System Oriented Development of Fusion Propulsion Concepts

16 Pages·2015·1.93 MB·English

by  Roland Antonius Gabrielli

System Oriented Development of
Fusion Propulsion Concepts
IEPC-2015-23 /ISTS-2015-b-23

Presented at Joint Conference of 30th International Symposium on Space Technology and Science
34th International Electric Propulsion Conference and 6th Nano-satellite Symposium,
Hyogo-Kobe, Japan
July 4 – 10, 2015

Roland A. Gabrielli,* Adam S. Pagan,† Eloi Ferrer-Gil,‡ and Georg H. Herdrich§
Institute of Space Systems, University of Stuttgart, Stuttgart, Baden-Wuertemberg, 70569, Germany
Abstract: Fusion promises to induce a paradigm shift in space transportation. Deriving
from generic thermal propulsion concepts, a Thermal Fusion Thruster is made up of the fu-
sion plasma as a heat source, a working medium, and a nozzle. Various combinations of fuels,
propulsion concept, and magnetic geometry are possible. Two common confinement geome-
tries, Field Reveresed Configurations and Spherical Tokamaks are discussed in the present
contribution. To this end two-dimensional, axisymmetric analytic models for the investigation
of the respective magnetic confinement of the fusion plasma are selected, implemented and
expanded to include the modelling of thermodynamic plasma quantities and energy transport
phenomena. The introduction of a simple fusion criterion allows for the study of local ignition
conditions and facilitates the establishment of an energy balance. The results for Spherical
Tokamaks are then compared to contemporary projects in terrestrial fusion research.
Nomenclature
A = aspect ratio n = particle density
j
A = area of plasma column cross section n = electron density
C e
a = semi-major axis P = bremsstrahlung (power loss)
BrS
a = acceleration P = Fusion power yield
k F
B = magnetic flux density P = Power gained
ϕ G
B = toroidal component of the magnetic P = Power lost
ϕ L
flux density P = Power recuperated
R
b = semi-minor axis P = synchrotron radiation (power loss)
Syn
c = effective exhaust velocity P = thermal power loss
e th
F or f = generic Function p = plasma pressure
f = bootstrap current fraction Q = ignition factor
BS
j = ion particle species R = radial axis
k = Boltzmann constant r = major radius of torus cross section
B 0
k = feedback fractions r = minor radius of torus cross section
back a
m = dry mass T = electron temperature
D e
m = propellant mass T = all ion species’ temperature
P i
N = number of individual Z = longitudinal axis / axis of revolution
ion particle species <Zj> = average charge state of ion species j

* Research Associate, Space Transportation Technology, [email protected].
† Research Associate, Space Transportation Technology, [email protected].
‡ Research Student, Institute of Space Systems, University of Stuttgart, [email protected].
§ Head of Plasmatechnology Group, Space Transportation Technology, [email protected].
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
α = mass specific jet power τ = burn duration
j B
β = ratio between the kinetic pressure and τ = energy confinement time
E
the applied magnetic pressure Ψ = magnetic flux function
Δv = velocity increment Ψ = particle multiple
tot
δ = Kronecker’s delta Ω = abbreviation according to term
ik
μ = vacuum permeability
0
<σν> = thermal fusion reactivity

I. Introduction
E
VEN after decades of space flight, destinations such as the planets of the Solar System have been explored with
robotic systems only. No manned mission has ever entered space beyond Earth’s Moon. This is due to the fact
that available propulsion systems cannot provide for flexible and expeditious transfers which would reduce the risks
and inconveniences astronauts face in interplanetary space.1 The mission requirement of flexibility suggests a contin-
uously powered transfer with an acceleration a sufficient to build indefinite velocity increments
k
(cid:543)(cid:1874) (cid:3404) (cid:3505)(cid:1853) (cid:3)(cid:1856)(cid:1872) (1)
(cid:3038)
(cid:3099)(cid:3251)
during a sufficiently short burn duration τ . However, available high acceleration thrusters provide bad dry mass frac-
B
tions for large Δv due to their limited specific exhaust velocity c, as shown by Tsiolkovsky’s equation
e
(cid:1865) (cid:3397)(cid:1865)
(cid:3005) (cid:3017)
(cid:543)(cid:1874) (cid:3404)(cid:1855) (cid:3)(cid:142)(cid:144)(cid:3436) (cid:3440) (2)
(cid:3032) (cid:1865)
(cid:3005)
in which m is the dry mass and m the mass of the propellant used for the manoeuvre. Thus, flexible mission designs
D P
also necessitate a high specific exhaust velocity c. However, the indefinite raise of both parameters is precluded due
e
to the power supply’s available mass specific power
(cid:2869)
(cid:2009) (cid:3)(cid:3410)(cid:2009) (cid:3404) (cid:3)(cid:1853)(cid:3)(cid:1855) . (3)
(cid:3017)(cid:3020) (cid:3011) (cid:2870) (cid:3032)
always exceeding the mass specific jet power α. This substantiates that with a given power supply the acceleration
J
can only be augmented at the expense of the specific exhaust velocity and vice versa, and that, eventually, both pa-
rameters can only be augmented simultaneously, if a very dense power source was provided.
This rationale has already motivated the investigation of many Nuclear Thermal Propulsion concepts among which
the notorious fission rocket engine NERVA which has even undergone ground testing and thus constitutes the most
mature example.2 But while fission rocket engines yield promising results in their own right, they are expected to be
vastly outperformed by nuclear thermal fusion propulsion systems.3,4 Besides promising better power levels, acceler-
ations, and specific exhaust velocities fusion thrusters may also alleviate some of the radiologic concerns typically
associated with the former.5 Much research has been conducted in the USAFehler! Textmarke nicht definiert.-6 and, more re-
cently, in Europe.
The Institute of Space Systems of the University of Stuttgart (IRS) has contributed to the topic since 2006.7 The
studies aim at identifying the most promising setup of a nuclear fusion thruster considering both the physics – begin-
ning with the ignitability of the fusion fuels8,9 and extending over plasma properties and potential neutron background
radiation10 – and engineering key technologies applicable to subsystems.11,12 The concept and parameters studies were
accompanied with the development of an engineering instrument named Space Vehicle Code (SVC).13 The present
contribution summarises some of the recent results related to the introduction of the Magnetic Field Geometry finally
enabling a comparison to terrestrial fusion reactor studies, encompassing cases as ARIES, ITER and others.
II. Fusion Propulsion Concepts
A. A Brief Overview on Fusion and Magnetically Confined Fusion Reactors in Space

Nuclear fusion is a process in which atomic nuclei are combined into a new, heavier one, entailing a mass defect
which is then released as energy surplus if the educts’ masses were sufficiently small.14 To achieve this, the nuclei
need to be approached close enough to each other to allow the strong interaction to apply. Due to the strong repulsive
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
electrostatic forces – the Coulomb wall – this would require extremely high kinetic energies. If it was attempted to
provide this energy by heating, this would require temperatures far above 100 Million degrees Kelvin, which would
make the process impractical. However, a significant fusion rate was already observed at temperatures of about
10 Million degrees Kelvin.15 The cause of this alleviation is the quantum physical tunnel effect identified by George
Gamow.16 For practical reasons, such temperatures are indicated in kilo Electron-Volt (1 keV = 11.609×10(cid:898) K).
If it comes to harnessing this source of energy, one of the most commonly discussed approaches is thermal fusion
which consists in generating a high enthalpy plasma in which fully ionised atoms would attain respective temperatures
and in which fusion would occur. The plasma needs to be around 10 to 100 keV and quench severely upon contact
with solid materials. This makes a case for a magnetic plasma confinement. The thermal ions in the plasma can be
deviated and thus kept from walls by magnetic fields. Several geometries and respective open and closed reactor
designs have been proposed during the past decades. The most prominent current research project on this topic is the
International Thermonuclear Experimental Reactor, or ITER, based on a toroidal magnetic confinement configuration
known as the Tokamak.17 A more compact version of the Tokamak is the Spherical Tokamak (ST),18 implemented in
the National Spherical Torus Experiment (NSTX).19 Spherical Tokamaks promise not only a comparatively compact
construction but also an increased plasma confinement efficiency, and have therefore already been considered for
space propulsion systems.6,20 Besides these concepts with closed magnetic geometries, also designs with open mag-
netic fields were considered, like Gas Dynamic Mirrors (GDM),21 or the Field Reversed Configuration (FRC).22 In
absence of a central hole, the latter – likewise toroidal geometry –works without magnetic field coils running through
a central column. If applied to space propulsion, this open geometry also entails an intrinsic divertor which may be
utilised as a magnetic nozzle.
In principle, there is little conceptual difference between terrestrial and space borne fusion reactors based on ST
or FRC geometries. However, there is a number of relevant systemic differences proper to the space borne designs:
1. Self-sustaining fusion reactors in space yield power abundantly, ranging up to several thousand MW.3 Even
if a major part of this energy is converted to thrust, the remaining waste power will be excessive and will have
to be removed by the means of large radiators.11
2. The reactor must be able operate continuously for several weeks to enable disruptive missions.11,23
3. The reactor requires auxiliary subsystems – for example a fission reactor or secondary batteries – providing
for autarchic start-up and stabilisation.23
4. Compact construction fitting to available launchers or modular construction allowing for in-space assembly.23
5. In case of neutron abundant reactions or fission auxiliaries, shielding of crew and payload is required.11
6. A vacuum system may be obsolete due to the availability of the near-vacuum of space. This can enable ap-
proaches such a skeletal design, both resolving waste heat issues (heat contributions may escape directly into
space), and reducing the system mass. This may however entail another arduousness, the unimpeded poisoning
of the plasma with debris.
In the following, thermal thrusters, are discussed. An additional system difference arises from the requirement to
heat a propellant efficiently, preferably utilising the fusion products and/or reactor waste heat.11 Due to the expected
heat of the downstream propellant mass flow beyond thermal limitations of materials, a magnetic nozzle may be
necessary to exhaust ionised propellants and fusion products.23
B. (Variable) Working Gas Fusion Propulsion Systems

Fusion (Variable) Working Gas Drives (V)WGD are a particular case of Nuclear Thermal Propulsion (NTP). NTP
concepts utilise the heat yielded by nuclear processes in the system core to convectively energise a coolant/propellant
medium, typically gaseous hydrogen due to its excellent caloric properties, which is then expanded through a nozzle
to generate thrust.2 A decisive constraint which is limiting these systems’ performance in terms of exhaust velocity is
the material thermal load limit, as the structure may not be heated beyond its solidus line. For illustration, the exhaust
jet of NERVA-I had a temperature of 2700 K, restricting the exhaust velocity to 8093 m/s.2 Other Fission Thermal
Propulsion concepts like Gas Core Reactors are aimed at resolving this issue by allowing for phase changes. The
gaseous fission core is maintained in a cavity pressurised by an injected working medium. A drawback of these sys-
tems was however, that the major part of the core’s heat was transferred inefficiently by thermal radiation, motivating
the seeding of the working medium, gaseous hydrogen, with opaque solid particles.
In WGD, a similar concept is realised, making use of the presence of a so-called scrap-of layer in magnetically
confined fusion reactors which is a stream of ejecta from the core plasma. It is into this layer that the coolant is injected.
As a result, the principal heat transfer mechanism is the mixing of the coolant and the ejecta, assumed to be more
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
Figure 1. Schematic cross section of a Spherical Tokamak Working Gas Drive burning D-3He (Adapted
from ref. 6).
efficient than by thermal radiation. It also permits to reap a significant part of the thermal losses of the fusion core
caused by the loss of hot particles for the generation of thrust. Furthermore, a large fraction waste heat is continuously
removed of it from the spacecraft, greatly relieving the radiators. The working gas drive concept also enables a con-
tinuous removal of fusion products, which is necessary to maintain a steady-state operation of the reactor and of the
fusion propulsion system as a whole. A cross-section of a hypothetical working gas fusion propulsion system utilising
a Spherical Tokamak reactor core is pictured in figure 1.
While the magnets are the most important subsystem of working gas drives confining the fusion plasma, further
subsystems are also important. Figure 2 depicts the generic setup for both D-T and D-3He fuelling:
The innermost is the blanket with its first wall facing the plasma column. While it can serve as a shield against
potential neutron radiation its primary purpose is to
deal with the waste heat. On one hand, some of the
radiative heat contributions emanating from the D-T D-3He
plasma may be reflected back, contributing to the sus-
tainment of the fusion reaction, on the other hand,
some of it may be absorbed by a coolant passed
through its stipulated pores. Subsequently, the cool-
ant is effused into the scrap-off layer between the first
wall and the plasma. It is assumed to be sufficiently
ionised to be kept out of the fusion plasma by the
magnetic field. Instead it would mix with the particle
flow of the scrap off layer and be further heated, fi-
nally being exhausted by the divertor, generating
thrust.
Note, that the mass flow of the coolant is critical. rMag rBlt,in rP Fusion rP rBlt,in rMag
If it was too large with respect to the fusion plasma, d Plasma d
Blt Blt
it could risk not being fully ionised, entailing coolant
pbioneaut rsest reiircvnulevepreste siplntyiegg nq ateuhttieerona nctcisohn,n1eg1td iin tnahunteood uc tisoht soef rlufaesunaisotci tnoma nnra etps amslca tifsixlmootnuwa.r. e IwI ntd aiwplsur oeltuaevildidd-, DEC Outer Blanket Magnets Inner Blanket Halo Halo Blanket DEC Magnets
to the end of recovering most of the waste heat for Figure 2. Generic diagram of subsystem layers sur-
propulsion purposes, yielding an approximate 90 % rounding the fusion plasma core for D-T and D-3He re-
efficiency similar to NERVA,2 but this also very large actant couples.
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
and therefore unpractical mass flows. However, the same investigations also revealed, that Ash Drives, the variant
waiving the coolant mass flow altogether, was also unpractical. It arises that the most realistic approach would be to
allow for a scalable coolant mass flow varying between both extremes trading off efficiency for feasibility; this con-
cept is called Variable Working Gas Drive.
The remaining subsystems are a Direct Energy Conversion (DEC) layer serving to harvest waste heat dissipating
to the outside for the electric provision of the reactor or the space-craft, and a cryogenic cooler required to maintain
the operating temperature of the magnetic coils, especially with respect to superconducting variants. As stated above,
radiators are important. For the sake of simplicity, these are omitted in figures 1 and 2.
C. The Spherical Tokamak (ST)

The Spherical Tokamak or Spherical Torus (ST) configuration, is a variant of the well-studied Tokamak design.
In contrast to the latter, STs have a realtively low aspect ratio A, relating the major radius r to the minor radius r :
0 a
(cid:1870)
(cid:2868)
(cid:1827)(cid:3404) (4)
(cid:1870)
(cid:3028)
The plasma column of STs consequently describes the shape of a cored apple. A typical plasma column cross
section is shown in figure 3. Just as in regular Tokamaks, STs provide for confinement with a strong toroidal field
created by external toroidal magnetic coils. Moving charged particles are principally trapped in this field. Neverthe-
less, there are some loss mechanisms that have to be answered by applying a poloidal magnetic field induced by a
toroidal electric plasma current I . The superposed field’s lines are twisted into a helical shape shifting the gyrating
P
plasma plasma particles around the minor axis of the torus, assuring a very good, efficient confinement,24 outmatching
the one of regular Tokamaks by achieving a similar de-
gree of confinement with lesser magnetic fields, and
hence simpler coils. Besides, a substantial fraction of the
plasma current required to induce the poloidal field (the
so-called bootstrap current fraction f ), occurs intrinsi-
BS
cally due to the low aspect ratio, alleviating the require-
ments for an external current drive.18
As is depicted in figure 1, an ST fusion propulsion
system operating by the working gas drive principle
could make use of a set of ITER-like poloidal divertors
to continuously remove propellant and product particles
from the scrape off layer and to divert them towards the
magnetic nozzle. Other approaches to utilise the ST as a
propulsion system have been suggested, one being to use
divertors to remove energetic plasma particles from the
SOL and feeding these to the propellant within a mag-
netic mixing reservoir directly upstream of the magnetic
nozzle.6 An alternative concept makes use of magnetic
field ripples, causing highly energetic fusion products to

be deliberately expelled from confinement into a prefer- Figure 3. Outline of smooth ST plasma column
ential direction, allowing for a smooth and uninterrupted cross section with characteristic magnetic flux sur-
plasma surface and thus completely dispensing with the faces (for Ψ = const.) and definition of geometric pa-
need of a magnetic divertor, somewhat simplifying the rameters (κ: elongation, δ: triangularity).
reactor design.20
D. The Field Reversed Configuration (FRC)

The Field Reversed Configuration (FRC) is a magnetic configuration that can be generated inside a cylindrical
coil without the need of invasive toroidal field coils running through the centre of the plasma volume. Configurations
such as this, in which there is no central hole, are known as Compact Toroids (CT). A variety of CT can be generated
by combining varying components of toroidal and poloidal magnetic fields. The FRC is a particular case of a CT, in
which the toroidal magnetic field is negligible in relation to the poloidal field.25
Due to low magnetic flux densities within the FRC plasma, its stabilisation cannot be explained by using the ideal
magneto hydrodynamics (MHD) theory and is instead attributed to non-ideal effects.22
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
The absence of a central column and the effectively non-existent toroidal magnetic field result in a geometric and
magnetic simplicity that contribute to making the FRC a promising concept for the confinement of fusion plasmas 25.
The purely poloidal magnetic field results in a very high ratio β between the kinetic pressure and the applied magnetic
pressure. In consequence, high plasma pressures can be achieved, allowing highly energetic aneutronic fusion reac-
tions to take place.22
Two important regions can be distinguished in an FRC, as shown in figure 4. In the outer region, the magnetic
field lines are open in the domain of observation, i.e., the interior of the magnetic coil. The magnetic field lines are
closed within the inner region, allowing
for plasma to be confined. The frontier
between these two regions is known as
separatrix and is central to defining the
geometry of an FRC.26 The separatrix
can be defined by its semi-axes a and b,
and by its relative radius, which is the ra-
tio between the semi-minor axis of the
separatrix a, and the radius of the coil r.
c
The shape of the magnetic field lines out-
side of the separatrix conveniently result
in a natural divertor that can be shaped
into a magnetic nozzle, exhausting the
fusion products expelled by the plasma
together with the heated propellant in or- Figure 4. Outline of FRC cross section with characteristic mag-
der to provide thrust or to generate en- netic flux surfaces (for Ψ = const.) and definition of geometric pa-
ergy.25 rameters (a: minor axis, b: major axis, RCoil: coil radius).
III. Analytic Modelling

In an attempt to achieve results suitable to compare both systems with regards to their overall performance, a
simple analytic model is constructed by combining existing solutions regarding the modelling of the respective mag-
netic confinement environment with a simple model yielding approximate values of the thermodynamic properties
and transport phenomena expected to arise within the plasma. A coordinate system is defined in accordance with
figures. 3 and 4. By assuming axial symmetry around Z, the problem is effectively reduced to a two-dimensional one.
A. Confinement Modelling

The first step in order to model the magnetic confinement environment of both configurations is to obtain a math-
ematical model of the distribution of the magnetic flux Ψ throughout the fusion plasma. For a 2D system, the magnetic
flux function Ψ is required to comply with the Grad-Shafranov (GS) equation. The GS equation describes an ideal
magnetohydrodynamic (MHD) equilibrium in a two-dimensional plasma27 and is given as
(cid:2034) (cid:883)(cid:2034)(cid:2006) (cid:2034)(cid:2870)(cid:2006) (cid:1856)(cid:1868) (cid:883)(cid:1856)(cid:1832)(cid:2870)
(cid:1844) (cid:3436) (cid:3440)(cid:3397) (cid:3404)(cid:3398)(cid:2020) (cid:1844)(cid:2870) (cid:3398) (5)
(cid:2034)(cid:1844) (cid:1844)(cid:2034)(cid:1844) (cid:2034)(cid:1852)(cid:2870) (cid:2868) (cid:1856)(cid:2006) (cid:884) (cid:1856)(cid:2006)
where μ is the vacuum permeability, p is the plasma pressure and F = RB . B (R,Z) is the toroidal component of the
0 ϕ ϕ
magnetic flux density, which is generated by the toroidal field coils in the ST and negligible for the FRC configuration.
Both p and F are considered functions of Ψ(R,Z).
A variety of analytic solutions to the GS equation exist based on different approaches to the problem, often specific
to the observed magnetic configuration. Both solutions selected for modelling the magnetic confinement environment
within the ST and the FRC are based on Solov’ev profiles, where p and F are assumed to be in linear dependence of
Ψ.28 While this corresponds to an unrealistic description of the toroidal current profile, which in turn has a detrimental
effect on the accuracy of the resulting distributions of p and F, such solutions still retain many crucial physical prop-
erties of the configurations in question, are simple to implement and yield fast results.29 The Spherical Tokamak is
modelled using a solution strategy presented in ref. 29. This solution allows for a full definition of the geometric shape
parameters of the fusion plasma column with little or no need to determine operational parameters through iteration.
If so desired, single or double x-divertors can be considered in the solution.
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
The solution selected to model the Field Reversed Configuration is the Steinhauer Analytical Equilibrium,30 which
constitutes a superposition of two solutions of the GS equation. The internal solution models the region inside of the
separatrix and is based on the Solov’ev solution. It includes a term matching it to an external solution, which models
the magnetic configuration outside of the separatrix, fitting it to the cylindrical coil.
Once the magnetic flux function Ψ(R,Z) has been established, both the magnetic flux density distribution B(R,Z)
and the distribution of the plasma pressure p(R,Z) are defined. This allows for the further extraction of plasma figures
of merit such as the plasma beta β, which relates the plasma pressure to the magnetic pressure exerted by the magnetic
environment. For the sake of brevity, the calculation of these figures of merit will not be discussed here, whereas they
are explained in refs. 29 and 30.
B. Plasma Performance Model

The ideal gas law14
(cid:3015)
(cid:1868)(cid:3404)(cid:1863) (cid:4684)(cid:1866) (cid:1846) (cid:3397)(cid:1846) (cid:3533)(cid:1866) (cid:4685) (6)
(cid:3003) (cid:3032) (cid:3032) (cid:3036) (cid:3037)
(cid:3037)(cid:2880)(cid:2869)
applies to a fusion plasma with N being the number of individual ion particle species j confined to the plasma volume.
Furthermore, n is the particle density and the temperature of j, whereas all ion species are assumed to have the same
j
temperature T. The electron density and temperature are given through n and T respectively. The Boltzmann constant
i e e
is k . Assuming a given temperature distribution, the respective unknown property may be calculated from the pressure
B
p yielded by the confinement solution. As only charged particles may be influenced through electromagnetic forces,
neutral particles such as neutrons are considered to be almost immediately lost from the confinement volume.
The two reactant species’ ions are denoted through the indices ‘i’ and ‘k’ and are usually assumed to exist in
identical quantities (n = n ). Neither fusion products, sometimes called ash, nor contamination through heavier ions
i k
are considered at any depth for the purpose of this study. However, their effects on the electron and reactant densities
are accounted for through simplistic, but nonetheless effective means.
The charge neutrality condition requires that the plasma remains in state of quasi-neutrality by assuring that
(cid:3015)
(cid:1866) (cid:3404)(cid:3533)(cid:1866) (cid:1731)(cid:1852)(cid:1732) (7)
(cid:3032) (cid:3037) (cid:3037)
(cid:3037)(cid:2880)(cid:2869)
where <Zj> is the average charge state of the ion species j and may be considered equal to its proton number, as most
species likely to be encountered in a fusion plasma at temperatures of T ≥ 8 keV are fully ionised.31 Thus a certain
fraction of ash and impurities within the plasma shifts the overall balance towards a noticeable majority of electrons.
This is accounted for by simply requiring that
(cid:1866) (cid:3404)(cid:882)(cid:484)(cid:887)(cid:887)(cid:3)(cid:1866) (8)
(cid:3032)
The direct displacement of reactant ions through ash and impurities is approached in a similar fashion, by setting
a fixed ratio between the reactant species and electron densities:
(cid:1866)
(cid:3032)
(cid:1866) (cid:3404)(cid:1866) (cid:3404) (9)
(cid:3036) (cid:3038) (cid:884)(cid:484)(cid:887)
with the residual density assumed to be an unspecified composition of ash and impurities. Other, more sophisticated
and accurate ash and impurity models have been implemented, but while eqns. (8) and (9) appear to be overly sim-
plistic, trials in which results obtained using this method were compared to reliable values from previous works on
experimental fusion devices such as JET32,33 showed a remarkable agreement. This simple method was thus imple-
mented in an effort to find a common denominator for the differing ST and FRC models, enabling a direct comparison.
Of course, this holds true only when assuming that the rate of ash removal as well as the degree of contamination
in an FRC plasma is comparable to that of a (Spherical) Tokamak.
Under the assumption of a similarity to fusion plasmas in regular Tokamaks,17 peaked temperature profiles, often
with T (cid:1222) T, are assumed for the ST analysis, whereas the corresponding density profiles are comparatively flat.
e i
Profiles are prescribed in a “piggy-back” manner by assuming a similar, albeit inverted distribution as the magnetic
flux Ψ and exploiting the solution for the flux function accordingly.
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
For the FRC, a constant temperature T = T = const. is assumed throughout the plasma volume, based on observa-
e i
tions made in previous studies of experimental FRC confinement devices.34 The shape of the resulting density profiles
thus resembles the pressure profile yielded by the confinement solution.
Once the individual particle density and temperature distributions have been established, energy transport phe-
nomena may be modelled. Reference 8 presents expressions enabling the calculation of the fusion power
P = f(n,n,T), the thermal losses P = f(n,T,τ ) and the Bremsstrahlung and Synchrotron radiation losses,
F i k i th E
P = f(n,n,n,n ,T) and P = f(n,T,B), respectively, for a generic 0D fusion reactor. The energy confinement
BrS e i k ash e Syn e e
time τ is approximated individually for the ST and the FRC in accordance with literature data.34,35
E
The transport power expressions can be adapted for the multidimensional model by applying the formulae to all
discrete data points and numerically integrating the thus obtained power density values over the whole of the plasma
volume utilising the Guldinus theorem. As an example, the total fusion power P is obtained thus:
F,tot
(cid:1866) (cid:1866)
(cid:1842) (cid:3404)(cid:884)(cid:3)(cid:2024) (cid:3505) (cid:3036) (cid:3038) (cid:1731)(cid:2026)(cid:1874)(cid:1732)(cid:1831) (cid:1844)(cid:3)(cid:1856)(cid:1827) (10)
(cid:3007)(cid:481)(cid:3047)(cid:3042)(cid:3047) (cid:883)(cid:3397)(cid:3)(cid:2012) (cid:3007)
(cid:3036)(cid:3038)
(cid:3002)(cid:3252)

in which A represents the area of the plasma column’s cross section and δ is Kronecker’s delta, preventing identical
C ik
reactants from being counted twice. <σν> is the thermal fusion reactivity, which is calculated in dependence of T and
i
in accordance with reference 36. Last, E represents the energy released through an individual fusion event.
F
C. Lawson Criterion

A generic energy balance equation is established in ref. 8. Its time derivative is

(cid:1842) (cid:3397)(cid:1842) (cid:3404)(cid:1842) , (11)
(cid:3008) (cid:3019) (cid:3013)
in which P , P and P refer to the power gained, recycled and lost, respectively. The energy transport expressions
G R L
are multiplied with individual feedback fractions k for those power losses which may be partially recuperated
back
through reflective walls or other means in order to reheat the plasma. After substituting these expressions into the
energy balance eqn. (11) and rearranging the result, the generalised burn criterion is obtained:
(cid:1866) (cid:2028) (cid:1846) (cid:3404)(cid:3)(cid:2871)(cid:3435)(cid:2869)(cid:2879)(cid:3038)(cid:3277)(cid:3276)(cid:3278)(cid:3286)(cid:481)(cid:3295)(cid:3283)(cid:3439)(cid:3038)(cid:3251)(cid:3021)(cid:3284)(cid:3118)(cid:3077)(cid:3178)(cid:3173)(cid:3178). (12)
(cid:3036) (cid:3006) (cid:3036) (cid:2870) (cid:2960)(cid:3255)(cid:2879)(cid:2960)(cid:3251)(cid:3293)(cid:3268)(cid:3127)(cid:3208)(cid:3268)(cid:3300)(cid:3289)
Here, Ψ is a particle multiple, relating the sum of all confined particle densities to the first species reactant density
tot
n while also compensating for temperatures deviating from T. The abbreviation Ω represents the power gain through
i i F
fusion processes, while Ω and Ω represent the losses through bremsstrahlung and synchrotron radiation. The
BrS Syn
parameter k is the thermal feedback factor, while the equivalent factors pertaining to the recuperation of neutron
back,th
power and radiation are included in the abbreviations Ω. Moreover, n τ T is known as the triple product and is
i E i
supplied by the plasma performance model for every discrete point throughout the plasma volume. The right-hand
side of eqn. (12) is evaluated accordingly.
The result may be interpreted in that all regions in which the burn criterion is exceeded may be regarded as con-
tributing to the net power obtained from the reactor, whereas the areas where the criterion is not met are essentially
losing power. However, bearing in mind that eqn. (12) has been adapted from a generic 0D model, this corresponds
to the potentially unrealistic case in which recuperated power losses are fed back directly into their point of origin.
IV. Analyses
A. Fuel selection considerations

A variety of nuclear fusion reactions of different reactant pairings have been considered for application in reactors
and propulsion systems.11 The use of two distinct fuel pairings is considered in this work, both of which are considered
the most likely candidates for the realisation of working fusion reactors within the foreseeable future:
D + T (cid:1037) n + 4He
0
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
D + 3He (cid:1037) p+ + 4He
The first fusion reaction, referred to here as the Deuterium-Tritium or D-T reaction, imposes relatively low re-
quirements on the reactor system with plasma temperatures around 10 keV usually being sufficiently high for a sus-
tained reaction. This makes it the fusion reaction of choice for many conceptual and experimental terrestrial fusion
devices such as JET32 and ITER.17 The supply of fuel components is considered ample. However, since a considerable
fraction (approx. 80%) of the fusion power is yielded to the produced neutrons, shielding of the payload and of reactor
materials prone to radioactive activation through neutrons are of considerable concern, often requiring an excessive
amount of additional mass.8,11 Furthermore, as the neutrons cannot contribute to the reheating of the fusion plasma,
some additional external plasma heating may be required, further complicating the system.
The Deuterium-Helium-3 (D-³He) fusion reaction greatly alleviates this problem, as neutrons are produced merely
in a side reaction (not shown here) at a fraction of approx. 5%, thus considerably reducing the shielding and probable
reheating requirements. However, temperatures of around 100 keV are required to sustain such a reaction, imposing
high requirements on the effectiveness of the magnetic confinement. Helium-3 (³He) is considered to be extremely
rare and difficult to obtain in significant quantities on Earth, while abundant supplies are expected to be found else-
where in the Solar System, most notably on the Moon and the gas giants. Further, more advanced fusion reactions
yielding few or no neutrons have been considered for fusion space propulsion systems:
11B + p+ (cid:1037) 3 4He
3He + 3He (cid:1037) 2p+ + 4He
However, the requirements imposed on adequate magnetic confinement systems are immense. Early trials con-
ducted for this work have suggested that the magnetic flux densities required to uphold the necessary plasma pressures
would cause excessive energy losses through bremsstrahlung radiation, surpassing even the energy gained through the
fusion reaction. Thus, no analysis was conducted incorporating these advanced reactant pairings.
B. Design point selection

The ST and the FRC propulsion systems are compared by determining the range of the mass specific power α that
can be achieved under practical circumstances. Reactor design points are selected both for D-T and for D-3He reactions
with the intention of encompassing the respectively largest possible range of operation. The volume-averaged net
power P of the propulsion system is zero, which corresponds to the case in which the Lawson criterion eqn. (12) is
net
met for the plasma volume as a whole. A practicable fusion propulsion system would operate close to this point without
exceeding it, thus guaranteeing a mostly self-sustaining fusion reaction while retaining full control over the reactor.
The working gas drive concept applied to both confinement configurations considered in this study requires the
absorption of much of the waste energy emitted in the form of electromagnetic radiation and neutrons by the plasma
column within the blanket.8 All of the thus absorbed waste energy is assumed to be efficiently used to preheat the
coolant conductively prior to it being seeded into the halo. A significant fraction of the thermal losses, however, are
absorbed by the preheated coolant within the halo before reaching the blanket.
The maximum heat and neutron loads P ≈ 5 MW/m3 and P ≈ 20 MW/m3 are not to be exceeded at the
bth,max bn0,max
first wall for any configuration, effectively enforcing a lower limit with regards to the reactor size, as wall loads grow
with a reduced plasma volume. For added safety, the thermal loading at the first wall was asserted disregarding the
prior heat absorption by the coolant within the halo.
Parameters concerning the recuperation of energy losses such as wall reflectivities and blanket absorption effi-
ciencies for the individual energy loss phenomena are prescribed at common values as given in ref. 8 so as to simplify
a direct system comparison. The power gain Q is set to Q → ∞, assuming a fully self-sustaining reaction.
The Spherical Tokamak system design evaluated here is based on the ARIES-ST reactor study.37 While maintain-
ing identical geometric and magnetic confinement field proportions, the desired design points were met by rescaling
both the volume of the plasma column and the toroidal vacuum magnetic flux density B . The corresponding energy
0
confinement time τ is calculated using the adapted ITER-89P scaling mode as given in ref. 35. A normalisation
E
towards the baseline values of the ARIES-ST study yields
(cid:2028) (cid:3404)(cid:3)(cid:884)(cid:484)(cid:883)(cid:886)(cid:883)(cid:890)(cid:886)(cid:3)(cid:149)(cid:3)(cid:942)(cid:4672) (cid:3010)(cid:3265) (cid:4673)(cid:2868)(cid:484)(cid:2876)(cid:2873)(cid:942)(cid:4672) (cid:3003)(cid:3116) (cid:4673)(cid:2868)(cid:484)(cid:2870)(cid:942)(cid:4672) (cid:1731)(cid:3041)(cid:3280)(cid:1732) (cid:4673)(cid:2868)(cid:484)(cid:2869)(cid:942)(cid:4672)(cid:3019)(cid:3116)(cid:4673)(cid:2869)(cid:484)(cid:2873)(cid:942)(cid:4672) (cid:3017)(cid:3269)(cid:3293) (cid:4673)(cid:2879)(cid:2868)(cid:484)(cid:2873), (13)
(cid:3006)(cid:481)(cid:3020)(cid:3021) (cid:2870)(cid:2876)(cid:484)(cid:2872)(cid:2877)(cid:3)(cid:2897)(cid:2885) (cid:2870)(cid:484)(cid:2868)(cid:2876)(cid:3)(cid:2904) (cid:2869)(cid:484)(cid:2873)(cid:2876)(cid:3400)(cid:2869)(cid:2868)(cid:3118)(cid:3116)(cid:2923)(cid:3127)(cid:3119) (cid:2871)(cid:484)(cid:2870) (cid:2875)(cid:2872)(cid:2872)(cid:484)(cid:2871)(cid:3)(cid:2897)(cid:2907)
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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015
with P the total of the power required to reheat the plasma. This scaling law applies only to ST reactors of identical
TR
geometric proportions as ARIES-ST. For simplicity, similar or identical operational parameters to those given for
ARIES-ST are prescribed where possible.
Operational boundaries such as the Greenwald density limit38 were not exceeded. An effective upper limit to the
reactor size was somewhat arbitrarily enforced by not allowing the propulsion system diameter to exceed a total value
of D = 12m, catering to a generous upper limit of contemporary launcher capabilities. The most prominent geometric
tot
and operational parameters for the proposed ST propulsion system are given in table 1.
The FRC configurations given in table 2 have been selected by taking into account that higher values for the
specific power can be achieved with minimal core radii. However, a minimum value of the coil radius below which it
is impossible to operate the FRC without exceeding the physical limitations of the first wall exists and was taken into
consideration. The length of the magnetic coils has been constrained to a maximum value of 20m due to space launch
system limitations. The size of the separatrix inside the coil has been optimised for maximum power generation with-
out exceeding the physical constraints.
The energy confinement time scaling law for the FRC propulsion system is determined in analogy to the ST by
normalising the scaling law provided by ref. [22] to the ARTEMIS-L fusion reactor study, obtaining
(cid:2028) (cid:3404)(cid:3)(cid:888)(cid:484)(cid:889)(cid:3)(cid:149)(cid:3)(cid:942)(cid:4672) (cid:3028) (cid:4673)(cid:2870)(cid:484)(cid:2875)(cid:942)(cid:4672) (cid:3003) (cid:4673)(cid:2869)(cid:484)(cid:2871)(cid:2873)(cid:942)(cid:4672)(cid:1731)(cid:3027)(cid:1732)(cid:4673)(cid:2869)(cid:484)(cid:2871)(cid:2873)(cid:942)(cid:4672) (cid:3021) (cid:4673)(cid:2868)(cid:484)(cid:2871)(cid:2870)(cid:2873), (14)
(cid:3006)(cid:481)(cid:3007)(cid:3019)(cid:3004) (cid:2869)(cid:484)(cid:2874)(cid:2876)(cid:3)(cid:2923) (cid:2873)(cid:484)(cid:2871)(cid:2874)(cid:3)(cid:2904) (cid:2869)(cid:484)(cid:2873) (cid:2876)(cid:2871)(cid:484)(cid:2873)(cid:3)(cid:2921)(cid:2915)(cid:2906)
where <Z> is the average proton charge number of the reactants. It is assumed that the volume-averaged magnetic
flux density within the separatrix <B> roughly scales proportionately to the externally applied field B.
While the criteria concerning the power output of both the ST and the FRC propulsion system may be satisfied
with a variety of configurations, some of which may prove more efficient than the designs discussed here, the given
configurations can be regarded as typical examples of the respective systems.
Fuel D-T D-³He Fuel D-T D-³He
A 1.60 L / m 10.8 20 20 9 20 20
Coil
κ 3.40 R / m 2 2 3 1.5 1.5 2
Coil
δ 0.64 a / m 0.72 0.72 0.6 0.6 0.6 0.46
R / m 0.6 1.6 2.2 1.6 2.2 2.7 b / m 3.6 6.67 6.67 3 6.67 6.67
0
V / m3 4.80 91.1 237 91.1 237 438 V / m3 9.32 17.36 12.06 5.40 12.1 7.09
Plasma Plasma
B / T 4.31 2.21 1.78 6.37 5.13 4.46 B / T 4.50 5.11 8.41 10.19
0
I / MA 11.2 15.3 16.9 44.0 48.7 52.0
P
f / % 84.5
BS
βt / % 50 β / % 91 97 89 96
qa 12.2
<T> keV 16 50 <T> keV 23.7 110.4
T0/<T> 2 T0/<T> 1
<ne>/1020m-3 7.13 1.88 1.22 4.98 3.23 2.44 <ne>/1020m-3 10.6 14.7 7.78 12.37
n0/<n> 1.26 n0/<n> 1.10 1.03 1.13 1.04
τE / s 0.30 1.14 1.76 1.49 2.30 3.04 τE / s 0.21 0.15 0.84 0.53
PF,tot / MW 456.0 600.3 656.2 904.2 989.5 1046 PF,tot / MW 2654 4762 6287 284 636 940
Pth,tot / MW 88.7 116.8 127.7 737.7 807.0 853.9 Pth,tot / MW 493 955 1209 239 536 893
PBrS,tot / MW 4.87 6.40 7.00 190.2 208.1 220.1 PBrS,tot / MW 21 39 52 36 80 118
PSyn,tot / MW 0.08 0.11 0.12 8.75 9.57 10.1 PSyn,tot / MW 0.02 0.03 0.01 0.19 0.43 0.20
PT / MW 430.6 566.7 619.5 835.6 914.1 967.1 PT / MW 2409 4476 5909 258 576 852
Pdn0 / MWm-2 16.1 2.99 1.73 0.28 0.16 0.11 Pdn0 / MWm-2 20 16 0.25
Pdth / MWm-2 3.93 0.73 0.42 4.59 2.66 1.87 Pdth / MWm-2 4.56 4 5

Table 1. Geometric and operational parameters for a Table 2. Geometric and operational parameters for
an ST fusion propulsion reactor. a FRC fusion propulsion reactor.

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Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan
July 4 – 10, 2015