Journal of Fusion Energy

, Volume 29, Issue 6, pp 561–566

Macron Formed Liner as a Practical Method for Enabling Magneto-Inertial Fusion

Authors

Special Issue: ICC2010

DOI: 10.1007/s10894-010-9314-y

Cite this article as:
Kirtley, D. & Slough, J. J Fusion Energ (2010) 29: 561. doi:10.1007/s10894-010-9314-y

Abstract

To take advantage of the smaller scale, higher density regime of MIF an efficient and repeatable method for achieving the compressional heating required to reach fusion gain conditions is needed. The macro-particle (macron) formed liner compression of the field reversed configuration (FRC) provides such a method. The approach to be described employs an assemblage of small, gram scale, macrons to form a more massive liner that both radially and axially compresses and heats the FRC plasmoid to fusion conditions. The large liner energy (several MJ) required to compress the FRC is carried in the kinetic energy of the full array of macrons. The much smaller energy required for each individual macron is obtained by accelerating the macron to ~3 km/s which can be accomplished remotely using conventional inductive techniques. 3D numerical calculations demonstrate that macron convergence can form a coherent liner provided minimum velocity and timing accuracy is met. Experimental results have demonstrated that a cylindrical or spherical macron can be accelerated to velocity within 2 m/s and timing less than 1 microsecond. Initial testing of a 6-stage launcher yielded 280 m/s at a final coupling efficiency of greater than 40%.

Keywords

Magneto-inertial fusionMagnetic liner compressionFRC

Introduction and Methodology

To take advantage of the smaller scale, higher density regime of magneto-inertial fusion (MIF) an efficient method for achieving the compressional heating required to reach fusion gain conditions must be found. The method needs to be simple and capable of repetitive operation. The macroparticle (macron) formed liner compression of the FRC is such a method, and uniquely addresses the major challenges facing magneto-inertial fusion. Specifically, it embodies a compression scheme that can very efficiently and repetitively generate the kinetic liner energy required to reach fusion gain. It provides for both the target plasma and liner energy to be generated remote from the reactor vessel. This is critical as the reactor environment is likely to be incompatible with the specialized pulse power equipment employed in conventional liner approaches. The timescale for forming and accelerating both the FRC and liner can be much longer than the time that the energy is thermalized in the implosion. This avoids the need for the very high voltages required to produce multi-megampere compression currents. In fact the switching requirement for the macron formed liner (MFL) is well within the range of currently available solid state devices. The time and cost for testing the concept is negligible when compared to alternative approaches. The process required for target FRC formation is already routinely produced at the MSNW laboratory [1].

The key to attaining a workable pulsed fusion reactor is achieving the required standoff for the liner driver as is illustrated in Figs. 1 and 2. The target plasma is assembled by producing two FRCs remote from the burn chamber, and translating them into the center where they merge and form a stable target FRC. The merged FRC must have a decay time long compared to the FRC assembly and compression time—but not the liner formation time which can be initiated much earlier. This aspect of the concept—the FRC formation, acceleration and merging process—has already been successfully demonstrated in the inductive plasma accelerator (IPA) experiments at MSNW [2]. In the reactor application a small (~0.1 T) guide field maintains the FRC against radial expansion as illustrated in Fig. 1.
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Fig. 1

Reactor illustration based on the macron formed liner (MFL) compression of the FRC (R-Z plane). Scale of the FRC and liner has been enlarged for purpose of illustration

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Fig. 2

Midplane cross sectional view of reactor based MFL. The macron size was increased and reactor scale reduced for illustration

The metallic liner is also produced by merging as well, however, in this case by a cluster of small masses (macrons) that are injected into the reactor by an array of launchers, also situated outside the reactor vessel. In this way the energy required to compress the FRC can be delivered on a timescale that is much slower than the FRC compression, and produced by a source that is highly efficient as well as broadly distributed spatially. Most importantly, the liner formation system can be located outside the reactor in a manner compatible with long-term repetitive operation.

The macron launchers are initiated in such a manner that a large array of small metallic masses arrive at the central section of the reactor (see Fig. 2) and converge to form a contiguous liner at smaller radius (rL ~ 0.1 m). The FRC is then introduced within the liner by merging two translated FRCs.

The shape of the macron will be considered later, but for this discussion its radial size and mass are the relevant parameters. The scale and number of macron launchers required for the liner can be readily estimated by what was achieved in the MTF liner experiments performed at AFRL [3]. A cylindrical aluminum shell liner of roughly 300 g mass was imploded using the 4.5 MJ Shiva Star capacitor bank. The 5 cm radius liner was compressed with an axial current of 12 MA at 84 kV and produced a final liner kinetic energy of ~1 MJ in 22 μs [4]. Employing the macron launcher it should be possible to exceed this liner energy by a factor of three for a similar stored energy. For example, the target parameters for each launcher are a macron mass of 4 g accelerated to a velocity of 3 km/s (3 mm/μs). The kinetic energy of each macron would thus be ~20 kJ. For a 1 cm radius macron, the 30 macron launchers depicted in Fig. 1 would converge to make contact at a radius of roughly 10 cm and be well merged at 5 cm. Five such rings arranged axially would initially form a liner of length comparable to the AFRL liner, and have roughly twice the mass (0.6 kg). The liner kinetic energy, however, would total 3 MJ. And it would compress from 5 cm radius to the minimum (r < 2 mm) in less time (~15 μs). In addition to a higher kinetic energy, the macron liner can be made to converge axially thereby increasing the compressional energy even further. This is an significant advantage as the FRC equilibrium length contracts axially (l − r2/5) as it is compressed. With a fixed liner length, the efficiency of compression is greatly diminished by this effect. Without electrical contacts, the experimental apparatus and vacuum system can also be greatly simplified. Difficulties with the post implosion vacuum integrity are also much easier to avoid. A unique benefit of macron assembled liners is the ability to impart a specified amount of rotation. The rotating inner surface will be stabilized to instabilities during the final compression stages. The amount of rotation to be added to the liner must balance stability, shear, and overall energy efficiency.

It should be noted that the small aperture in the chamber wall required for macron introduction is quite small (~3 cm2), so that even with several hundred such holes, the portion of reactor wall area exposed would be less than a small fraction of a percent of the total area (<0.1% of a 3 m diameter chamber). The time from launch to the arrival of detritus from the liner implosion will be several milliseconds providing sufficient time to aperture the small ports closed. The neutron fluence to the launcher structure is also significantly reduced by being completely outside the blanket with only a small acceptance angle for exposure due to the long connection tube length through the blanket. The distributed launcher array greatly reduces the demand on the power delivery systems. With a constant acceleration of the macron to 3 km/s over a distance of 3 m, the liner energy input can occur over a time span as long as 2 ms—two orders of magnitude slower than the conventional Z pinch method. As will be seen, the energy input from each macron launcher is low enough and slow enough that it can be provided with commercial solid state electronics.

There are three key issues that must be solved for the macron formed liner concept to be validated. First, the timing accuracy and velocity spread must be able to deliver the required liner behavior. Secondly, the dynamical behavior of the liner during compression must be understood, specifically how it is influenced by non-uniformities or rotation. Finally, the optimal projectile geometry, trajectory, and number must be fully characterized. The answer to these three questions from the initial results of the 3D modeling indicates that a suitable liner can be formed. The modeling results are discussed in more detail in section “Modeling Results”. The most critical issue is to demonstrate the ability to produce the desired macron kinetic behavior in a consistent and repeatable manner. It is clear from the considerations above; the launcher must be simple, reliable and robust. It is believed that the prototype design possesses all these features. Initial testing of the launcher is described in section “Experimental Results”.

There is a wide array of possible methods to achieve the desired velocities and masses. The velocity requirement is rather modest compared to that attained by a large range of projectile acceleration methods. There are both electromagnetic (e.g. rail guns) and gas dynamic approaches (e.g. light gas guns) that have achieved velocities far in excess of the 3 km/s target velocity. Pulsed Inductive acceleration employing a series of sequenced coils as illustrated in Fig. 3 turns out to be the preferred method for several reasons. For methods that rely on physical motion for initiation such as light gas guns and pistons, it is highly unlikely that an array of such devices that could achieve the timing jitter required for the liner assembly. As a crude estimate of what timing would be required, consider the arrival of a 2 cm long macron at the point of contact. If a position accuracy of 2 mm was desired, the firing time jitter at a macron closing speed of 3 km/s would be ~0.6 μs. While larger variations may prove to be acceptable, it will certainly be of this order. While this is an easily achieved jitter for EM launch techniques, achieving velocity control as well eliminates techniques such as rail guns where rail contact and other issues limit the accuracy that can be achieved in terminal velocity. With the inductive accelerator, the motion of the macron is controlled by the propagating magnetic wave which can be made identical in each accelerator. Variation in macron response to acceleration due to variations in size or mass are largely self-correcting due to the axial gradient in the force imparted by the propagating field. If the macron moves out ahead of the wave the axial driving force experienced by the macron diminishes bringing the macron back into alignment. The same effect corrects for a macrons that fall behind requiring greater force.
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Fig. 3

Pulsed inductive acceleration of a cylindrical macron

The force on the projectile can be easily is derived from Lens’ law with an equation of motion that depends on the coil-projectile geometry and currents:
$$ F(z,t) = I_{\text{c}} (t) \cdot I_{\text{p}} (t){\frac{{{\text{d}}M_{\text{cp}} }}{{{\text{d}}z}}}, $$
(1)
where Mcp is the mutual coupling between coil and projectile. Integrating the above equation along the axis of motion yields the work done on the projectile and thus the final kinetic energy of the projectile:
$$ \frac{1}{2}m_{\text{p}} v_{\text{p}}^{2} = I_{\text{c}} I_{\text{p}} \int\limits_{0}^{\infty } {M_{\text{cp}} {\text{d}}z} = {\frac{{I^{2} }}{2}}M_{{{\text{cp}}0}} $$
(2)
where Mcp0 is the value of the average mutual coupling to the array of coils experienced by the projectile along the length of the accelerator. It was assumed during the projectile acceleration that the current in the coil is oscillatory, and that the macron image currents completely screen the solenoidal coil fields (Ip = Ic ≡ I). To maximize the efficiency of the driver, the drive coil circuits execute a full period before the circuit is opened. This allows any energy not transferred into macron motion to be returned to the capacitor for future use. The direction of the axial force is independent of field direction so that the axial force is continued with current reversal. By using an oscillatory current the magnetic field is kept from penetrating the macron and reducing the accelerating magnetic gradient. The polarity of each coil is reversed and timed so that the field is in alignment with the field produced during the previous coil’s second half cycle. The stability of the macron is also much improved with this acceleration method as the direction of the dipole moment alternates as the macron moves from coil to coil preventing a possible tilt instability from growing. A multipole (quadrupole) guide field can also be applied to assure alignment of the macron and to maintain clearance with the accelerator wall.

Modeling Results

The 3-D colliding macron liner was simulated using ANSYS Multiphysics which has the capability of calculating the behavior of the colliding aluminum macrons well into the non-linear plastic deformation regime. The initial calculations were performed for macrons comprised of hollow aluminum spheres, 6.7 g in mass which were launched radially inward at 3 km/s. The spheres were observed to collide, deform, and finally form a complete, uniform liner. The entire process occurred with little diminution in the radial velocity. For the target velocity of 3 km/s the macron merging and compression is sub-sonic. The simulation was 3 dimensional and both stabilizing rotation (initiated with small non-radial velocity component) and 3-D compression (using 2–3 arrays of macron liners) were modeled. Finally, a complete treatment of the transient structural physics of the macron collision and integration was performed, including full plastic deformation stress–strain relationships, detailed surface roughness, and interaction studies.

Studies of liner uniformity and stability showed that the readily compressive nature of the hollow macrons themselves allows for a fairly large tolerance in initial velocity and timing, and still yield a uniform liner compression. It was found that initial positioning of up to one half macron radii (25% of total scale) is tolerable. For a 3 m diameter chamber this corresponds to an initial velocity and timing accuracy of 15 m/s and 2.5 μs. Surprisingly, it was found that even a fully missing macron is compensated for during the compression process. It was observed that greater than 95% of initial kinetic energy is maintained during maximum compression. A key advantage to this method of liner formation is the ability to aim the macrons to converge axially matching the FRC contraction and increasing the compressional energy on target. The macrons can also be aimed slightly off the axis of symmetry in order to provide rotational stability to the liner. The calculation shown in Fig. 4 illustrates the liner formation process employing 20 hollow aluminum macrons. In addition to the timing and velocity perturbations, the modeling showed that the high compressibility of the macrons allowed for significant rotation and axial variations to be added to the liner while still maintain maximum radial velocity.
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Fig. 4

Macron velocity profiles during formation of a 1.5 MJ liner formed employing 20 hollow spherical aluminum macrons moving radially inward at 3 km/s with the center of mass 5 mm off-axis to impart angular momentum to the liner for stabilization

Several aspects of the macron liner dynamics still need to be resolved. A significant issue concerns the behavior of the inner surface of the merged macron liner during field compression and stagnation. With macrons there is the possibility of a more irregular inner surface which could be prone to spiking. There are two effects that should ameliorate this tendency. As has been mentioned, it is trivial to add a large rotational motion to the inner wall by a coherent non-radial displacement of the macron initial trajectory. As can be seen in Fig. 4 this causes a rapid winding up and smearing of the inner boundary. The rotational motion is also stabilizing to the Rayleigh–Taylor modes that are the primary cause of the spiking. With a compressed megagauss magnetic field, there will also be large wall currents that would lead to rapid melting of small protuberances on the inner wall.

It should be possible to resolve the full macron liner behavior during magnetic field compression and stagnation with the same ANSYS Multiphysics code used in the initial simulations. To achieve this result, future code work will incorporate an axial magnetic field along with modifications to include material phase change and a temperature dependent resistivity. It will also require a finer mesh than was used in these calculations. With these improvements it should be possible to establish the dynamic stability and compressive efficiency of the MFL.

Experimental Results

A six-stage macron launcher was constructed in order to determine the efficiency and accuracy of gram-scale pulsed-inductive launchers. The initial launcher was constructed with six separate coil stages. Each 1.3 cm long stage consisted of a 5-turn 1.9 cm diameter coil made from 13 gauge Litz wire each potted in a low-viscosity epoxy. Each coil was driven with a 600 uF electrolytic capacitor bank switched with six paralleled 1,700 V Isolated Gate Bipolar Transistor (IGBT) switches producing a peak current of 24 kA at 1 kV resulting in an axial magnetic of up to 25 Tesla with a cylindrical macron. Between the coils a high-intensity, 660 nm fiber transmitter/receiver was positioned for velocimetry measurements (see Fig. 5). The initial launcher is shown in Fig. 5.
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Fig. 5

6 stage macron launcher

The six stage-launcher successfully demonstrated the pulsed-inductive acceleration of cylindrical macrons with masses ranging from 1 to 4 g. Variable aspect ratio cylinders and spheres were also examined. The six stage launcher demonstrated a peak velocity of 280 m/s and validated performance expectations for the limited energies tested. Several important characteristics were exhibited. The jitter from sequential launches with different macrons was observed to be near the resolution of the digitizer at a fraction of a microsecond. The macron arrival times varied by less than 1 μs corresponding to a velocity accuracy of 2 m/s. Additionally, velocity could be pre-programmed to less than 5 m/s. An initial position study was performed as it was found that maintaining the initial position of the macron was important in minimizing the jitter as well as the final macron velocity. This is primarily due to the accelerator coil inefficiency at low macron velocities where the magnetic field oscillations were not synchronized well with the macron passage. A key parameter in coupling efficiency is the mutual inductance between macron and the driver coil [see (2)]. As this quantity is difficult to calculate accurately, it was found the most straight forward (as well as most appropriate) way was to simply measure it. This was done for both the cylindrical and spherical macrons with the results shown in Fig. 6. The coupling was higher for the cylindrical macron, but the spherical macron produced a better force gradient for position and velocity control. As mentioned, it is difficult to properly drive the macron during the startup phase. The energy coupling efficiency, ηe, is defined as the ratio of the kinetic energy acquired by the macron to the coil magnetic energy. As indicated in Fig. 7, the energy coupling steadily improves for later stages at higher macron velocity where ηe approaches the theoretical maximum of 45% inferred by (2) and the measured mutual inductance for the prototype coils and cylindrical macrons. Higher efficiencies are achievable with larger coils. Different macron shapes will also be tested such as a bullet shaped macron that combines the advantages of both the cylinder and spherical macrons considered so far. Future work will increase the number of stages to bring the macron velocity closer to that desired for the reactor application.
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Fig. 6

Coil and macron total inductance for cylindrical and spherical macrons, measured as a function of position relative to the coil center

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Fig. 7

Macron velocity and energy coupling efficiency at each stage

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© Springer Science+Business Media, LLC 2010