Smoothed Particle Hydrodynamics Modeling of Charged
Particle and Neutron Yield in Plasma Driven MagnetoInertial Fusion
Kevin Schillo1 and Jason Cassibry2
University of Alabama in Huntsville, Huntsville, AL 35899
Magneto-inertial fusion (MIF) concepts may offer the key to yielding economic fusion
reactor designs. Using plasma liners as a standoff driver may offer a solution to some of the
technical challenges that MIF faces. This research may eventually lead to the development of
small, lightweight fusion reactors that could be used in spacecraft propulsion systems. This
study explores the parameter space for plasma liners, which includes the liner species,
kinetic energy, number of plasma jets, plasma density, and fusion fuel target density and
diameter.
Nomenclature
A
dr’
h
m
r
W
δ
κ
ρ
=
=
=
=
=
=
=
=
=
Particle property
Differential volume element
Radius of influence
Particle mass
Particle position
Smoothing kernel
Dirac delta function
Smoothing function constant at a given location
Density
I. Introduction
Research into magnetic confinement and inertial confinement fusion reactor concepts have been explored
extensively over the past several decades. But these efforts have still failed to result in a fusion reactor that generates
more energy than what is required to operate the reactor.
One potential path to developing a breakeven fusion reactor is magneto-inertial fusion (MIF). MIF uses a strong
magnetic field is used to implode a liner onto a fusion fuel target until fusion reactions occur [1, 2].
This confinement method may result in significantly smaller reactors [3] and enable near term development of
propulsion systems [4]. Because of the potential for lower compression power and total stored energy in the fusion
plasma, MIF may hold potential as being a low-cost development path for fusion for terrestrial power production.
Smaller fusion reactors would also be far more economical for terrestrial power production than the larger reactor
concepts associated with magnetic confinement or inertial confinement reactors. Additionally, fusion propulsion
using MIF approaches as the reactor have been shown by Cassibry et al [4], Miernik et al [5], and Adams et al [6] to
rapidly reduce interplanetary trip times compared to convential propulsion systems.
1
Graduate Research Assistant, Department of Mechanical and Aerospace Engineering, 301 Sparkman Dr NW,
Huntsville, AL 35899, AIAA Student Member.
2
Associate Professor, Department of Mechanical and Aerospace Engineering, 301 Sparkman Dr NW, Huntsville,
AL 35899, AIAA Senior Member.
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Figure 1. HOPE conceptual vehicle design [6].
Plasma-jet-driven magneto-inertial fusion (PJMIF) is an MIF concept developed by Thio [7] in which a series of
plasma jets are fired from plasma guns and form a cylindrical or spherical liner. This liner then implodes on a
magnetized fusion fuel target and brings it to fusion conditions. Formation of a spherically imploding plasma liner
via merging plasma jets is now being pursued on the Plasma Liner Experiment (PLX) at Los Alamos National
Laboratory [8] .
The principal objective of this study is to determine what parameters are needed in order for a plasma liner to
achieve breakeven fusion, in which the energy generated by the fusion reactor is equivalent to the energy needed to
operate the reactor.
The plasma liner parameter space includes different plasma species, plasma density, initial kinetic energy,
number of plasma jets, and plasma temperature. The fusion fuel target examined in this study is a sphere of
deuterium-tritium. The density and diameter of the target are other parameters explored.
II. Smoothed Particle Hydrodynamics
Smoothed particle hydrodynamics (SPH) is a meshless Lagrangian method that simulates fluid flows by dividing
a fluid into a set of particles and using a summation interpolant function to calculate the properties and gradients for
each of these particles. Particle properties are calculated by using a kernel function that adds up the properties of the
particles that lie within the kernel. The properties assigned to a specific particle are determined based on the density
and proximity of other nearby particles [9] [10] [11].
For the kernel approximation, an integral interpolant can be used to obtain the value for any property in a fluid,
and is defined as [12]
𝐴(𝑟) = ∫ 𝐴(𝑟′)𝛿(𝑟 − 𝑟′)𝑑𝑟′
The Dirac delta function is given by
1
𝛿(𝑟 − 𝑟′) = {
0
𝑟 = 𝑟′
𝑟 ≠ 𝑟′
By replacing the Dirac delta function with a smoothing function, the kernel approximation can be expressed as:
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𝐴(𝑟) = ∫ 𝐴(𝑟′)𝑊(𝑟 − 𝑟′, ℎ)𝑑𝑟′
The smoothing function is required to meet several constraints. The first is the normalization condition, which can
be expressed as
∫ 𝑊(𝑟 − 𝑟 ′ , ℎ)𝑑𝑟 ′ = 1
The second constraint is the Delta function property in the limit as h approaches 0, given by
𝑙𝑖𝑚 𝑊(𝑟 − 𝑟 ′ , ℎ) = 𝛿(𝑟 − 𝑟 ′ )
ℎ→0
The third constraint is the compact support condition, and is given by
𝑊(𝑟 − 𝑟 ′ , ℎ) = 0 𝑤ℎ𝑒𝑛 |𝑟 − 𝑟 ′ | > 𝜅ℎ
A particle approximation is used following the kernel function application. In the particle approximation, the
system is represented by a finite number of particles characterized by a specific mass and location. The continuous
integral representations in the kernel approximation are converted to discretized forms of summation over all of the
particles that lie within the support domain. This allows the integral interpolant to be approximated with a
summation interpolant given by:
𝐴(𝑟) = ∑ 𝑚𝑏
𝑏
𝐴𝑏
𝑊(𝑟 − 𝑟𝑏 , ℎ)
𝜌𝑏
The spatial gradient for this quantity can be computed using the equation summation
𝛻𝐴(𝑟) = ∑ 𝑚𝑏
𝑏
𝐴𝑏
𝛻𝑊(𝑟 − 𝑟𝑏 , ℎ)
𝜌𝑏
The single fluid equations of motion are solved in SPFMax. Conservation of mass is given by
∂ρ
+ 𝛻 ∙ (𝜌𝑢) = 0
∂t
In our code, conservation of mass is solved exactly by replacing the continuity equation above with
𝜌𝑎 = ∑ 𝑚𝑏 𝑊𝑏
The x, y, and z momentum equations are given as
ρ
ρ
ρ
𝑏
∂u
∂p ∂
∂u ∂v ∂w
∂u
∂
∂v ∂u
∂
∂u ∂w
=−
+ [λ ( +
+
) + 2μ ] + [μ ( + )] + [μ ( +
)]
∂t
∂x ∂x
∂x ∂y ∂z
∂x
∂y
∂x ∂y
∂z
∂z ∂x
+𝛱𝑥
∂v
∂p ∂
∂v ∂u
∂
∂u ∂v ∂w
∂v
∂
∂w ∂v
=−
+ [μ ( + )] + [λ ( + +
) + 2μ ] + [μ (
+ )]
∂t
∂y ∂x
∂x ∂y
∂y
∂x ∂y ∂z
∂y
∂z
∂y ∂z
+𝛱𝑦
∂w
∂p ∂
∂u ∂w
∂
∂w ∂v
∂
∂u ∂v ∂w
∂w
= − + [μ ( +
)] + [μ (
+ )] + [λ ( +
+
) + 2μ ]
∂t
∂z ∂x
∂z ∂x
∂y
∂y ∂z
∂z
∂x ∂y ∂z
∂z
+𝛱𝑧
The bulk viscosity coefficient is defined as being
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2
λ=− 𝜇
3
The single temperature energy equation is given by
ρ
De
∂(𝑝𝑢) ∂(𝑝𝑣) ∂(𝑝𝑤)
∂
∂u ∂v ∂w
∂u
∂u ∂v ∂w
∂u ∂u
=−
−
−
+ u [λ ( +
+
) + 2μ ] + [λ ( +
+
) + 2μ ]
Dt
∂x
∂y
∂z
∂x
∂x ∂y ∂z
∂x
∂x ∂y ∂z
∂x ∂x
∂
∂v ∂u
∂v ∂u ∂u
∂
∂u ∂w
∂u ∂w ∂u
+ u [μ ( + )] + [μ ( + )]
+ u [μ ( +
)] + [μ ( +
)]
∂y
∂x ∂y
∂x ∂y ∂y
∂z
∂z ∂x
∂z ∂x ∂z
∂
∂v ∂u
∂v ∂u ∂v
∂
∂u ∂v ∂w
∂v
+ v [μ ( + )] + [μ ( + )] + v [λ ( +
+
) + 2μ ]
∂x
∂x ∂y
∂x ∂y ∂x
∂y
∂x ∂y ∂z
∂y
∂u ∂v ∂w
∂v ∂v
∂
∂w ∂v
∂w ∂v ∂v
+ [λ ( +
+
) + 2μ ] + v [μ (
+ )] + [μ (
+ )]
∂x ∂y ∂z
∂y ∂y
∂z
∂y ∂z
∂y ∂z ∂z
∂
∂u ∂w
∂u ∂w ∂w
∂
∂w ∂v
∂w ∂v ∂w
+ w [μ ( +
)] + [μ ( +
)]
+ w [μ (
+ )] + [μ (
+ )]
∂x
∂z ∂x
∂z ∂x ∂x
∂y
∂y ∂z
∂y ∂z ∂y
∂
∂u ∂v ∂w
∂w
∂u ∂v ∂w
∂w ∂w
+ w [λ ( +
+
) + 2μ ] + [λ ( +
+
) + 2μ
]
− 𝜌𝛻 ∙ (𝑘𝛻𝑇)
∂z
∂x ∂y ∂z
∂z
∂x ∂y ∂z
∂z ∂z
4
− 4𝜌𝜎𝑇 𝜒𝑃𝑙𝑎𝑛𝑐𝑘
III. Plasma Liner Simulations
Make a table that is rougly 10 cases or so long that show how the parameters will be varied (MW, jet velocity
perhaps?, jet density?)
Indicate that the primary figures of merit will be the charged particle energy yield and neutron yield
IV. Results
Multiple simulations of plasma jet liner formation and implosions have already been conducted with the SPH
code. A representative example of a case study with 60 plasma jets is shown in Figure 2:
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Figure 2. 3D implosion of 60 plasma jets.
Multiple case studies examined the effects of parameters such as molecular weight, jet velocity, and plasma density.
Figure 3 shows the effect that different molecular weights have on the ram pressure of the imploding plasma liner.
Figure 3. Effect of molecular weight on ram pressure.
Additional results will be obtained that show the effects that plasma jet molecular weight, implosion velocity,
plasma density, and target size have on the charged particle and neutron yield.
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V. Conclusions
Acknowledgments
This work was supported by the ARPA-E ALPHA program, and the authors wish to acknowledge discussions
with the members of the PLX modeling team, including Scott Hsu, Roman Samulyak, Peter Stolz, Kris Beckwith,
and Samuel Langendorf.
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