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\begin{document}

\title{\bf 4. CONCEPTUAL DESIGN OF SUPERCONDUCTING MAGNET COILS}
\author{
\autlist{Thomas Kupiszewski}{}{Owen R.~Christianson}
}
\date{ }
\maketitle

\begin{titlepage}
{
\tableofcontents
}
\end{titlepage}

\setcounter{chapter}{3}
\setcounter{page}{1}

\chapter{CONCEPTUAL DESIGN OF SUPERCONDUCTING MAGNET COILS}

\section{INTRODUCTION}
\label{s-4-1}

Stellarators are steady-state magnetic confinement fusion devices.  Unlike
tokamaks, which utilize a combination of ac and dc magnetic fields,
stellarators use only dc magnets since poloidal-field modulation is not
required during plasma start-up or for position control.  From a magnet design
perspective, the advantages inherent to the stellarator are the suitability of
conventional voltage comparison techniques for quench detection and device
operation without superconductor ac losses.

In this work, a conceptual magnet internal design is developed to meet the
operating requirements representative of stellarator power plants.
These requirements, established by the Stellarator Power Plant Study (SPPS) for
the modular Helias-like heliac (MHH), are presented in
\hbox{Table~\ref{t-4-1-1}}. Key design drivers are the 14.5~T peak magnetic
field intensity and 34.7~MA/m$^2$ effective coil current density, along with a
maximum nuclear heat input to the coil of roughly 800~W/m$^3$.  This
combination of requirements provides an unprecedented cooling design challenge.
It implies magnet design feasibility is strongly coupled to mitigation of hot
spot temperature rise for this type of application.

Also noteworthy, as illustrated in \hbox{Fig.~\ref{f-4-1-1}}, is the highly
non-planar, non-symmetric geometry of the magnet coils as they are presently
envisioned for stellarator-based power plants.  The most significant geometric
characteristic of these coils is the presence of at least one, small radius,
out-of-plane bend.  Since the minimum bend radii are extremely small fractions
of the gross packaging dimensions, magnet internal design is impacted
significantly by stellarator topology.  This impact manifests itself in the
necessary design tradeoff between quench voltage and quench temperature rise.
As bend radius decreases, allowable conductor cross-section area and current
capacity decrease, causing an increase in total turn count to maintain a
constant ampere-turns values.  To achieve acceptable quench temperature rise,
small time constants for field collapse are desirable, so dump resistance
increases with coil inductance according to the square of the total number of
conductor turns.  The resulting increase in peak quench voltage must be
reconciled against the voltage standoff capability of the winding pack
insulation design.

\begin{table}[tb]
\caption{}
\centering{\bf SPPS Magnet Design Requirements} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l @{\hspace{2.cm}} c}
\hline \\
Number of magnets                      & 32   \\
Average coil radius                    & 4.0  \\
Radial coil height (m)                 & 0.78 \\
Lateral coil width (m)                 & 0.51 \\
Minimum coil curvature radius (m)      & 0.34 \\
Minimum distance between coils (m)     & 0.12 \\
Plasma average major radius (m)        & 14.0 \\
Total coil current (MA)                & 13.8 \\
Total inductance (one-turn) ($\mu$H)   & 816  \\
Stored magnetic energy (GJ)            & 78   \\
Max magnetic field at coil (T)         & 14.5 \\
Magnetic field at plasma axis (T)      & 5.0  \\
Peak coil nuclear heat load (W/cm$^3$) & 807  \\
\\ \hline
\end{tabular}
\label{t-4-1-1}
\end{table}

\begin{figure}[b]
\centerline{\epsfig{file=spps-4-1-1.ps,width=6.5in,bbllx=80pt,bblly=170pt,%
            bburx=514pt,bbury=322pt,clip=}}
\caption[f-4-1-1]
{Elevation view of an 8 magnet quadrant as viewed from the central axis.}
\label{f-4-1-1}
\end{figure}

The coil concept development philosophy underlying this study was to
extrapolate and adapt the superconductor designs for the International
Thermonuclear Experimental Reactor (ITER) and Toroidal Plasma experiment (TPX)
to the performance regime defined by the stellarator power plant magnet
requirements.  This philosophy assumed a design baseline consisting of an
Incoloy-908-jacketed Nb$_3$Sn cable-in-conduit conductor (CICC), with forced
convection heat removal using sub-cooled supercritical helium, and a
wind/react/vacuum-pressure-impregnation (VPI) coil manufacturing sequence.
From this baseline, a conceptual point design for the SPPS magnet winding pack
was postulated and a theoretically-feasible preliminary design implementation
was developed.

Section~\ref{s-4-2} summarizes the design effort applied to the SPPS magnet
internal arrangement and presents a description of the present embodiment of
the winding pack design concept as it has evolved to date.  Section~\ref{s-4-3}
presents the results of design analyses which support concept feasibility.  The
scope of these analyses was limited to coil grading, steady-state
thermal-hydraulic simulation, quench performance, and conductor operating
margins.

\section{DESIGN OF MAGNET INTERNALS}
\label{s-4-2}

\subsection{Description}
\label{s-4-2-1}

The proposed magnet architecture is based upon a graded winding configuration,
as depicted in Fig.~\ref{f-4-2-1}.  Grading is achieved in this context by
partitioning the winding window into two regions such that a different
ampere-turn coil resides within each region.  This partitioning consist of a
low ampere-turn inner coil (aka insert) nested within a high ampere-turn outer
coil (aka outsert).  They are energized by separate power supplies injecting
different amounts of electrical current.  A theoretically-feasible design
implementation of this graded winding concept is depicted by the dimensioned
layout illustration shown in Fig.~\ref{f-4-2-2}.  The following discussion
refers to this figure.

\begin{figure}[t]
\centerline{\epsfig{file=spps-4-2-1.eps,height=4.1in}}
%\vspace*{4.3in}
\caption[f-4-2-1]
{Comparison of uniform current density and graded coil concepts.}
\label{f-4-2-1}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-2-2.eps,width=6.5in}}
%\vspace*{7.7in}
\caption[f-4-2-2]
{Design implementation of graded coil concept.}
\label{f-4-2-2}
\end{figure}

The outsert portion of the winding pack is further subdivided into two coils,
designated as outsert coils A and B.  Both outer coils are layer-wound with
144 turns in 8 nested layers ({\it i.e.,} 18 turns per layer) for a total
current of 6.55~MA per coil, or 45.5~kA per turn.  Each outsert coil is
center-tapped and slaved to its own dedicated resistive dump circuit and
charging power supply.  Internal heating resulting from neutron attenuation is
dissipated convectively by ventilation of the conductors with supercritical
helium flowing at a rate of 3.5~g/s per layer at inlet temperature and
pressure conditions of 3~K and 1.25~MPa, respectively.

The insert portion of the winding pack is also layer-wound for a total turn
count of 240 turns.  With a 10 layer radial build, the number of turns per
layer is 24.  Like the outsert coils, the insert coil is also forced-flow
cooled with supercritical helium.  However, key differences exist.  First,
because of a smaller minimum bend radius, this coil uses a smaller conductor.
Second, the close proximity of the insert to the plasma implies higher mass
flow rates are needed to remove greater energy deposition by neutrons
penetrating beyond the radiation shields.  Results of thermal-hydraulic
simulations suggest peak helium mass flow rates as great as 8~g/s per
layer at inlet conditions of 1.25~MPa and 3~K are needed to achieve temperature
margins greater than 0.5~K.  A third difference is that the insert coil
includes a square auxiliary coolant conduit with a 13~mm diameter circular
channel.  This channel is located between the first and second layers of CICC
and rasters in the same direction as the first CICC layer.  It is flushed with
supercritical helium at a flow rate of 0.1~kg/s at inlet conditions of 3~K and
1.25~MPa.  It augments heat removal from conductor layers 1 and 3 by conduction
across turn insulation barriers to the auxiliary coolant stream.

Both the insert coil conductors and auxiliary coolant conduit are wound
``two in-hand'' so that temperature rise and pressure drops can be
maintained within tolerable limits.  Here, the ``two-in-hand'' winding
technique means that, within any given 24 turn coil layer, 2 sets of 12
continuous turns of conductor are connected electrically in series, but
hydraulically in parallel.  A design ramification is complex electrical
joint and hydraulic plumbing layouts within highly confined clearance
spaces between adjacent magnets.  A design solution is illustrated in
Fig.~\ref{f-4-2-3}.  It entails staggering the winding pack entry and
exit points of the conductors by one half of a turn.  Accordingly, two
pairs of header boxes would be mounted at the ``sides'' of the insert coil
and would contain helium inlet and outlet manifolds as well as layer-to-
layer electrical joints as needed.  This approach reduces the number of
components within a given insert coil header box by 50\%.  Since the
outer coil does not require ``two in-hand'' winding, it is equipped with
one pair of header boxes.  Because header boxes compete for in-vessel
space with other fusion core equipment, like beam ducts and instrumentation
feedthroughs, their actual placement is a system integration issue.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-2-3.eps,height=7.5in}}
%\vspace*{7.5in}
\caption[f-2-3]
{A concept for positioning of electrical and hydraulic connection header
boxes.}
\label{f-4-2-3}
\end{figure}

The two notional conductor designs are presented in Fig.~\ref{f-4-2-4}.
As envisioned, all coils would employ Nb$_3$Sn CICC.  Both CICC designs
use chrome-plated wires, to prevent sintering during the reaction
process, and Incoloy 908 conduits.  Selection of Incoloy 908 as the
conduit material increases manufacturing cost, but mitigates against
degradation of critical current density due to differential thermal
contraction strain during cool-down.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-2-4.eps,height=7.7in}}
%\vspace*{7.7in}
\caption[f-2-4]
{Notional conductor design configurations.}
\label{f-4-2-4}
\end{figure}

The insert coil CICC is a reduced scale, modified version of the ITER model
coil conductor.  It consists of a square 16 mm conduit drawn down around an
annular cable of superconducting wires tightly compacted around a 7~mm OD $\X$
5~mm ID coil spring running the entire length of conductor.  This central
channel provides a low impedance path for the helium mass flows required to
dissipate nuclear heat.  Without this central channel, supercritical helium
moving through the cable void space would eventually enter the subcritical
state and unreliable conductor performance would probably occur as a
consequence of two phase flow.  A triplex-based, 3 or 4 stage cabling process
could be used to spiral 6 to 8 subcables around the central channel.  With a
cable space void fraction of 0.3, wire diameter would be comparable to the
Westinghouse LCT wire~\cite{r1}, ranging from 0.62~mm to 0.7~mm.  However,
Nb$_3$Sn processing yields would have to be greater than those for LCT since a
Cu to non-Cu ratio of 1:1 would be needed to achieve margin.  The ratio of
minimum bend radius to conductor dimension is \hbox{$340/16 \simeq 21.3$}.

The outsert coils CICC is a larger version of the Westinghouse LCT
conductor.  It consists of a square 25.7~mm conduit, or jacket, drawn
down around a 5 stage cable of 486 superconducting wires.  With a cable
space void fraction of 0.35 and conduit wall thickness of 2~mm, wire
diameter would equal approximately 0.9~mm, roughly 32\% greater than the
LCT wire.  No central channel is needed since adequate mass flow rates
can be developed without decreasing pressure below the critical value
near 0.23 MPa.  The ratio of minimum bend radius to conductor dimension
is \hbox{$609/25.7 \simeq 23.7$}.

Both conductors would be wrapped with turn insulation possessing a nominal
compressed build of 1.25~mm ({\it i.e.,} four, 0.012'' half-lapped tapes).
If the coils were fabricated using present technology and techniques, turn and
ground wall insulation would probably default to fiberglass or ceramic fabric
tapes.  This approach presents a tractable level of risk for the insert coil
since predicted quench voltages are less than 1~kV.  To assure reliable
operation of the outsert coils, void-free vacuum pressure impregnation (VPI)
with epoxy and a center-tapped winding would be absolute necessities since the
predicted quench voltage drop to ground is 15~kV for a non center-tapped
configuration (30~kV if both outserts are consolidated into a single non
center-tapped coil).  If coils are inserted within support cases after reaction
of the conductor, then application of polyimide film sheets as a ground wall
barrier becomes an option.  Our key operating assumption has been that a
cost-effective, reliable insulate-wind-react-VPI manufacturing sequence is
available if coil fabrication were to begin at some future time, perhaps as
much as twenty years hence.

\begin{figure}[p]
\vspace*{0.5in}
\centerline{\epsfig{file=spps-4-2-5.eps,width=6.5in}}
%\vspace*{7.7in}
\caption[f-2-5]
{Magnet case concept without coils.}
\label{f-4-2-5}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-2-6.eps,height=7.6in}}
%\vspace*{7.7in}
\caption[f-2-6]
{Cross-section view of magnet case and coils.}
\label{f-4-2-6}
\end{figure}

A candidate case structure design concept is illustrated in Figs.~\ref{f-4-2-5}
and \ref{f-4-2-6}.  Figure~\ref{f-4-2-5} depicts the magnet case without coils.
Figure~\ref{f-4-2-6} depicts the coils inserted within the case.  The case is
constructed from four circumferential bands and three pairs of side wall plate
assemblies.  The circumferential bands also function as mandrels for coil
winding, so they are referred to as mandrel bands.  All case material is 316LN
stainless steel or, alternatively, Incoloy 908.  The insert coil mandrel band
is overwrapped with a 24 turn coil of conductor conduit (w/o cable) on its
outermost surface.  Supercritical helium flows in this conduit, intercepting
nuclear heat within the band before it conducts to the first conductor layer.
Usage of CICC conduit as a coolant tube negates the need for extensive
machining or welding operations performed on the mandrel band to interface
cooling hardware.  A similar arrangement exists between the insert coil and the
adjacent outsert coil.  Accompanying flange surfaces for all mandrel bands are
provided by side wall plates joined to the bands.  As presently envisioned,
this joining is accomplished by full plug welds located at through-running
holes in the side wall plates.  The proposed usage of plug welds is intended as
a potential means of eliminating or greatly reducing plate distortion due to
weld shrinkage.  Since the primary load transfer path passes through these plug
welds, their number and size must be chosen to provide sufficient section area
such that weld zone stresses do not reach yield values.  These plug welds are
supplemented with continuous fillet ``seam'' welds which traverse the case
circumferentially, parallel to the direction of current flow.  Although the
seam welds improve case rigidity and overall load-bearing strength, their
primary function is to seal the case for VPI during manufacture and to contain
conductor coolant in the event of conduit leakage.  Closing the case is a
7.5~cm thick outer belt which contains the outsert coil.  Again, plug and seam
welds would be used for joining.

In general, considerable care must be taken during welding to prevent thermal
damage.  For mitigating against thermal damage to the winding pack, each
mandrel to flange plate interface surface undergoes a right angle bend to
reduce exposure of the ground wall insulation material to weld flash.  In
addition, although not depicted in either Fig.~\ref{f-4-2-5} or
Fig.~\ref{f-4-2-6}, the use of integral, water-cooled, chill blocks is
envisioned to shield the winding pack from weld-related heat effects.  These
chill blocks would be located within shallow channels milled into the
interior-facing sides of all mandrel and side wall plates.

\subsection{Rationale}
\label{s-4-2-2}

\subsubsection{Cooling Design}
\label{s-4-2-2-1}

To produce magnetic fields in the $>$14~T regime, recent magnet development
efforts have relied upon coil operation at 1.8~K for achieving adequate
superconductor temperature margin.  At the National High Magnetic Field
Laboratory (NHMFL), for example, the design of the 14-T, NB$_3$Sn, CICC
outsert~\cite{r2} of a 45-T hybrid magnet system was based on static HeII
cooling.  And in Japan, a pool-boiled, fully-superconducting, (Nb,Ti)$_3$Sn
magnet system~\cite{r3} successfully achieved 20~T while operating in a bath of
saturated HeII.

Although both of the aforementioned cooling design approaches are
suitable for laboratory applications, they are not readily scalable to
power-plant operating environments.  Consider, for example, a hypothetical
CICC coil stabilized by static HeII cooling at 1.8~K with a peak nuclear
heat input of roughly 1000~W/m$^3$ per unit length of cable.  With a
maximum critical channel heat flow limit~\cite{r4} of 7.4 W/cm$^{5/3}$ for
HeII at atmospheric pressure, the spacing of 1.8~K coolant ports along
the conduit wall could not exceed 16 m without causing fluid temperature
to exceed the lambda transition value at the halfway point between the
ports.  Alternatively, if magnets are HeII bath-cooled, winding packs
must be provided with a multiplicity of internal coolant channels to
dissipate the nuclear heat load adequately and maintain near isothermal
conditions.  These channels increase structural compliance of the
winding packs, thereby increasing the difficulty of reliable load
transfer between adjacent magnets.  Bath-cooling of non-planar
stellarator coils also presents the nontrivial challenge of preventing
vapor lock along horizontal portions of the coolant channels.  To
prevent bubble formation, these channels must be sized also according to
the critical channel heat flow limit. Consequently, effective current
density would tend to decrease.

Based on the aforementioned considerations, forced-flow cooling with
supercritical helium was selected as the preferred option for the design
baseline.  Subcooling the helium to 3~K was motivated by results of conductor
operating margin analysis, as documented in Section 3.3. Since the flow path
lengths are very long (approximately 25~m per turn on average), helium inlet
pressure was set to 1.25~MPa to provide sufficient pressure head for sustaining
single phase flow at the flow rates required for removing large nuclear heat
loads.  Furthermore, these inlet conditions are approximately equivalent to
those of the Westinghouse LCT coil~\cite{r1}, so they are therefore not a far
extrapolation from known refrigerator design space.  Selection of the CICC
approach to forced-flow cooling also provides the benefits of a distributed
winding support and coolant vessel structure as well as minimum quench energies
greater than 150~mJ/cm$^3$ of strand material~\cite{r5}.

Three winding options were considered from the perspective of cooling design.
These options were single-pancake winding, double-pancake winding, and layer
winding. Layer winding was chosen for two reasons.  First, in comparison to
double-pancake CICC coils, layer-wound CICC coils do not suffer from the
significant increase~\cite{r6} in hot spot temperature, near the midpoint of
the flow path, beyond that which would occur if the turns were adiabatically
isolated from one another.  In layer wound geometries, coolant temperature
increases monotonically along the flow path, attaining the hot spot value at
the outlet.  Second, due to the aspect ratio of the nominally-rectangular
cross-section of the winding window, layer winding translates to shorter flow
paths than those associated with double-pancake or even single-pancake winding.
Hence, pumping power and quench pressure rise are reduced.

\subsubsection{Coil Grading and Conductor Size}
\label{s-4-2-2-2}

As illustrated in Fig.~\ref{f-4-2-7}, the benefit of a graded winding pack
versus an ungraded winding pack ({\it i.e.,} uniform $J$) is that current can
be shifted away from the high field region, where conductor operating margins
are small, to a low field region where the margins are comparatively large.
This approach increases margin within the high field region at the expense of
relatively small percentage decreases in margin within the low field region.
However, the redistribution of current must be determined with care so magnetic
field at the plasma axis is maintained at roughly 5~T.  Magnetic field
calculations were undertaken to determine that the change in plasma axis field
attributable to the chosen grading scheme would not exceed 5\% of the nominal
value.  Results of these analyses are presented in Sec.~\ref{s-4-3-1} and
suggest the plasma field impact of the chosen grading scheme is limited to
about 3\%.  The effect on field quality ({\it i.e.,} harmonics), however, was
not assessed.

\begin{figure}[t]
\centerline{\epsfig{file=spps-4-2-7.eps,height=5.in}}
%\vspace*{5.in}
\caption[f-2-7]
{Benefit of graded winding pack vs a uniform current density
configuration.}
\label{f-4-2-7}
\end{figure}

The most significant coil geometric characteristic is the presence of at least
one, small radius, out-of-plane bend.  Since the coils minimum bend radii are
extremely small fractions of the gross packaging dimensions, magnet internal
design is impacted significantly by stellarator topology.  This impact
manifests itself in the necessary design tradeoff between quench voltage and
quench temperature rise.  As bend radius decreases, allowable CICC
cross-section area and current capacity must decrease, causing an increase in
total turn count to maintain a constant ampere-turns value.  To achieve
acceptable quench temperature rise, small time constants for field collapse are
desirable, so dump resistance increases with coil inductance according to the
square of the total number of conductor turns.  The resulting increase in peak
quench voltage must be reconciled against the voltage standoff capability of
the winding pack insulation design.  In general, fewer turns ({\it i.e.,}
larger CICC dimensions) translates to lower quench voltage, hence less
dielectric stress on the insulation design.

Based on results of ITER model coil CICC bending tests~\cite{r7}, the bend
radius limit is estimated to be no less than 20 times the conduit external
dimension.  This criterion was applied to size both insert and outsert coil
CICC conduits.  The larger bend radius of the outsert coils allowed a larger
CICC, hence a reduction in self-inductance.  However, this reduction was not
sufficient for achieving quench voltages less than 20~kV to ground.  To limit
outsert coil quench voltage to less than 20~kV while simultaneously limiting
quench temperature to less than 300~K, the outsert coil was subdivided such
that each outsert ``subcoil'' would possess the same number of turns and would,
in the event of a detected quench, discharge into a separate dump resistor.
Results of quench performance analyses presented in Section 3.4 suggest this
coil partitioning scheme reduces quench temperature to roughly 150~K at a
quench voltage of 15~kV, or $\pm$7.5~kV if each outsert winding is
center-tapped.  Such quench performance was judged as providing a reasonable
balance between design conservatism, system complexity, and technical risk.

\subsection{Cooling Design Options for Insert Coil}
\label{s-4-2-3}

At present, the predicted minimum temperature margin of the insert coil is
0.7~K.  Three cooling design options have been identified for increasing this
minimum temperature margin.  These options are:

\begin{enumerate}

\item
Change present routing of helium through ``two-in-hand'' wound layers
from parallel flow to counterflow.

\item
Remove electrical insulation wrap surrounding each turn of the auxiliary
coolant conduit layer, thereby reducing the transverse heat conduction
impedance between the first three coil layers by nearly 50\%.

\item
Add a second layer of auxiliary coolant conduit.  This can be done by
replacing the existing auxiliary coolant conduit layer with a CICC layer
and replacing the resulting first and third CICC layers with auxiliary
coolant conduit.  (Here, the tradeoff centers around doubling the local
heat removal capacity to reduce hot spot temperature versus slightly
decreasing temperature margin as conductor current increases by 12.5\%).

\end{enumerate}

Analysis of these design optimization measures was beyond the scope of the
present feasibility study.  However, the availability of various approaches to
increasing design conservatism is another favorable indicator of overall magnet
design feasibility.

\section{DESIGN ANALYSIS}
\label{s-4-3}

\subsection{Coil Grading}
\label{s-4-3-1}

\begin{sidewaysfigure}[p]
\centerline{\epsfig{file=spps-4-3-01.eps,height=8.0in,angle=-90,clip=}}
\caption[f-3-1]
{Candidate schemes for coil grading.}
\label{f-4-3-1}
\end{sidewaysfigure}

Four alternative schemes for coil ({\it i.e.,} current) grading were
considered.  These alternative schemes are illustrated in
\hbox{Fig.~\ref{f-4-3-1}}.  Scheme 1 employs three coils in an attempt to
reduce manufacturing cost by utilizing NbTi superconductor in 30\% of the
current window.  The other schemes all employ two coils.  Schemes 2 and 3 are
geometrically identical, but operating temperature and superconductor differ.
Scheme 2 operates with superfluid helium, allowing the large outer coil to be
wound with the less expensive NbTi conductor.  Scheme 3 operates at subcooled
supercritical helium temperatures and would utilize all Nb$_3$Sn
superconductor.  Scheme 4 is essentially a geometric variation of Scheme 3.
Also shown on \hbox{Fig.~\ref{f-4-3-1}} are estimated values and locations for
magnetic field intensity at the coil boundaries.  These estimates were made by
visual inspection of magnetic field plots provided~\cite{r8} to Westinghouse.
It is important to note that it was assumed for expediency that the fields at
the coil do not change significantly by grading.  This simplifying assumption
is clearly not correct in a rigorous sense, but it uncouples the problem,
allowing the conductor temperature margin calculations to proceed for purposes
of down-selecting a preferred scheme.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-02.eps,height=7.5in}}
%\vspace*{7.5in}
\caption[f-3-2]
{Plan and elevation views of circular toroid model used to predict
relative changes in plasma magnetic field due to coil grading.}
\label{f-4-3-2}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-03.eps,height=7.5in}}
%\vspace*{7.5in}
\caption[f-3-3]
{Plan and elevation views of 4-lobed toroid model used to predict
relative changes in plasma magnetic field due to coil grading.}
\label{f-4-3-3}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-04.ps,width=6.5in,bbllx=80pt,bblly=324pt,%
            bburx=514pt,bbury=732pt,clip=}}
\vspace*{1.in}
\caption[f-4-3-04]
{Plan view of SPPS magnet array.}
\label{f-4-3-4}
\end{figure}

Magnetic Field Code (MAFCO) was used to develop two simple Biot-Savart
electromagnetic models for evaluating these schemes (\hbox{Figs.~\ref{f-4-3-2}}
and \ref{f-4-3-3}).  The first model, presented in Fig.~\ref{f-4-3-2}, is a
circular toroid consisting of a total of 32 solenoid coils whose centers are
coplanar and located at 14~m from the global origin.  These coils have the same
4~m average radius as the stellarator coils.  The coil centers are spaced at
equal angular increments around a 28~m diameter circular path.  The coils are
oriented so that their mid-planes all intersect at a vertical line running
through the global origin.  The second model, presented in Fig.~\ref{f-4-3-3}
and called the 4-lobed toroid, is derived from the first model by co-locating
the solenoid coil centers in Cartesian ($X,Y,Z$) space with the corresponding
centers for the stellarator coils.  This refinement was done to simulate the
quadrant layout of the stellarator coil array, which is illustrated for
comparison purposes in the plan view in \hbox{Fig.~\ref{f-4-3-4}}.  Clearly,
both models are gross, first order approximations to the stellarator magnet
array geometry.  The intent, however, was not to calculate magnetic field with
a high degree of accuracy, but to estimate relative changes in magnetic field
due to coil grading.

Graphs of the predicted relative field change versus global radius due to coil
grading are presented for Coil 1 (a coil at the corner of the array) and Coil 4
(a mid-side coil between corners) in Figures \ref{f-4-3-5} and \ref{f-4-3-6},
respectively.  These results were generated by the 4-lobed model and have been
normalized to a parameter designated as $B^*$, which is the field for the
uniform current density baseline case.  Our results suggest that all four
grading schemes should have negligible impact ($<3\%$) on the field magnitude
at the plasma axis.  This is not surprising, however, since the winding window
dimensions are less than one fourth of the mean coil radius.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-05.eps,height=3.1in}}
%\vspace*{3.3in}
\caption[f-3-5]
{Relative changes in coil 1 magnetic field distribution at $Z=0$ and
$\Theta=4.463^\circ$ due to alternative coil grading schemes ($B^*$ is the
field with a uniform current density).}
\label{f-4-3-5}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-06.eps,height=3.1in}}
%\vspace*{3.3in}
\caption[f-3-6]
{Relative changes in coil 4 magnetic field distribution at $Z=0$ and
$\Theta=37.541^\circ$ due to alternative coil grading schemes ($B^*$ is the
field with a uniform current density).}
\label{f-4-3-6}
\end{figure}

The circular toroid model was used also to predict the product of overall
current density times circumferential field in the global $Z=0$ plane.  This
product is the local Lorentz force density and, on a relative basis, is an
indicator of which coil grading scheme should incur the least mechanical
stress.  The results for grading schemes 2 (same as 4) and 3 are plotted versus
global radius at the inboard leg in \hbox{Fig.~\ref{f-4-3-7}} and are compared
against those for uniform current density.  The relative differences in peak
force density magnitude suggest that grading scheme 4 would be the most benign
from a structural engineering perspective.  However, scheme 3 was chosen
because it is characterized by the greatest combined conductor temperature
margin for both inner and outer coils.  This margin was calculated according to
the methodology discussed in \hbox{Sec.~\ref{s-4-3-3}}.  The results for
isothermal coil conditions, corresponding to no internal nuclear heating, are
presented in Table~\ref{t-4-3-1}.  A Nb$_3$Sn wire strain of 0.35\% was
assumed.

\clearpage

\begin{table}[t]
\caption{}
\centering{\bf Preliminary Temperature Margin Predictions \\
for Coil Grading Schemes (No Nuclear Heating)} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c}
\hline
Temperature  & Coil A   & Coil B    & Coil C    \\
Margin (K)   & (insert) & (outsert) & (outsert) \\
\hline \\
Scheme 1     & 7.6      & 4.2       & 2.25      \\
Scheme 2     & 9.2      & 2.6                   \\
Scheme 3     & 7.6      & 6.4                   \\
Scheme 4     & 6.0      & 3.9                   \\
\\ \hline
\end{tabular}
\label{t-4-3-1}
\end{table}

\begin{figure}[b]
\vspace*{3.5in}
\caption[f-3-7]
{Product of current density times theta component of magnetic field
versus radius as predicted by circular toroid model.}
\label{f-4-3-7}
\end{figure}

\clearpage

\subsection{Thermal Hydraulic Performance}
\label{s-4-3-2}

The steady-state thermal hydraulic performance of the stellarator coils
determines hot spot temperature rise due to nuclear heating.  This hot
spot temperature value has the effect of translating the superconducting
critical surface towards the design operating point along the magnet
load line since peak magnetic fields and peak nuclear heating rates are
co-located within the same region of the stellarator coil.
Consequently, coil design feasibility is strongly dependent upon the
extent of temperature rise.

To model the fluid mechanics of supercritical helium, the Navier-Stokes
equations including the effects of finite compressibility are used.  Helium
pressure and temperature profiles along the CICC flow path are obtained from
integrating the following nonlinear differential equations, derived from an
enthalpy balance~\cite{r9} on a control volume:
%
\begin{eqnarray}
\label{e-4-3-1}
 \frac{dP}{dx} & = & \frac{-2fG^2/(\rho{D}) + 4qG\beta/[\rho D (C_p -
    u^2\beta)]}{1 - (G^2/\rho)(\kappa + B\phi)} \: , \\
\label{e-4-3-2}
 \frac{dT}{dx}  & = & \frac{-2fG^2\phi/(\rho{D}) + 4q/[GD(C_p -
   \mu^2\beta)]}{1 - (G^2/\rho)(\kappa + \beta\phi)} \: ,
\end{eqnarray}
%
where the parameter $\phi$ is defined as
%
\begin{eqnarray}
\label{e-4-3-3}
\phi = \frac{\mu_jC_p + u^2\kappa}{C_p - u^2\beta} \: .
\end{eqnarray}
%
Here, $P$ is pressure, $T$ is temperature, $x$ is position along the flow path
mass flow rate, $\rho$ is density, $D$ is hydraulic diameter, $q$ is the heat
input flux, $C_p$ is constant pressure specific heat, $u$ is the bulk flow
velocity, $\beta$ is bulk expansivity, $\kappa$ is bulk compressibility, and
$\mu_j$ is Joule-Thomson coefficient.  Helium properties were provided by
Cryodata HEPAK software.

For the CICC cable space, relevant friction factors are given by Hooper's
correlation~\cite{r10} to Lue's data~\cite{r11}.  The correlation for the
friction factor in the central channel~\cite{r12} of the insert coil CICC is
given by:
%
\begin{eqnarray}
\label{e-4-3-4}
 f & = &  0.092 \, Re_D^{0.2} \: ,
\end{eqnarray}
%
where $Re_D$ is the Reynolds number of the central channel.  This correlation
yields friction factors roughly twice that of smooth tubes.  Consequently,
almost all of the helium flow in the insert coil CICC occurs in the central
channel.  For these scoping studies, it was assumed all of the flow can be
allocated to the central channel.

The heat input flux, $q$, along the flow path was determined from the
volumetric nuclear heat generation profile plotted in Fig.~\ref{f-4-3-8} from
data supplied~\cite{r13} to Westinghouse.  This was done by multiplying the
volumetric heat generation rate for a given coil layer by the ratio of CICC
cross-section area to wetted perimeter.  The effect of turn-to-turn heat
conduction between adjacent coil layers was included via an equivalent thermal
circuit approach based on the conduit insulation turn geometry and an assumed
thermal conductivity comparable to that of G10/11 epoxy-glass composite.

\begin{figure}[b]
\centerline{\epsfig{file=spps-4-3-08.eps,height=3.7in}}
%\vspace*{3.7in}
\caption[f-3-8]
{Distribution of volumetric nuclear heating within magnet.}
\label{f-4-3-8}
\end{figure}

Results for the outsert coil A layer closest to the plasma are presented in
\hbox{Fig.~\ref{f-4-3-9}}.  The predicted hot spot temperature is 5.2~K and is
located at the helium outlet. The pressure drop is roughly 4 atmospheres at a
flow rate of 3.5~g/s.  Hot spot temperature could be reduced by reducing inlet
pressure to increase helium specific heat capacity, by increasing mass flow
rate, by subcooling to 2.5~K, or a combination of all three.
Figure~\ref{f-4-3-10} presents the thermal-hydraulic response of the secondary
auxiliary case coolant flow stream.  Case coolant exits at nearly 4.7~K, which
is less than the aforementioned 5.2~K hot spot temperature, so it clearly is
also dissipating some of the heat deposited in the outsert coil as well as
intercepting case heat load.  Despite the large flow rate of 100~g/s, the
pressure drop is roughly 2.3~atm.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-09.eps,height=3.3in}}
%\vspace*{3.5in}
\caption[f-3-9]
{Predicted worst case thermal-hydraulic performance of outsert coil
(heat input is 40~W/m$^3$ and  flow rate is 3.5~g/s).}
\label{f-4-3-9}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-10.eps,height=3.3in}}
%\vspace*{3.5in}
\caption[f-3-10]
{Predicted helium temperature and pressure profiles in secondary case
cooling conduit (heat input is 32~W/m$^3$ and flow rate is 100~g/s)..}
\label{f-4-3-10}
\end{figure}

Results for the insert coil layer closest to the plasma are presented in
\hbox{Figs.~\ref{f-4-3-11}} and \hbox{\ref{f-4-3-12}}.
\hbox{Figure~\ref{f-4-3-11}} indicates the predicted hot spot temperature is
5.7~K.  The hot spot occurs upstream of the outlet due to expansion cooling of
the helium (Joule-Thomson effect).  A similar effect is predicted for the third
layer (second CICC layer).  However, the auxiliary coolant stream (layer 2)
exhibits monotonically increasing temperature. \hbox{Figure~\ref{f-4-3-12}}
presents the pressure profiles for these three layers.  The predicted CICC
outlet pressure in the 0.3 MPa to 0.4 MPa range suggests inlet pressures cannot
be greatly reduced while safely maintaining single phase flow everywhere along
the flow path.  The great disparity between CICC flow rates and auxiliary
coolant conduit (layer 2) flow rates is due to greater friction factors and
smaller hydraulic diameters of the CICC.  These results were based on the
assumption the primary auxiliary case coolant flow stream intercepts all of the
roughly 1~kW of nuclear heat deposited in the insert coil mandrel band.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-11.eps,height=3.3in}}
%\vspace*{3.5in}
\caption[f-3-11]
{Predicted worst case conductor temperature profiles in the insert coil
(heat input is 800~W/m$^3$).}
\label{f-4-3-11}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-12.eps,height=3.3in}}
%\vspace*{3.5in}
\caption[f-3-12]
{Predicted helium pressure profiles in the insert coil (heat input is
800~W/m$^3$).}
\label{f-4-3-12}
\end{figure}

\subsection{Conductor Stability}
\label{s-4-3-3}

The stability of the stellarator conductor is described by how close the
operating point is to the critical surface in terms of distance in ($I,B,T$)
space.  Field margin, temperature margin, quench point, minimum propagating
zone, and minimum quench energy are useful figures of merit in evaluating
design feasibility.

The superconducting critical surface is determined by using temperature
and magnetic field dependent critical current correlations.  Morgan's
correlation~\cite{r14} and a Westinghouse correlation~\cite{r15,r16}
similar to Green's formulation~\cite{r17} is used for NbTi, and Summer's
formulation~\cite{r18} is used for Nb$_3$Sn.

The quench point is the intersection of the load line with the critical
surface.  The distance along the load line, $LL$, is given by:
%
\begin{eqnarray}
\label{e-4-3-5}
 LL & = & \frac{I_{op}}{I_q}  \: \: =  \: \: \frac{B_{op}}{B_q} \: .
\end{eqnarray}
%
The temperature margin, $TM$, is given by:
%
\begin{eqnarray}
\label{e-4-3-6}
 TM & = & T_q - T_{cond} \: .
\end{eqnarray}
%
The field margin, $FM$, is given by:
%
\begin{eqnarray}
\label{e-4-3-7}
 FM & = & \frac{I_q - I_{op}}{I_{op}}  \:\: = \:\: \frac{B_q - B_{op}}{B_{op}}
     \: ,
\end{eqnarray}
%
where $I_q$ is the quench current, $I_{op}$ is the operating current, $B_{op}$
is the magnetic field at the operating point.  It is assumed that the load line
is linear ({\it i.e.}, no materials are present which exhibit saturation
effects).  The temperature margin is found by numerically finding the
temperature at which the critical surface intersects a particular point on the
load line.

Stability is also described by the minimum propagating zone (MPZ) and the
minimum quench energy (MQE)~\cite{r19}.  A normal zone larger than the MPZ will
grow in length.  A normal zone smaller than the MPZ will recover.  In a one
dimension adiabatic approximation, the MPZ is estimated by $l$,
%
\begin{eqnarray}
\label{e-4-3-8}
 l = \left[ \frac{2k (\Theta_c - \Theta_o) } { J_c^2 \rho} \right]^{1/2} \: ,
\end{eqnarray} % where k$ is the thermal conductivity, $\Theta_c$ is the
critical temperature, $\Theta{_o}$ is the bath or operating temperature, $J_c$
is the superconducting critical current density, and $\rho$ is the resistivity
of the strand.  The MQE is the energy necessary to create a normal zone of
length equal to the MPZ, {\it i.e.,}
%
\begin{eqnarray}
\label{e-4-3-9}
 {\rm MQE} & = & \int_{\Theta_o}^{\Theta_c}l \, A \, C(T)dT \: ,
\end{eqnarray}
%
where $A$ is the conductor area and $C$ is the conductor specific heat. While a
one dimensional approximation for MQE is used here for simplicity, a more
detailed design effort should consider two-dimensional geometry that is more
appropriate to the stellarator geometry.  The one dimensional approximation is
used for simplicity.  The one dimensional approximation is invalid when the MPZ
is smaller than the strand diameter.  In this case the strand diameter is used
for the MPZ instead of the calculated value.

The effects of cooling and enthalpy of helium are included in the MQE.  The
dual stability region separating the well cooled regime from the ill-cooled
regime is found from
%
\begin{eqnarray}
\label{e-4-3-10}
 i_{lim} & = & \left[ \frac{A_{cu} \, p \, h \, (T_c - T_b)}
                           {\eta{I_c}^2} \right]^{1/2} \: ,
\end{eqnarray}
%
where $A_{cu}$ is the copper area, $p$ is the wetted perimeter, $h$ is the heat
transfer coefficient, $T_c$ is the critical temperature, and $\eta$ is the
copper resistivity~\cite{r20}.  The physical constants used in the MPZ and MQE
are Kohler plot for the resistivity of copper~\cite{r21}, Lorentz ratio is used
to determine thermal conductivity, and the computer code Cryodata~\cite{r22} is
used for the specific heat of the elements comprising the conductor and helium.

The stability is actually described by comparing the MQE to the disturbance
spectrum of the magnet.  Prediction of the disturbance spectrum is beyond the
scope of this work.  Typical energy deposition due to ac losses from plasma
disruptions in Tokamak machines will range from 100 to 150~mJ/cm$^3$ of strand
materials~\cite{r23}. A MQE of about 100~mJ/cm$^3$ is assumed to be a minimum
value for stability in these large conductors.

\subsubsection{Insert Coil Operating Point and Margin}
\label{s-4-3-3-1}

The operating point for the insert coil without nuclear heating is 84\% along
the load line, shown in Fig.~\ref{f-4-3-13}.  The temperature margin is 3.4~K
at the operating point.  Including nuclear heating results in a temperature
rise of 2.7~K with the conductor operating at 5.7~K.  The margin is reduced to
0.7~K, and the conductor operates closer to the critical surface, 95\% along
the load line, see Fig.~\ref{f-4-3-14}.  (In both figures, the unit for the
vertical axis is ampere-turns.)

\begin{figure}[b]
\centerline{\epsfig{file=spps-4-3-13.eps,height=3.2in}}
%\vspace*{3.3in}
\caption[f-3-13] {The critical surface, load line, operating point, quench
point, and temperature margin for the proposed insert coil conductor without
the nuclear heating temperature rise. (For $LL=84\%$, $I_{op}/I_c=43\%$,
$TM=3.4$~K, and $k_f=0.09$.)}
\label{f-4-3-13}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-14.eps,height=3.2in}}
%\vspace*{3.4in}
\caption[f-3-14]
{The critical surface, load line, operating point, quench point, and
temperature margin for the proposed insert coil conductor with the
nuclear heating temperature rise included. (For $LL=96\%$, $I_{op}/I_c=78\%$,
$TM=0.7$~K, and $k_f=0.09$.)}
\label{f-4-3-14}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-15.eps,height=3.3in}}
%\vspace*{3.4in}
\caption[f-3-15]
{The minimum quench energy of the insert coil derived from the minimum
propagating zone.}
\label{f-4-3-15}
\end{figure}

The MQE is calculated using MPZ theory.  Inclusion of the cooling and enthalpy
of the helium in the conduit results in a MQE of about 2,000~mJ/cm$^3$, as
shown in Fig.~\ref{f-4-3-15}.  The well-cooled regime extends past the
operating point.  Experimentally, the LCP conductor had a MQE of about 1.5 to
1.8~J/cm$^3$~\cite{r24}.  This MQE exceeds the 100~mJ/cm$^3$ design goal.

\clearpage

\subsubsection{Outsert Coil Operating Point and Margin}
\label{s-4-3-3-2}

The operating point for the outer coil without nuclear heating is 80\% along
the load line, shown in Fig~\ref{f-4-3-16}.  The temperature margin is 4.3~K at
the operating point.  Including nuclear heating results in a temperature rise
of 2.2~K with the conductor operating at 5.2~K.  The margin is reduced to
2.1~K, and the conductor operates closer to the critical surface, 88\% along
the load line as is shown in Fig.~\ref{f-4-3-17}.

The MQE is calculated using MPZ theory.  Inclusion of the cooling and enthalpy
of the helium in the conduit results in a MQE of about 3,000~mJ/cm$^3$, as
shown in Fig.~\ref{f-4-3-18}.  This MQE exceeds the 100~mJ/cm$^3$ design goal.
The well-cooled regime extends well past the operating point.

\begin{figure}[b]
\centerline{\epsfig{file=spps-4-3-16.eps,width=6.3in}}
%\vspace*{3.3in}
\caption[f-3-16]
{The critical surface, load line, operating point, quench point, and
temperature margin for the proposed outsert conductor without the
nuclear heating temperature rise. (For $LL=80\%$, $I_{op}/I_c=53\%$,
$TM=4.3$~K, and $k_f=0.12$.)}
\label{f-4-3-16}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-17.eps,height=3.2in}}
%\vspace*{3.4in}
\caption[f-3-17]
{ The critical surface, load line, operating point, quench point, and
temperature margin for the proposed outsert coil conductor with the
nuclear heating temperature rise. (For $LL=88\%$, $I_{op}/I_c=67\%$,
$TM=2.1$~K, and $k_f=0.12$.)}
\label{f-4-3-17}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-18.eps,height=3.2in}}
%\vspace*{3.4in}
\caption[f-3-18]
{The minimum quench energy of the outsert coil derived from the minimum
propagating zone.}
\label{f-4-3-18}
\end{figure}

\clearpage

\subsubsection{Discussion of Radiation Effects and Wire Development}
\label{s-4-3-3-3}

The Nb$_3$Sn superconductor critical current is sensitive to strain in the
conductor.  Strains over 0.35\% result in unrecoverable degradation in the
critical current.  In addition, the Nb$_3$Sn filaments will break if the stress
in the conductor exceeds about 250 to 300~MPa.  Using typical stress-strain
values for the materials used in the conductor and coil pack, the stress for a
particular superconductor fill factor can be found at a given strain.  Results
are shown in Fig.~\ref{f-4-3-19}.

Preliminary estimates of the stress in the conductor indicate values
around 250~MPa.  The coil design is in the correct stress-strain and
fill factor region for a feasible design.

The critical currents used in the margin and stability sections are typical
values in use today, resulting in a magnet design that will work with today's
superconductors.  Improvements in the critical current are possible, see for
example an increase by a factor of two described by Pourrahimi~\cite{r23} or
Foner~\cite{r24} for laboratory powder metallurgy processing.  It is also
possible a wire development program today with the wire manufacturers might
allow optimization of the superconductor at the magnetic fields required for a
stellarator.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-19.eps,height=3.5in}}
%\vspace*{3.7in}
\caption[f-3-19]
{Acceptable fill factors for different amounts of stress at a strain of
0.35\%.}
\label{f-4-3-19}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-20.eps,height=3.5in}}
%\vspace*{3.7in}
\caption[f-3-20]
{Change in critical current in Nb$_3$Sn as a function of neutron
fluence.}
\label{f-4-3-20}
\end{figure}

The radiation environment of the stellarator will be intense with a neutron
fluence of 10$^{23}$~n/m$^2$.  Radiation creates defects in the superconducting
lattice.  For weakly pinned superconductors typical of bulk superconducting
filaments, the radiation defects can actually add pinning sites and increase
the critical current initially ({\it e.g.,} see Ref.~\cite{r25}).  Summers
estimated the effect of radiation upon the critical current.  Calculations for
Nb$_3$Sn superconductor are shown in Fig~\ref{f-4-3-20}.  Initially the
critical current increases, but then the critical current decreases.  This
figure suggests that the radiation environment will not be a problem in the
stellarator.  However, the change in critical current will be dependent upon
the initial pinning state of the superconductor.

\subsection{Quench Performance}
\label{s-4-3-4}

The quench performance of the cable-in-conduit conductors used in the inner and
outer coils is modeled by two different techniques.  One technique calculates
the hot spot temperature in an adiabatic, one dimensional approximation.  This
simple calculation is typically called a MIITS calculation.  The second
technique solves the one dimensional conservation relations for the helium mass
and momentum together with energy conservation relations for the helium, the
conductor, and conduit wall.

\subsubsection{MIITS Calculation}
\label{s-4-3-4-1}

The quench temperature rise is predicted using an adiabatic, one
dimensional approximation of a conductor carrying a current.  A simple
heat balance is
%
\begin{eqnarray}
\label{e-4-3-11}
 \frac{I^2(t) \, \rho(T)} {{A_{cu}}} \, dt  & =  & A \, C(T)dT \: .
\end{eqnarray}
%
Rearranging and integrating gives~\cite{r26}
%
\begin{eqnarray}
\label{e-4-3-12}
 \int_o^t I^2dt  & = & {A_{cu}} \, A \, \int_{T_o}^{T} \frac{C(T)}{\rho(T)}
  \, dT \: .
\end{eqnarray}
%
The units are MIITS (10$^6$~A$^2-$s).  The left hand integral is a function of
temperature.  To simplify the problem to save time, it is assumed that the
current decay is entirely controlled by the dump resistor.  The current decays
with a time constant of $L/R$ where $L$ is the inductance of a single magnet
and $R$ is the resistance of a single dump resistor which is slaved to a single
magnet.  The inductance is derived from the total stored energy and scaled by
1/32 to estimate the self inductance of a single typical magnet.  Mutual
inductance between adjacent coils is ignored.  The constant $L/R$ current decay
approximation ignores propagating normal zones.

Evaluation of the right hand integral requires values for the physical
parameters.  The specific heat copper, iron, and the elements comprising
Nb$_3$Sn are obtained from the computer program Cryodata~\cite{r22} (Values
beyond 300~K are extrapolated).  The resistance of copper as a function of
field is derived from a Kohler plot with Fickett's empirically derived
coefficients~\cite{r21}, and the temperature dependence follows the
equation~\cite{r26}
%
\begin{eqnarray}
\label{e-4-3-13}
 \rho(T,RRR)  & = & 1.545\X 10^{-8} RRR^{-1} + \nonumber \\
   & & \left[ 2.32547 \X 10^{17} T^{-5} + 9.57137\X 10^{13} T^{-3} +
         1.62735\X 10^{10} T \right]^{-1} . \hspace{1.5cm}
\end{eqnarray}
%
These quantities are plotted in Fig.~\ref{f-4-3-21}.

\paragraph{Insert Coil.}

The estimated stored energy of the insert coil is 13~MJ which, at a current of
4.63~KA, gives an inductance of 1.213 H.  Setting the quench voltage as 500~V
yields a dump resistor of 0.108~$\Omega$ and a decay constant of roughly 11~s.
It is assumed that the current decays exponentially, controlled entirely by the
dump resistor.  Evaluating the MIITS integrals gives a value of 117~MIITS, and
a hot spot temperature of 73~K, as shown in Fig.~\ref{f-4-3-22}.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-21.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-3-21]
{The physical properties used in the MIITS calculation.}
\label{f-4-3-21}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-22.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-3-22]
{The maximum temperature of the hot spot following a quench for the
proposed insert coil conductor and dump resistor.}
\label{f-4-3-22}
\end{figure}

Since it is assumed the current dump is controlled by the dump resister, it is
possible to obtain the hot spot temperature as a function of time.  The current
in the magnet decreases exponentially.  The MIITS integral can be evaluated at
different times.  Assuming the energy is deposited into the conductor, the
conductor hot spot temperature versus time is found.  Next it is assumed that
energy is transferred from the conductor to the conduit wall through a heat
transfer coefficient.  Thus, the conduit wall temperature as a function of time
is also found.  Results are shown in Fig.~\ref{f-4-3-23}.

Varying the quench voltage and dump resistance results in different
decay times and different maximum temperature rises.  The conductor
quench voltage is varied to display the variation in quench temperature,
as is shown in Fig.~\ref{f-4-3-24}.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-23.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-3-23]
{Projected transient response of the insert coil during a quench.}
\label{f-4-3-23}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-24.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-3-24]
{The insert coil quench hot spot temperature as a function of quench
voltage.}
\label{f-4-3-24}
\end{figure}

\paragraph{Outsert Coil.}

The outsert coil is subdivided into two equal coils to keep the quench
temperature rise and the quench voltage at reasonable values less than 200~K
and 20~kV, respectively.  The estimated stored energy of the outsert coil is
2.4~GJ which, at a current of 45~kA, gives an inductance of 2.37~H or roughly
1.2~H after subdivision.  Setting the quench voltage as 15~kV yields a dump
resistor of 0.33~$\Omega$ and a decay constant of 3.6~s.  Again, it is assumed
that the current decays exponentially, controlled entirely by the dump
resistor.  Evaluating the MIITS integrals gives a value of 3681~MIITS, and a
hot spot temperature of 140~K, as shown in Fig.~\ref{f-4-3-25}.

Since it is assumed that the current dump is controlled by the dump resister,
it is possible to obtain the hot spot temperature as a function of time.  The
current in the magnet decreases exponentially.  The MIITS integral can be
evaluated at different times.  Assuming the energy is deposited into the
conductor, the conductor hot spot temperature versus time is found.  Next it is
assumed that energy is transferred from the conductor to the conduit wall
through a heat transfer coefficient.  Thus the conduit wall temperature as a
function of time is also found.  Results are shown in Fig.~\ref{f-4-3-26}.

Varying the quench voltage and dump resistance results in different decay times
and different maximum temperature rises.  The conductor quench voltage is
varied to display the variation in quench temperature, as shown in
Fig.~\ref{f-4-3-27}.

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-25.eps,height=3.4in}}
%\vspace*{3.6in)
\caption[f-3-25]
{The maximum temperature of outsert coil hot spot following a quench for
the proposed outer coil conductor and dump resistor.}
\label{f-4-3-25}
\bigskip
\bigskip
\centerline{\epsfig{file=spps-4-3-26.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-3-26]
{Projected transient response of the outsert coil during a quench.}
\label{f-4-3-26}
\end{figure}

\begin{figure}[t]
\centerline{\epsfig{file=spps-4-3-27.eps,width=6.5in}}
%\vspace*{3.5in}
\caption[f-3-27]
{The outsert coil quench hot spot temperature as a function of quench
voltage.}
\label{f-4-3-27}
\end{figure}

\subsubsection{Result Verification with QUENCHER}
\label{s-4-3-4-2}

The QUENCHER code was developed by A.~Shajii at MIT~\cite{r27,r28}.  The code
solves the one dimensional conservation relations for the helium mass and
momentum together with energy conservation relations for the helium, the
conductor, and conduit wall.  Helium inertia is neglected, and it is assumed
that the supercritical helium coolant and conductor strands are at the same
temperature.

The basic equations describing the quench process are:
%
\begin{eqnarray}
\label{e-4-3-16}
  \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho \, v)
     & = & 0   \: , \\
\label{e-4-3-17}
  \frac{\partial p}{\partial x}  & =  & \frac{f \rho v|v|}{2d} \: , \\
\label{e-4-3-18}
  \rho C_t \, \frac{\partial T}{\partial t} + \rho C_h \gamma \, 
    \frac{\partial T}{\partial x} +
    \rho C_\beta T \, \frac{\partial v}{\partial x}
    & = & \frac{\partial}{\partial x} 
    \left( \kappa_c \, \frac{\partial T}{\partial x} \right) + 
    \frac{A_h}{A_c} \, \frac{f\rho \, |v|v^2}{2d} \nonumber \\
  & & + \eta_c \, J^2 \, H (T - T_{cs}) + \frac{hP_w}{A_c} (T_w - T) \: ,
\hspace{1cm} \\
\label{e-4-3-19}
  \rho_w C_w \, \frac{\partial{T_w}}{\partial{t}} & = & 
         \frac{hP_w}{A_w}(T - T_w) \: .
\end{eqnarray}
%
The quantities $A_h$, $A_c$, $A_w$ represent the cross sectional areas of the
helium, conductor, and conduit wall respectively, and $C_c$ and $C_w$ are the
heat capacities of the conductor and wall.  Also,
%
\begin{eqnarray}
\label{e-4-3-20}
  C_h & = & \left( \frac{A_h}{A_c} \right) \left( 
    \frac{T \, \partial S}{\partial T}  \right)_\rho \: , \\
\label{e-4-3-21}
 C_\beta & = & - \left( \frac{A_h}{A_c} \right)
    \left( \frac{\rho \, \partial S}{\partial \rho} \right)_T \: , \\
\label{e-4-3-22}
 C_t & = & C_h + \left( \frac{\rho_c}{\rho} \right) \, C_c \: .
\end{eqnarray}

Results for outsert coil are shown in Figs.~\ref{f-4-3-28} to \ref{f-4-3-31}
and for insert coil are shown in Figs.~\ref{f-4-3-32} and \ref{f-4-3-33}.  In
general, the inner coil has a quench temperature of 70~K, quench
pressure of 1.9~MPa, a normal zone length of 45~m at 30~s, and an inlet mass
flow of 5~g/s with an outlet mass flow of 12~g/s at 30~s.  It is interesting
that the normal zone does not propagate upstream, and actually begins to
translate downstream at large times.  The outer coil has a quench temperature
of about 170~K (slightly higher than the MIITS calculations), a quench pressure
of 4~MPa, a normal zone length of 35~m at 15~s, an a mass flow efflux of 15~g/s
out of the cable.  The quench propagation is symmetrical around the quench
initiation site in the outer coil.  Also, in the outer coil the temperature
difference between the conductor and conduit wall is about 100~K at about 5~s
into the quench.  The conductor temperature rises above 200~K at its maximum.

\begin{figure}[b]
\centerline{\epsfig{file=spps-4-3-28.eps,height=2.8in}}
%\vspace*{2.9in}
\caption[f-4-3-28]
{Predicted current and normal zone voltage transients when the coil and
dump resistor are connected (outsert coil).}
\label{f-4-3-28}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-29.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-4-3-29]
{Predicted maximum wire and conduit wall temperature rise transient
during a simulated quench (outsert coil).}
\label{f-4-3-29}
\bigskip
\centerline{\epsfig{file=spps-4-3-30.eps,height=3.4in}}
%\vspace*{3.6in}
\caption[f-4-3-30]
{Time evolution of symmetric wire temperature profiles along coolant
flow path during the first 15 seconds of a simulated quench (outsert coil).}
\label{f-4-3-30}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=spps-4-3-31.eps,height=3.4in}}
%\vspace*{3.5in}
\caption[f-4-3-31]
{Predicted pressure rise transient during a simulated quench (outsert coil).}
\label{f-4-3-31}
\centerline{\epsfig{file=spps-4-3-32.eps,height=3.4in}}
%\vspace*{3.5in}
\caption[f-4-3-32]
{Time evolution of symmetric pressure profiles along coolant flow path
during the first 15 seconds of a simulated quench (outsert coil).}
\label{f-4-3-32}
\end{figure}

\begin{figure}[t]
\centerline{\epsfig{file=spps-4-3-33.eps,height=3.4in}}
%\vspace*{3.5in}
\caption[f-4-3-33]
{Time evolution of asymmetric pressure profiles along coolant flow path
during the first 15 seconds of a simulated quench (insert coil).}
\label{f-4-3-33}
\centerline{\epsfig{file=spps-4-3-34.eps,height=3.4in}}
%\vspace*{3.5in}
\caption[f-4-3-34]
{Time evolution of asymmetric wire temperature profiles along coolant
flow path during the first 15 seconds of a simulated quench (insert coil).}
\label{f-4-3-34}
\end{figure}

\clearpage

\section{SUMMARY AND CONCLUSIONS}
\label{s-4-4}

A superconducting, forced-flow cooled, magnet internal design concept has been
developed to meet the requirements of a stellarator power plant.  Results of
electromagnetic, thermal-hydraulic, quench and stability analyses suggest a
layer-wound, graded CICC winding pack configuration with two nested coils
offers the promise of design feasibility.  A conceptual point design was
developed as a preliminary baseline configuration.  The characteristics of this
conceptual point design are summarized in Table~\ref{t-4-4-1}.

Several magnet concept development issues beyond the scope of this
present study should be addressed.  From a coil design perspective,
these open issues are:

\begin{enumerate}

\item
The minimum bend radius to allowable conduit dimension ratio was assumed to be
20:1 based on best engineering judgment.  This ratio needs validation for
stellarator-relevant conductor designs and may require an upward adjustment due
to winding pack twist.

\item
A conductor strain of 0.35\% was assumed in computing operating margins.
Actual strains may be significantly greater due to combined effects of Lorentz
forces, differential thermal contraction upon cool-down, and stress risers
related to local topology.  Consequently, present operating margin predictions
may be optimistic.

\item
Conduit wall thickness may be undersized, given the extent of conductor
bending combined with quench pressure rise effects.

\item
The kinematics of non-planar winding need to be analyzed for this
topology and conductor architecture, and factored into the design of the
external case structure.

\item
Predicted temperature margins may be under-predicted significantly due to
the fact the cooling design is not optimized.

\item
Variability in the local angle of neutron incidence with respect to the
coil may effect peak-to-average nuclear heating factors such that
outsert coil hot spot temperature may increase significantly beyond
present values.

\item
Electrical transient response of the inductively-coupled, nested coils
and case structure during quench needs to be analyzed from the
perspective of how electromagnetic imbalance forces between coils affect
design of the magnet structure.

\end{enumerate}

\begin{table}[p]
\caption{}
\centering{\bf SPPS/MHH Magnet Internal Design Data Summary} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c}
\hline
                                            & Insert & Outsert A & Outsert B \\
\hline \\
Winding window equivalent inside radius (mm) & 3,610 &   3,869\D &   4,145.2 \\
Winding window equivalent outside radius (mm)& 3,819 &   4,113.8 &   4,390\D \\
Winding window lateral width (mm)            & \D505 &   \D555\D &   \D555\D \\
Minimum radius of curvature (mm)             & \D340 &   \D609\D &   \D878\D \\
Case-to-coil clearance (mm) for groundwall   & 12    &    9.6    & 9.6 \\
\vspace*{0.4cm}
Total number of conductor layers             & 9      & 8        & 8 \\
Total number of auxiliary coolant conduit layers & 1  & 0        & 0 \\
Total number of layers in winding window     & 10     & 8        & 8 \\
Total number of conductor turns per layer    & 24     & 18       & 18 \\
Avg. Total conductor length (m) per year     & 560    & 451.4    & 482.7 \\
Avg. Coolant flow path length (m) per layer  & 280    & 451.4    & 482.7 \\
\vspace*{0.4cm}
Total winding window current (MA-turns)      & 1      & 6.55     & 6.55 \\
Total number of turns                        & 216    & 144      & 144 \\
Estimated  total stored energy (MJ per coil) & 13     & 1200     & 1,200 \\
Estimated total inductance (H)               & 1.2    & 1.2      & 1.2 \\
Dump resistance ($\Omega$)                   & 0.1    & 0.33     & 0.33\\
Center-tapped winding quench voltage (kV) & $\pm$0.25 & $\pm$7.5 & $\pm$7.5 \\
\vspace*{0.4cm}
Number of CICC strands                       & 216    & 486      & 486 \\
Nb$_3$Sn strand diameter (mm)                & 0.62   & 0.895    & 0.895 \\
Cu/non-Cu ratio                              & 1:1    & 2.27:1   & 2.27:1\\
Non-Cu area (mm$^2$)                         & 32.6   & 93.5     & 93.5\\
\\ \hline
\end{tabular}
\label{t-4-4-1}
\end{table}

\begin{table}[p]
\caption{}
\centering{\bf Table~\ref{t-4-4-1} (Cont'd)} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c}
\hline
                                            & Insert & Outsert A & Outsert B \\
\hline \\
Number of cabling stages                    & 4      & 5         & 5 \\
Cable space void fraction                   & 0.3    & 0.35      & 0.35 \\
Central flow channel diameter (mm)          & 5      & n/a       & n/a \\
Peak magnetic field (T)                     & 14.5   & 9.2       & 7.3 \\
Operating current (kA)                      & 4.63   & 45.5      & 45.5 \\
Predicted minimum temperature margin (K)    & 0.7    & 2.1       & TBD \\
QUENCHER predicted quench temperature (K)   & 70     & 170       & TBD \\
QUENCHER predicted quench pressure (MPa)    & 1.9    & 4         & 4 \\ \\
Square outer conduit dimension (mm)         & 16     & 25.7      & 25.7 \\
Conduit minimum wall thickness (mm)         & 1.5    & 2         & 2 \\
Conduit inner diameter (mm)                 & 13     & n/a       & n/a \\
Conduit material type                       & I908   & I908      & I908 \\
Insulation wrap thickness (mm)              & 1.25   & 1.25      & 1.25 \\
Turn-to-turn spacing (mm)                   & 2.5    & 2.5       & 2.5 \\
Layer-to-layer spacing (mm)                 & 2.5    & 2.5       & 2.5 \\
\\					      	       	      
Helium maximum mass flow rate (g/s per layer) & 8    & 3.5       & TBD \\
Helium maximum inlet pressure (MPa)         & 1.25   & 1.25      & TBD \\
Helium inlet temperature (K)                & 3      & 3         & 3 \\
Helium maximum outlet temperature (K)       & 5.7    & 5.2       & TBD \\
\\ \hline
\end{tabular}
\end{table}

Steps which can be taken during a subsequent design development effort
to address the aforementioned issues include:

\begin{enumerate}

\item
Perform experimental conductor bending trials with twist to revise bend
radius limits.

\item
Develop a smeared property finite element structural model of an encased
non-planar coil to predict strains due to cool-down and electromagnetic
loads.

\item
Review recent wire developments, such as tertiary alloys and artificial
pinning center (APC) technologies, to determine potential gains in
critical current density and applicability to the stellarator power
plant.

\item
Conduct a manufacturing study of the winding and insulating process.

\item
Develop a structural model of conduit and refine wall thickness to
account for strains due to bending and quench pressure rise.

\item
Perform a more detailed, self-consistent cooling design study based on
steady-state thermal-hydraulic simulations of the entire winding pack,
including case conduction~\cite{r29}.  Investigate the effects of counterflow
coolant routing, case cooling with conductor effluent, subcooling to 2.5K, and
more efficient utilization of the helium heat capacity via optimal selection of
coolant inlet pressure and temperature combinations.

\item
Refine MIITS calculation of quench performance, using a nonlinear coupled
electrical circuit model of nested coil and case assembly to predict case eddy
currents as well as revise quench voltage and temperature rise.  Use prediction
of case eddy currents to determine impact on magnet structural design.

\item
Obtain input from wire vendors to assess the extent to which further increases
in Nb$_3$Sn critical current density and processing yields are possible and how
such increases would impact the magnet internal design.

\end{enumerate}

\clearpage

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\end{document}

