flux tubes near the mirrors (in the thin direction) and may for strongly elliptic regions give alpha particle gyro radii exceeding this thin width. Also, highly elliptic plasmas will produce a slightly angular-dependent neutron flux, which may be somewhat unfavorable for the fission mantle, although the fission mantle will smear this out.
- 3.
Too strong gradients in the field components in the z direction has to be avoided to find a practical coil set residing outside the fission mantle that produces those field components in the plasma confinement region. The thick fission mantle makes the coil design problem much harder since the distance between the coils and the plasma increases significantly.
There are also additional parameters that are relevant for the optimization. Neoclassical radial plasma losses are often addressed [14]. However, since the confinement time in a fusion-fission device can be much shorter than in a pure fusion reactor, axial losses are expected to dominate and radial losses are neglected. Large magnetic expanders are required to ensure tolerable plasma axial power load on the wall. A list with numerous requirements to be addressed for tandem mirror design has been given by Baldwin [15]. However, the present study is limited to the three properties stated above. These requirements are somewhat contradicting, and the task is to find a solution that is a suitable compromise for all requirements.
A field satisfying the stability and ellipticity criteria, the Straight Field Line Mirror (SFLM) which is a marginal minimum B vacuum field, has been derived in Ref. [25] which satisfies marginal minimum-B stability has been derived by Ågren and Savenko [25]. This SFLM field have however inconveniently strong gradients near the mirror ends for moderately high values of the mirror ratio. Also, that field does not have a natural end of the mirror, since \( \partial B/\partial z^{2} \) is monotonically increasing away from the midplane. Thus, this field can not constitute the mirror field over the entire domain and has to be concatenated with some other field at some region before the mirror ends. Although such a concatenated field has beneficial properties at the central section, it is not obvious that the total solution has optimal properties with respect to our criteria listed above (or even satisfies them).
Calculation for Coil System
To reproduce the magnetic field selected in the previous section, a set of superconducting coils with specified currents are defined. The problem is simplified by using a filamentary set of line currents for each coil. A current density value of J _{max} = 1.5 kA/cm^{2} is used to determine the size of the superconducting coils. The spatial distribution of coils is limited by the condition that the coils must reside outside the fission mantle and neutron shielding due to heating from neutrons. The distance to the z axis from the coils are therefore larger than in a fusion reactor (with the same fusion device), which makes it harder to produce sharp relative derivatives \( \partial /\partial z(\ln \tilde{B}) \) and \( \partial /\partial z(\ln g) \). It is in principal possible to create sharp relative derivatives by cancelling fields (with opposing currents in nearby coils), but such a method would be uneconomic and less accurate [27]. Also, the coils are not allowed to intersect.
Since the field components are defined near the z axis, the optimization has been made along this line. Noting that the magnetic field is defined by the two functions g(z) and \( \tilde{B}(z) \), a convenient way to reproduce the field is to separate the problem into one optimization problem for g(z) and another for \( \tilde{B}(z) \). This is arranged by selecting a suitable coil set, where axisymmetric coils reproduces \( \tilde{B}(z) \) and a particular set of quadrupolar coils reproduces g(z). Yin-Yang coils [28] may be capable of reproducing the optimized field, but a coil similar to a baseball coil [29] with quadrupolar symmetry which will not add to the \( \tilde{B}(z) \) field at the z axis is used instead to fully separate the problem. Since the quadrupolar field is harder to create, the thinner quadrupolar coils will be placed inside the circular coils.
The optimizations are heavily weighted for the confinement region, and the comparatively large relative errors in the recirculation and expander regions are assumed less important, since the design is aimed for a very low plasma density in this region. The field profiles for the recirculation region and the magnetic expander are only chosen by the loose criteria that g(z) shall change sign shortly beyond the mirror end to limit maximum ellipticity and that the field shall be recirculated and thus returning the ellipticity to about unity at the expander end. Also, g(z) has to be weak in the expander and \( \tilde{B}(z) \) shall be weak with positive curvature at the end of the expander to provide added stability. If a more detailed recirculation and expander region are modelled, the optimization should be weighted differently. Also, it is probably necessary to have in situ adjusted weak correction coils at the expander to distribute the heat load on the divertor plates correctly, since the magnetic field here is very weak and thus a large cancellation in magnetic field accuracy is present.
The optimization is made using a Nelder-Mead numerical local optimization method. During optimization, different approaches have been tried. First, r _{ i }, I _{ i } and z _{ i } for each coil were used as free parameters with certain constraints. It proved however difficult to get a spatially acceptable result. The best result was achieved by keeping r _{ i } constant and restraining z _{ i } to a small region around the initial value to prevent coils from intersecting. In order to find a fairly good local minimum, a search was made with randomized initial values and k _{ c } = 1 for hundreds of starting points. The initial values of the best result was then used as a seed for the optimization with k _{ c } = 100. This showed that a lot of minima existed, and that those minima differed at least with a factor of 3 for the value of the functional (29).
Results and Discussion
The magnetic field has been optimized for MHD stability, low ellipticity and low field gradients, where a stability margin has been requested since the details of the pressure profile is not known at present. A problem of designing the magnetic field for a fusion-fission reactor is harder that that for a fusion reactor. It is crucial to find a magnetic configuration with low field gradients, since the coils that should produce the field are placed outside the thick fission mantle and are located far from the plasma. Therefore, the gradient steepness along the z direction determines a minimum length of the machine. The initial idea was to use the SFLM field [25] concatenated with some other field near the mirror ends. However, it turned out that the SFLM field had inconveniently strong gradients near the mirror ends, and the concatenation point therefore ended up at \( \left| z \right| = 8.75 \). In another approach the axisymmetric and quadrupolar field was modelled with a spline representation for the optimization. With a constant pressure profile, e.g. using the average minimum B criterion, this approach worked well. However, when a representative sloshing ion distribution was used the optimizer created undesired “bumps” in the g and \( \tilde{B} \) functions in order to create good curvature at the sloshing ion peaks. The consequence of this is that the \( \tilde{B} \) function flattens out at the sloshing ion peak which makes it impossible to create such a pressure profile for equilibrium reasons. To address this problem the pressure profile could be modelled as some function of \( \tilde{B} \), which is left for future studies. The chosen method was to use the concatenated SFLM field and to model the ending fields with splines. The ending field was optimized manually by moving the spline points once the behavior had been examined.
The resulting optimized field has several beneficial properties. The field gradients are fairly low, the maximum ellipticity of about 20 is acceptable and there is a stability margin. The sloshing ion peak is expected to give a stronger contribution to the region of good curvature than to the region of bad curvature. Also, the expander region and line tying effects are expected to add to the stability margin. The SFLM field in the middle part of the confinement region is omnigenious, and although the ending field is not omnigenious the neoclassical radial losses are expected to be much smaller than the axial losses. Also, for a fusion-fission device, omnigenity is not as important as it would have to be for a pure fusion device, since the confinement time is expected to be determined by the axial loss. Concerning β-limiting ballooning modes, Ref. [30] indicates that they are probably not of great importance and this is left for future studies. Therefore, the SFLM field concatenated with a spline line optimized field is selected for the fusion-fission reactor in this study.
The circular coils on the positive z side, defined by inner radius, cross section center z coordinate, cross section width/height and current
Inner radius (m) |
z (m) |
Coil width (m) |
Current (kA) |
---|---|---|---|
2.45 |
0.900 |
0.473 |
2,335 |
2.45 |
1.833 |
0.184 |
352 |
2.45 |
2.701 |
0.393 |
1,609 |
2.50 |
3.604 |
0.323 |
1,086 |
2.50 |
4.769 |
0.445 |
2,061 |
2.57 |
6.497 |
0.513 |
2,740 |
2.66 |
7.391 |
0.286 |
852 |
2.89 |
9.381 |
0.209 |
456 |
3.09 |
10.441 |
0.750 |
5,852 |
3.09 |
11.470 |
1.023 |
10,905 |
3.09 |
12.650 |
1.258 |
16,497 |
2.87 |
14.000 |
1.384 |
19,956 |
2.77 |
17.289 |
0.241 |
−606 |
2.55 |
18.125 |
0.789 |
−6,482 |
The quadrupolar coils on the positive z side, defined by end z coordinate, cross section width/height of the straight bars, cross section width/height of the quarter-circle segments, current in the straight bars and current in the quarter-circle segments
End z (m) |
Width (m) |
Width c. (m) |
Current (kA) |
Current c. (kA) |
---|---|---|---|---|
2.000 |
0.315 |
0.074 |
1,032 |
58 |
3.781 |
0.332 |
0.108 |
1,147 |
121 |
5.416 |
0.365 |
0.147 |
1,390 |
225 |
7.000 |
0.420 |
0.216 |
1,840 |
484 |
8.700 |
0.519 |
0.314 |
2,809 |
1,030 |
9.602 |
0.684 |
0.488 |
4,869 |
2,485 |
10.957 |
0.972 |
0.074 |
9,840 |
57 |
12.500 |
0.978 |
0.633 |
9,954 |
−4,178 |
13.800 |
0.392 |
0.600 |
1,597 |
−3,751 |
16.250 |
0.753 |
0.248 |
−5,905 |
644 |
16.900 |
0.666 |
0.230 |
−4,617 |
549 |
17.600 |
0.581 |
0.411 |
−3,519 |
1,760 |
19.375 |
0 |
0.164 |
0 |
280 |
21.005 |
0.232 |
0.164 |
559 |
−280 |
The ripples produced by the coils are not expected to give problems. Ref. [31] indicates that a ripple larger than 1% in \( \tilde{B} \) in the central region would give rise to ballooning modes. The ripple in the central region is however smaller than 1% and since Ref. [31] was written before Refs. [30] and [31] also points out at that time unpublished results in Ref. [30] this is not expected to be a problem. If so, use of ferromagnetic materials [31] is an option to substantially reduce the ripple.
For producing real superconducting coils, some filamentary current distribution with high resolution has to replace the low resolution filamentary line currents, where care should be taken regarding the connections between the quadrupolar coil segments, and some sharp turns have to be rounded off. Also, a deeper analysis concerning maximum forces and magnetic field strengths for maintaining superconductivity needs to be made, and higher order fields should be calculated to investigate the magnetic well radius. Also, the finite β diamagnetic effects should be taken into account. A strength with the obtained solution is that there is flexibility to independently control the currents in the quadrupolar and circular coils to adjust for finite β effects.
All optimizations in this paper are performed using local optimizers. A local optimizer finds local minima, which may or may not be good solutions to the problem, and which minima the optimizer will find depends on the initial values of the variables. The functionals minimized in this paper depend of many variables and at least the coil functionals have a large number of local minimas. It would be desirable to search the entire space for the global minima. It is however too many variables involved to make a grid search or similar, which indicates that some heuristic approach has to be made. There are many methods for global optimization available [32]. For stellarators, even genetic algorithms have been applied successfully [20]. This study is however limited to use of local optimizers, with some simple random search algorithm in the coil optimizations to search for a reasonably good initial iteration point. Since the representation of the coils is based on some rather coarse assumptions, a global optimization may anyway not be worthwhile until those assumptions have been clearly specified.
Conclusions
In this study, a vacuum magnetic field and a coil set producing that field has been derived for a quadrupolar mirror hybrid reactor with sloshing ions. The device with a 25 m long confinement region is aimed for 10 MW power, which would correspond to a steady state 1.5 GW thermal power output. The magnetic field has been optimized for MHD flute stability, low flux tube ellipticity and low field gradients, and consists of a central part based on the Straight Field Line Mirror concatenated with another field to end the mirrors. A simple recirculation and magnetic expander region has been added to the confinement region. A set of circular and quadrupolar coils has been suggested to reproduce the optimized fields with satisfactory accuracy. The vacuum magnetic field produced by the coils has then been examined for flute stability and ellipticity. The obtained magnetic field satisfies the flute stability criteria with some margin, has a maximum ellipticity of about 20 and has smooth profiles (although with some ripple) for the axisymmetric and quadrupolar field components.
Acknowledgments
Johan Abrahamsson is acknowledged for assistance with Solid Works 3D images. The Swedish institute has supported V.M. Moiseenko with a grant. Prof. Mats Leijon is acknowledged for support.
Open Access
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