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\begin{document}

\title{\bf 8. ASSESSMENT OF LOW-ASPECT-RATIO TOKAMAK POWER PLANT}
\author{
\autlist{Charles~G.~Bathke}{Michael~Billone}{Leslie~Bromberg}
\autlist{David~A.~Ehst}{Laila~A.~El-Guebaly}{Stephen~C.~Jardin}
\autlist{Hesham~Y.~Khater}{Charles~E.~Kessel,~Jr.}{Tak-Kuen~Mau}
\autlist{Farrokh Najmabadi}{Michael Sidorov}{Don~Steiner}
\autlist{Igor~N.~Sviatoslavsky}{Dai-Kai~Sze}{Mark~S.~Tillack}
\autlist{Peter Titus}{}{Clement~P.~C.~Wong}
}
\date{}
\maketitle

\begin{titlepage}
{%baselineskip=1.8ex
\tableofcontents
}
\end{titlepage}

\setcounter{chapter}{7}
\setcounter{page}{1}

\chapter{ASSESSMENT OF LOW-ASPECT-RATIO TOKAMAK POWER PLANT}

\section{INTRODUCTION}
\label{s-8-1}

The low-aspect-ratio tokamak is one of several power plant candidates which
were assessed within the Starlite project.  For all of the assessments, the
existing literature and databases were relied upon heavily.  New analyses were
performed only for those features which are unique to the low-aspect-ratio
tokamak power plant, and which required additional work in order to properly
address the requirements and issues.

A methodology was developed for evaluating all fusion power plant design
options in a consistent manner~\cite{Wag95}.  This methodology is based on a
set of minimum requirements that must be met in order that a fusion power plant
will be judged sufficiently attractive such that it will be able to compete
with alternative energy sources and penetrate the energy marketplace during the
21st century.  The requirements relate to economic, safety and environmental
features that present-day utility and government representatives believe will
be necessary in the future.

Uncertainties in basic data and in component performance remain for all fusion
power plant options.  In some cases the uncertainties are modest and in other
cases they are large enough to cast serious doubts on the feasibility of the
concept.  It is essential in any assessment of options to factor these ``key
issues'' into the evaluation.  It is usually true that attractiveness and
feasibility trade off against one another;  new ideas which appear more
attractive than existing ``worked-out'' designs often carry with them
significant uncertainties.  Conversely, a larger data base always allows a
better understanding of the limitations and problems with a concept.

In this section, we present our assessment of low-aspect-ratio tokamak power
plants.  This assessment includes the results of detailed physics calculations,
preliminary engineering design considerations, and parametric systems
analyses.  Although our assessment of low-aspect-ratio tokamak power plants was
not comprehensive, we believe that we have identified and evaluated the key
issues associated with this concept.  In this regard, our study will provide an
excellent starting point of departure for further analysis and assessment of
the low-aspect-ratio tokamaks.

\section{PHYSICS ASSESSMENT}
\label{s-8-2}

The distinguishing feature of the low-aspect-ratio tokamak is that the Toroidal
Field (TF) coils are made of copper or copper alloy rather than of
superconductor, and that there are no ohmic-heating (OH) poloidal-field (PF)
coils on the inboard side of the torus.  These two features allow a design with
little or no shielding on the inboard side of the tokamak and with very low
aspect ratio ($A=R/a$ where $R$ and $a$ are the major and minor radii of
plasma, respectively).  In tokamak with superconducting TF coils, the natural
increase in $\beta$ with decreasing aspect ratio, ({\it i.e.,} \hbox{$\beta\sim
A^{-1}$} at fixed $\beta_N$ and $q_*$) is offset by the decrease in the
toroidal field strength at the plasma center set by the maximum allowable field
at the superconductor ({\it i.e.,} \hbox{$B_T \sim A^{1/2}$} for fixed
$B_{max}$ at the superconductor and with shielding present).  This trade-off
may not be present in tokamaks with normal-conducting coils in which the
inboard shielding requirements are greatly reduced from that in a
superconducting tokamak. Thus, in the low-aspect-ratio tokamak there is a net
gain in the fusion power density $\beta^2B_T^4$ as $A$ is decreased at fixed
$\beta_N$ and $q_*$.  If we combine this effect with the observed tendency for
$\beta_N$ to increase as $A$ decreases, and if we make use of a class of
high-bootstrap fraction equilibria (with high central safety factor $q_0$) to
minimize the current drive requirements, then we find a potentially attractive
configuration at low $A$ with an increased mass power density over conventional
designs~\cite{Bat95}.  The main drawbacks appear to be the large recirculating
power associated with the ohmic losses in the TF coils and, perhaps, with the
current-drive systems, and the lack of a substantial experimental database.  In
this section we describe the detailed equilibrium, MHD stability, and current
drive calculations that were done within the Starlite project in support of the
low-$A$ design.

Throughout this section, the toroidal $\beta$ is defined as \hbox{$\beta_T = 2
\mu_{o} \langle p \rangle / B_{T}^{2}$}, the circularized safety factor,
\hbox{$ q_* = 2 \pi a^2 B_T S / \mu_{o} R I_{p}$}, the normalized $\beta$
\hbox{$\beta_N=100 \times \beta_T/ (I_P/aB_T)$}, the ratio of TF coil currents
to the plasma current \hbox{$I_{TF}/I_P = A^2q_*/S $}, where the shape factor
\hbox{$S=(1+ \kappa^{2})/2$}.  Here, \hbox{$\langle p \rangle$} denotes the
volume averaged pressure, $B_{T}$ is the toroidal field in vacuum at the plasma
major radius $R$, and $I_{p}$ is the total plasma current.  Also, $\kappa$ is
the elongation, and $\delta$ is the triangularity, $\ell_i$ is the internal
inductance, and $\beta_p$ is the poloidal $\beta$ of the plasma.

\subsection{Equilibrium and Vertical Stability}
\label{s-8-2-1}

It is well known that reducing the plasma aspect ratio leads to larger plasma
elongation in a uniform vertical
field~\cite{Muk71,Reh75,Pen86,Hak90,Rob92,Gal93,Kal94}.  In a truly uniform
vertical field the decay index would be zero, so that the plasma would not be
vertically unstable even though it might have significant elongation.  Thus, so
long as the plasma takes on an elongation less than the value obtained in a
uniform magnetic field, it would not need a vertical control system.

As the aspect ratio is reduced the plasma distorts its shape to maintain force
balance.  In fact, the entire plasma shape changes, not just its elongation, as
has been the emphasis in previous studies.  In order to assess the plasma shape
changes as a function of aspect ratio, the Tokamak Simulation Code
(TSC)~\cite{Jar86} is used with a multi-polar field specification~\cite{Reu86}
that allows for a uniform vertical field to the accuracy of the computer
precision (10$^{-12}$ in this case), guaranteeing that all gradients in the
vertical magnetic field are negligible.  The influence of the current profile
($\ell_i$), the pressure ($\beta_p$), and the aspect ratio on the resulting
free-boundary plasma shape is examined.

As the aspect ratio is reduced both the plasma elongation and triangularity
increase, at rates which depend on the current profile and pressure.  For
example, for a set of equilibria with $\ell_i=0.5$ and $\beta_p=0.5$, as the
aspect ratio decreases through 2.0, 1.5, 1.25, and 1.15, the plasma elongation
increases sequentially through 1.4, 1.6, 1.8, and 1.9, and the triangularity
increases through 0.1, 0.2, 0.35, and 0.47, respectively.  The strong increase
in triangularity is particularly important, since it has a strong stabilizing
effect on $n>0$ ideal MHD modes.  Another very strong dependence of the plasma
shape occurs as the plasma current profile is changed.  At the fixed aspect
ratio of 1.25, the plasma elongation at zero pressure varies from 1.5 to 3.3 as
$\ell_i$ is varied from 1.35 to 0.15.  The triangularity shows a similar rapid
increase at very low values of $\ell_i$, although not as large as the
elongation.  Variations in the pressure have a weaker effect on the plasma
shape, decreasing the elongation and increasing the triangularity, as the
pressure is increased.  Finally, the effects of the detailed profiles were
briefly examined.  Two equilibria both with $\ell_i=0.26$ and $\beta_p=0.50$
but different current profiles were examined which had elongations of 1.75 and
1.83, and triangularities of 0.37 and 0.34, respectively.  These results show
that details of the plasma profiles will affect the plasma shape and should
give rise to a band of possible values rather than a line.  A number of these
equilibria are reported in Table~\ref{t-8-2-1}.

\begin{sidewaystable}
\caption{}
\centering{\bf Natural Shapes for Low-Aspect-Ratio Tokamaks}
\vspace{1.0cm}
\small{
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{l c c c l c c c l c c c l c c}
\hline
\multicolumn{3}{c}{$A=1.15$}& & \multicolumn{3}{c}{$A=1.25$}& &
\multicolumn{3}{c}{$A=1.50$}& & \multicolumn{3}{c}{$A=2.00$} \\
\cline{1-3} \cline {5-7}
\cline{9-11} \cline {13-15}
$\ell_i(3)$, $\beta_p(1)$ & $\kappa$ & $\delta$ & &
$\ell_i(3)$, $\beta_p(1)$ & $\kappa$ & $\delta$ & &
$\ell_i(3)$, $\beta_p(1)$ & $\kappa$ & $\delta$ &  &
$\ell_i(3)$, $\beta_p(1)$ & $\kappa$ & $\delta$  \\
\hline \\
0.43,1.00 & 1.907 & 0.539 & & 0.44,1.00 & 1.834 & 0.425 & &
0.48,1.00 & 1.629 & 0.260 & & 0.51,1.00 & 1.364 & 0.125 \\
0.52,1.00 & 1.762 & 0.507 & & 0.53,1.00 & 1.700 & 0.390 & &
0.56,1.00 & 1.533 & 0.238 & & 0.59,1.00 & 1.322 & 0.114 \\
0.63,1.00 & 1.641 & 0.488 & & 0.64,1.00 & 1.590 & 0.372 & &
0.67,1.00 & 1.451 & 0.226 & & 0.69,1.00 & 1.280 & 0.108 \\
0.82,1.00 & 1.507 & 0.476 & & 0.83,1.00 & 1.460 & 0.362 & &
0.83,1.00 & 1.364 & 0.217 & & 0.84,1.00 & 1.230 & 0.105 \\
0.98,1.00 & 1.437 & 0.472 & & 0.97,1.00 & 1.400 & 0.354 & &
0.99,1.00 & 1.308 & 0.215 & & 0.99,1.00 & 1.196 & 0.104 \\
1.13,1.00 & 1.389 & 0.471 & & 1.12,1.00 & 1.357 & 0.354 & &
1.13,1.00 & 1.270 & 0.214 & & 1.13,1.00 & 1.172 & 0.105 \\
1.34,1.00 & 1.337 & 0.471 & & 1.35,1.00 & 1.300 & 0.357 & &
1.33,1.00 & 1.232 & 0.214 & & 1.31,1.00 & 1.147 & 0.105 \vspace{0.2cm}\\
0.27,0.50 & 2.270 & 0.576 & & 0.31,0.50 & 2.110 & 0.440 & &
0.37,0.50 & 1.775 & 0.251 & & 0.46,0.50 & 1.402 & 0.106 \\
0.36,0.50 & 2.057 & 0.521 & & 0.40,0.50 & 1.930 & 0.397 & &
0.46,0.50 & 1.657 & 0.225 & & 0.54,0.50 & 1.361 & 0.097 \\
0.46,0.50 & 1.894 & 0.492 & & 0.49,0.50 & 1.793 & 0.373 & &
0.56,0.50 & 1.565 & 0.211 & & 0.63,0.50 & 1.323 & 0.092 \\
0.62,0.50 & 1.711 & 0.471 & & 0.65,0.50 & 1.640 & 0.352 & &
0.71,0.50 & 1.464 & 0.201 & & 0.77,0.50 & 1.275 & 0.089 \\
0.77,0.50 & 1.609 & 0.462 & & 0.81,0.50 & 1.538 & 0.343 & &
0.86,0.50 & 1.401 & 0.197 & & 0.91,0.50 & 1.242 & 0.089 \\
0.93,0.50 & 1.536 & 0.459 & & 0.96,0.50 & 1.475 & 0.341 & &
1.00,0.50 & 1.356 & 0.196 & & 1.04,0.50 & 1.219 & 0.089 \\
1.14,0.50 & 1.467 & 0.457 & & 1.16,0.50 & 1.414 & 0.338 & &
1.18,0.50 & 1.313 & 0.198 & & 1.22.0.50 & 1.195 & 0.091 \\
1.41,0.50 & 1.402 & 0.458 & & 1.56,0.50 & 1.337 & 0.345 & &
1.43,0.50 & 1.273 & 0.201 & & 1.44,0.50 & 1.171 & 0.094 \vspace{0.2cm} \\
0.15,0.00 & 3.334 & 0.530 & & 0.14,0.00 & 3.323 & 0.445 & &
0.31,0.00 & 1.961 & 0.205 & & 0.42,0.00 & 1.445 & 0.079 \\
0.25,0.00 & 2.634 & 0.484 & & 0.22,0.00 & 2.664 & 0.385 & &
0.37,0.00 & 1.859 & 0.191 & & 0.50,0.00 & 1.400 & 0.071 \\
0.41,0.00 & 2.117 & 0.447 & & 0.32,0.00 & 2.262 & 0.348 & &
0.45,0.00 & 1.726 & 0.176 & & 0.58,0.00 & 1.360 & 0.068 \\
0.56,0.00 & 1.882 & 0.428 & & 0.47,0.00 & 1.918 & 0.317 & &
0.60,0.00 & 1.585 & 0.164 & & 0.72,0.00 & 1.311 & 0.066 \\
0.72,0.00 & 1.750 & 0.423 & & 0.62,0.00 & 1.747 & 0.307 & &
0.74,0.00 & 1.501 & 0.161 & & 0.85,0.00 & 1.278 & 0.066 \\
0.92,0.00 & 1.639 & 0.422 & & 0.77,0.00 & 1.644 & 0.302 & &
0.87,0.00 & 1.446 & 0.161 & & 0.98,0.00 & 1.254 & 0.067 \\
1.18,0.00 & 1.546 & 0.425 & & 0.96,0.00 & 1.554 & 0.301 & &
1.05,0.00 & 1.395 & 0.163 & & 1.14,0.00 & 1.231 & 0.070 \\
1.52,0.00 & 1.468 & 0.430 & & 1.36,0.00 & 1.447 & 0.309 & &
1.28,0.00 & 1.349 & 0.168 & & 1.36,0.00 & 1.209 & 0.073 \\
\\ \hline
\end{tabular}
}
\label{t-8-2-1}
\end{sidewaystable}

Once the aspect ratio has been determined, the primary issue in designing the
poloidal-field (PF) system is the difference between the desired plasma shape
and the natural plasma shape.  If it is desired to have a larger elongation
than occurs in the uniform vertical field, then a quadrapole field must be
provided and a vertical-position control system will be required.  If a smaller
elongation is desired, then a quadrapole field with opposite polarity is
required.  The variations of plasma elongation with $\ell_i$ probably pose the
most difficult problem, since at higher $\ell_i$ a vertically destabilizing
quadrapole field would be necessary to give the plasma the same elongation as
is achieved naturally at low $\ell_i$.  The triangularity is relatively flat
for values of $\ell_i$ between 0.5 and 1.5.  However, if a higher value of
triangularity is desired then a hexapole field must be applied.  The hexapole
may itself destabilize the plasma position, thus necessitating the superposition
of a stabilizing quadrapole field.

\subsection{MHD stability and bootstrap current}
\label{s-8-2-2}

Initial stability studies of low aspect ratio tokamaks performed in support of
the Starlite project concentrated on conventional first-stability current
profiles with central safety factors near or slightly above unity, \hbox{$q_0
\approx 1$}~\cite{Men94}.  A scan of different pressure profiles for
configurations with aspect ratios $1.2<A<2.5$, with shaping parameters
$\kappa=1.6$, $\delta=0.35$ and monotonic current profiles such that
$q_*/q_0=1.5$ or $3.0$, revealed that configurations with broad pressure
profiles in the range \hbox{$p(0)/\langle p \rangle \sim 1.6-2.0$} maximize the
stable $\beta_N$, and that $\beta$ limit due to ballooning modes is smaller
than that due to kink modes.  In aspect ratio scans with these same parameters
in which the pressure peaking factor is held fixed at \hbox{$p(0)/\langle p
\rangle=1.8$}, we find that the critical $\beta_N$ for both low-$n$ (kink) and
high-$n$ (ballooning) modes increases with decreasing $A$, and that this effect
is particularly pronounced for the $n=1$ kink mode, both with and without a
conducting wall.

\begin{figure}[p]
\centerline{\psfig{figure=sti-8-2-1.ps,height=3.2in}}
\caption[sti-8-2-1]
{Parallel currents as functions of poloidal flux $\psi$ for a
high-bootstrap-fraction equilibrium which requires edge current drive.}
\label{sti-8-2-1}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{figure=sti-8-2-2.ps,height=3.2in}}
\caption[sti-8-2-2]
{Parallel currents as functions of poloidal flux $\psi$ for a
high-bootstrap-fraction equilibrium which does not require edge current drive.}
\label{sti-8-2-2}
\end{figure}

These initial studies were next extended to configurations with $q_0>1.1$ in
order to achieve higher ballooning $\beta$ limits and increased bootstrap
fraction~\cite{Men95} ({\it i.e.,} the ratio of the plasma current arising from
the bootstrap effect to the total plasma current).  As the computed bootstrap
fraction~\cite{Kes94} began to approach unity, it became necessary to develop
special techniques for computing low-$A$ equilibria to assure that the pressure
driven currents never exceeded the equilibrium currents, either globally or
locally.  (Note that by ``pressure driven current'', we refer to the
combination of bootstrap, diamagnetic, and Pfirsch-Schl\"{u}ter currents that
all combine to provide toroidal plasma current from the plasma pressure.  We
use $f_{\nabla p}$ to denote the fraction of the total toroidal plasma current
that is provided by all the pressure driven currents.)  These
high-bootstrap-fraction equilibria were further divided into two categories,
those that require edge current drive and those that do not, depending on
whether or not external current drive must be supplied outside the minor radius
where the bootstrap current peaks (see Figs.~\ref{sti-8-2-1}
and~\ref{sti-8-2-2}).  The maximum stable $\beta$ is largest for equilibria in
the first category, but the recirculating power associated with current drive
is minimized by equilibria in the second category (see Sec.~\ref{s-8-2-3}).
The high-bootstrap-fraction equilibria also tend to have very high edge $q$
values, making it difficult for numerical stability codes to resolve the
stability of intermediate-$n$ modes.  Some of the most interesting stable
low-$A$ equilibria found in these studies are listed in Table~\ref{t-8-2-2}.

\begin{table}[b]
\caption{}
\centering{\bf Equilibrium Parameters for Stable Low-$A$ Equilibria}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{1.truecm}} c c c c c c c c c}
\hline
       & & & & & & & & & Current-drive \\
CASE
         & $A$ & $\kappa$ & $\beta_N$ & $\beta_T$ & $q_*$ & $q_0$ &
               $f_{\nabla P}$ & $I_{TF}/I_{P}$ &  location     \\
\hline \\
STREAC06 & 1.25 & 2.00 & 8.8 & 43\% & 2.38 & 3.3 & 0.85 & 1.27 & center \& edge \\
STREAC09 & 1.25 & 2.00 & 5.8 & 12\% & 5.83 & 3.3 & 0.90 & 3.12 & center \\
STREAC13 & 1.25 & 2.00 & 4.6 & 80\% & 0.74 & 1.3 & 0.15 & 0.35 & center \& edge \\
STREAC15 & 1.25 & 2.20 & 6.5 & 17\% & 4.48 & 4.0 & 0.95 & 2.39 & center \\
STREAC16 & 1.25 & 2.20 & 6.5 & 21\% & 3.63 & 4.0 & 0.85 & 1.95 & center \\
STREAC17 & 1.40 & 2.20 & 7.0 & 16\% & 4.52 & 3.9 & 0.97 & 3.03 & center \\
STREAC18 & 1.40 & 2.20 & 7.0 & 21\% & 3.45 & 4.0 & 0.89 & 2.32 & center \\
STREAC19 & 1.25 & 3.00 & 7.0 & 40\% & 3.51 & 3.0 & 0.99 & 1.09 & center \\
\\ \hline
\end{tabular}
\label{t-8-2-2}
\end{table}

The first three equilibria in Table~\ref{t-8-2-2} clearly illustrate the
tradeoff between high pressure-driven current fraction (STREAC09 with
$\beta_T=12\%$, $f_{\nabla P}=0.90$), high $\beta$ (STREAC13 with
$\beta_T=80\%$, $f_{\nabla P}=0.15$), and a reasonable compromise (STREAC06
with $\beta_T=43\%$, $f_{\nabla P}=0.85$).  Cases requiring only central
current drive in addition to the bootstrap current are marked ``C'', while
those also requiring edge current drive are marked ``C,E''. STREAC06 and
STREAC13 equilibria are stable to $n=1,2,3$ with a conducting wall at
\hbox{$c/a \sim 1.24$} (where $c$ is the distance of the wall from plasma
center on mid-plane) and STREAC09 is stable to $n=1$ with the same wall.  The
high edge $q$ of STREAC09 makes the stability of $n=2$ and higher mode number
kinks exceedingly difficult to test.  For the same reason, STREAC15 through
STREAC19 equilibria have only been tested for ballooning stability but are
presumed to be kink stable since these cases are similar to STREAC09.  Without
the wall for stabilization, the best case found at aspect ratio 1.25 has
$\beta_T=27\%$ and $f_{\nabla P}=0.41$.

Since the power requirements associated with the edge current drive turned out
to be so large (Sec.~\ref{s-8-2-3}), our final studies concentrated on a series
of equilibria that require only central current drive.  We impose the
constraint that the surface averaged parallel current density \hbox{$\langle
{\bf J } \cdot {\bf B} \rangle / \langle {\bf B} \cdot \nabla \phi \rangle$} be
a linear function of the poloidal flux $\psi$ interior to the point where the
linear profile is tangent to the bootstrap current profile and equal to the
bootstrap current outside this point (Figure~\ref{sti-8-2-2}).  This constraint
completely specifies the profiles and insures that the first derivatives of
equilibrium quantities are continuous everywhere.

For these scans, we kept the shape fixed with $\kappa=2.2$ and $\delta=0.45$,
but looked at two aspect ratios, $A=1.25$ and $A=1.4$.  For each configuration
we considered four different pressure profiles with a range of peaking factors,
and for each pressure profile three density profiles were considered for a
total of 12 different profile combinations. The density and pressure profiles
were functions of the form $f \sim {(1-\hat{{\psi}}^{\beta})}^{\alpha}$, where
$\hat{\psi}$ is the normalized poloidal flux.  For the initial $A=1.25$ scan
for instance, the pressure profiles we considered had the following exponents:
$\alpha=2$ and $\beta=4$, 2.5, 1.5, and 1.0.  For each pressure profile, three
normalized density profiles used which were equal to the normalized pressure
profile raised to the 3/4, 1/2, and 1/4 power.  For each of these, we
determined the maximum ballooning-stable $\beta$ as a function of the central
safety factor, $q_0$.  After determining the optimum $q_0$ and pressure and
density profiles, we refined the profiles with $q_0$ fixed to further optimize
$\beta$.  The STREAC15 through STREAC18 equilibria in Table~\ref{t-8-2-2}
contain some of the most attractive low-$A$ equilibria found in this scan.

Finally, we performed a series of runs at aspect ratio $A=1.25$ and with plasma
elongations considerably above the natural elongation of about 2.2.  These runs
were specifically aimed at finding stable profiles that had near 100\%
self-driven current and high $\beta$ at the higher elongations.  In these runs,
we employed the built-in constraint of no edge current drive. The best
configuration found in this extensive scan is STREAC19.  This equilibrium has
an elongation of 3.0, which will certainly necessitate a vertical position
control system.  However, over 99\% of the plasma current is self-driven for
this configuration, and the $\beta$ is 40\%.  Because of these two features,
STREAC19 has been chosen as our baseline for assessment of low-$A$ tokamak
power plants. (Note that in the systems-code analysis, and in the engineering
design, the $\beta$ is reduced to 36\%, consistent with the Starlite guideline
of operating the power-plant plasma at 90\% of the theoretical stability
$\beta$ limit)

\subsection{Current drive}
\label{s-8-2-3}

In the Low-$A$ tokamak concept, externally generated RF waves must be used to
sustain the necessary seed current; {\it i.e.,} the difference between the
plasma current needed for equilibrium and stability and that provided by the
bootstrap and other pressure-driven effects.  The high plasma dielectric
constant, approximately equal to the squared ratio of the electron plasma
frequency to the electron cyclotron frequency in these configurations ,
$f_{pe}^2/f_{ce}^2$, precludes the use of either electron-cyclotron or
lower-hybrid waves from penetrating into the core of the plasma.  However, we
find that the superposition of three conventional RF current drive techniques
can provide the required seed current profile.  These are (1)~low-frequency
fast waves (LFFW) with $f<f_{cT}$ for central region ($\hat{\psi}<0.4$),
(2)~high-frequency fast waves (HFFW) with \hbox{$f \sim 20f_{cD}$} for
mid-plasma region (\hbox{$0.4 \leq \hat{\psi} \leq 0.8$}), and (3)~lower hybrid
(LH) waves for edge-plasma region (\hbox{$0.8 < \hat{\psi} < 1.0$}).  We
attempt to obtain the best possible match between the RF-driven and seed
current profiles by selecting the power spectra launched by the antenna and the
launcher poloidal location on the outboard edge.

The CURRAY ray tracing code was used for the analysis~\cite{Mau92}.  This code
utilizes the full Hirshman-Sigmar bootstrap current model~\cite{Kes94} and
includes three distinguishing features applicable to a low-$A$ tokamak power
plant: (a)~high-$\beta$ effects on HFFW dispersion and damping,
(b)~slowing-down $\alpha$-particle absorption in the unmagnetized
approximation, and (c)~perpendicular thermal ion Landau damping for HFFW and LH
waves.  The local current-drive efficiencies include the deleterious effect of
trapped electrons, which can be substantial in the outer part of the plasma.

Extensive current-drive calculations on four of the low-$A$ equilibria listed
in Table~\ref{t-8-2-2}, namely, STREAC06, STREAC15, STREAC16, and STREAC17,
reveals a large amount of power is needed to drive currents in the mid-plasma
region (\hbox{$0.4<\hat{\psi}\leq 1$}).  This is exemplified by STREAC06
equilibria given in Table~\ref{t-8-2-2}.  The following parameters were used
for current-drive calculations: $R=3.0$~m, $a=2.4$~m, $B_T=2.37$~T,
$I_p=27.9$~MA $q_0=3.36$, $\beta=0.426$, $\beta_p=1.23$, $p_0=1.479$~MPa,
\hbox{$n_0/\rangle n \langle =1.32$}, and \hbox{$T_0/\rangle T \langle =1.32$}.
We also assumed that $n_e/n_i=1.2$, $T_i/T_e=0.9$, ion concentrations of
$f_D=f_T=0.43$ $f_{He}=0.13$, and $f_{Oxygen}=0.01$.  The LFFW used for on-axis
drive had a frequency of $f=6$~MHz, the HFFW used for off-axis current drive
had a frequency of $f=236$~MHz, and the LH waves used for the edge drive had a
frequency of $f=1.1$~GHz.  A temperature scan for the required current drive
power in the three RF techniques used is presented in Table~\ref{t-8-2-3}.  As
an example, Fig.~\ref{sti-8-2-3} shows the profiles of the driven seed current
calculated by the CURRAY code for the \hbox{$\langle T_e\rangle =15$~keV} case
of Table~\ref{t-8-2-3}.

\begin{table}[tp]
\caption{}
\centering{\bf Temperature Scan of Current-Drive Power\\
for STREAC06 Low-$A$ Equilibria}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c c c c c c c c}
\hline
      &                &      &  & \multicolumn{4}{c}{Current-drive power (MW)} \\
\cline{5-8}
$\langle T_e \rangle$ (keV)
      & $\langle n_e \rangle$ (m$^{-3}$)
                         & $\beta_f/\beta$
                                & $I_{BS}/I_{P}$
                                        & $P_{FW}$
                                               &  $P_{HF}$
                                                       &  $P_{LH}$
                                                               & $P_{tot}$ \\
\hline
\hspace{1.4cm} & \hspace{1.4cm} & \hspace{1.4cm} & \hspace{1.4cm} &
\hspace{1.4cm} & \hspace{1.4cm} & \hspace{1.4cm} & \hspace{1.4cm} \\
\Z8.0 & $3.69\X 10^{20}$ & 0.02 & 0.769 & 76.6 & 807.5 &  131.7 & 1,015.8 \\
 12.0 & $2.37\X 10^{20}$ & 0.06 & 0.786 & 21.4 & 542.0 &  153.6 & \D617.0 \\
 15.2 & $1.82\X 10^{20}$ & 0.09 & 0.785 & 15.1 & 324.8 &  130.2 & \D370.1 \\
 20.0 & $1.32\X 10^{20}$ & 0.13 & 0.789 & 13.2 & 262.6 & \Z17.0 & \D292.8 \\
\\ \hline
\end{tabular}
\label{t-8-2-3}
\end{table}

\begin{figure}[bp]
\vspace{0.5cm}
\centerline{\psfig{file=sti-8-2-3.ps,height=3.2in,clip=}}
\caption[sti-8-2-3]
{Current density profiles calculated from the CURRAY code for the
\hbox{$\langle T_e\rangle=15$~keV} case of Table~\ref{t-8-2-3}.  Curves labeled
E, B, FW, LH and T correspond to equilibrium, bootstrap, fast-wave,
lower-hybrid, and total driven current densities, respectively.}
\label{sti-8-2-3}
\end{figure}

It is clear from examination of the table that the power requirements are
dominated by the high-frequency fast-wave component.  The reason for the high
current-drive power required (or low current-drive efficiency) in that region
of the plasma is twofold: (1)~the current carrying electrons are essentially
thermal resulting in severe reduction of the local current-drive efficiency due
to trapped electron effects, and (2)~there is significant absorption of the
wave power by the energetic $\alpha$-particles and thermal fuel ions.

One obvious way to avoid the excessive power required for bulk current drive is
to make use of equilibria that have current profiles that are well aligned with
the bootstrap current profile in the outer part of the plasma, thus requiring
only current drive on axis, even though they may have somewhat less attractive
plasma $\beta$ values.  We have examined such a case in detail (STREAC15
equilibrium of Table~\ref{t-8-2-2}).  The parameters used for current-drive
analysis were: $R=3.0$~m, $a=2.4~$m, $B_T=2.73$~T, $I_p=23.1$~MA, $q_0=4.0$,
$\beta=0.168$, $\beta_p=1.72$, $p_0=1.668$~MPa, \hbox{$n_0/\langle n \rangle
=1.48$}, and \hbox{$T_0/\langle T \rangle =1.48$}. All other parameters are the
same as in current-drive analysis of STREAC06 which are reported in
Table~\ref{t-8-2-3} ({\it i.e.,} $n_e/n_i=1.2$, $T_i/T_e=0.9$, ion
concentrations of $f_D=f_T=0.43$ $f_{He}=0.13$, and $f_{Oxygen}=0.01$).  Since
only on-axis current drive is required, only the LFFW system is required to
drive the seed current.  The frequency used in this case is 10~MHz.  A
temperature scan of the required current drive power is given in
Table~\ref{t-8-2-4}.  As an illustration, the matching of the RF-driven and
equilibrium current density profiles for the \hbox{$\langle T_e\rangle
=20$~keV} case for STREAC15 is displayed in Fig.~\ref{sti-8-2-4}.  As shown in
Table~\ref{t-8-2-4}, only 32.2~MW of current drive power is required in this
particular example, about one tenth of what is typically required in STREAC06.

\begin{table}[tp]
\caption{}
\centering{\bf Temperature Scan of Current-Drive Power \\
for STREAC15 Low-$A$ Equilibria}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c c c c c}
\hline
$\langle T_e \rangle$ (keV)
      & $\langle n_e \rangle$ (m$^{-3}$)
                         & $\beta_f/\beta$
                                & $I_{BS}/I_{P}$
                                        & $P_{tot}$ (MW) \\
\hline
\hspace{2.0cm} & \hspace{2.0cm} & \hspace{2.0cm} & \hspace{2.0cm} &
\hspace{2.0cm} \\
\Z8.0 & $3.34\X 10^{20}$ & 0.02 & 0.895 & 135.8 \\
12.0  & $2.10\X 10^{20}$ & 0.06 & 0.898 & \Z76.1 \\
16.0  & $1.53\X 10^{20}$ & 0.10 & 0.905 & \Z43.9 \\
20.0  & $1.18\X 10^{20}$ & 0.13 & 0.903 & \Z32.2 \\
\\ \hline
\end{tabular}
\label{t-8-2-4}
\end{table}

\begin{figure}[pb]
\centerline{\psfig{figure=sti-8-2-4.ps,height=3.2in}}
\caption[sti-8-2-4]
{Current density profiles calculated from the CURRAY code for the
\hbox{$\langle T_e\rangle=20$~keV} case of Table~\ref{t-8-2-4}.  Curves labeled
E, B, FW, and T correspond to equilibrium, bootstrap, fast-wave, and total
driven current densities, respectively.}
\label{sti-8-2-4}
\end{figure}

The STREAC19 equilibrium (Table~\ref{t-8-2-2}) appears to be most attractive as
a power-plant candidate equilibrium because it has a high $\beta$
($\beta=40\%$), and a low field-utilization factor ($T_{TF}/I_{P}=1.09$).  In
this equilibrium, the bootstrap current is almost perfectly aligned with the
equilibrium current, requiring only 0.15~MA of seed current to be driven on
axis. The plasma parameters were: $R=3.0$~m, $a=2.4$~m, $B_T=2.59$~T,
$I_p=35.4$~MA, $q_0=2.97$, $\beta=0.401$, $\beta_p=1.41$, $p_0=1.818$~MPa,
\hbox{$n_0/\langle n \rangle =1.62$}, and \hbox{$T_0/\langle T \rangle
=1.14$}. All other parameters are the same as in current-drive analysis of
STREAC06 which are reported in Table~\ref{t-8-2-3} ({\it i.e.,} $n_e/n_i=1.2$,
$T_i/T_e=0.9$, ion concentrations of $f_D=f_T=0.43$ $f_{He}=0.13$, and
$f_{Oxygen}=0.01$).  A LFFW system at 8~MHz is used to drive the on-axis seed
current.  A temperature scan of the required current-drive power is given in
Table~\ref{t-8-2-5}.  The current-drive power requirement for this case is in
the range of 10~MW or less.  We point out that the current-drive power will be
extremely sensitive to slight mismatches in the current density profiles due to
variations in the plasma density and pressure profiles and this issue should be
addresses in future studies.

\begin{table}[tb]
\caption{}
\centering{\bf Temperature Scan of Current-Drive Power \\
for STREAC19 Low-$A$ Equilibria}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c}
\hline
$\langle T_e \rangle$ (keV)
      & $\langle n_e \rangle$ (m$^{-3}$)
                         & $\beta_f/\beta$
                                 & $P_{tot}$ (MW) \\
\hline
\hspace{2.0cm} & \hspace{2.0cm} & \hspace{2.0cm} & \hspace{2.0cm} \\
12.0  & $3.20\X 10^{20}$ & 0.045 &  10.5 \\
14.0  & $2.70\X 10^{20}$ & 0.050 & \Z8.3 \\
16.0  & $2.33\X 10^{20}$ & 0.065 & \Z6.2 \\
18.0  & $2.03\X 10^{20}$ & 0.080 & \Z5.1 \\
20.0  & $1.82\X 10^{20}$ & 0.090 & \Z4.4 \\
\\ \hline
\end{tabular}
\label{t-8-2-5}
\end{table}

\subsection{Summary}
\label{s-8-2-4}

The low-$A$ tokamak may offer an alternative to the more conventional tokamak
with an aspect ratio in the range of three to five.  It has been shown that the
natural elongation of a low-$A$ tokamak increases with decreasing A and with
decreasing $\ell_i$ and $\beta_p$.  It has also been illustrated that there is
a tradeoff between high pressure-driven current fraction and high $\beta$.
However, configurations with elongations higher than the natural one offer the
possibility of very low current-drive requirements simultaneous with moderately
high $\beta$.

Equilibria that are stable to all MHD modes (with a conducting wall present)
and that require very little current-drive power were found at $A=1.25$,
$\kappa=3.0$, with $\beta=40\%$.  These would definitely require a vertical
feedback system.  All the cases shown here also require a conducting wall to
stabilize the external kink mode, and thus future work must consider
maintaining stability to the resistive wall mode~\cite{Bon94}.  In the absence
of wall stabilization, the fraction of pressure driven current to total current
would drop considerably.  The sensitivity of current-drive power requirement to
variations in the plasma profiles is an important issue that needs to be
addressed, particularly for equilibria where the pressure-driven current
fraction approaches unity.

\section{ASSESSMENT OF ENGINEERING ASPECTS}
\label{s-8-3}

\subsection{Introduction}
\label{s-8-3-1}

This section summarizes the engineering analyses and conclusions reached in
support of the assessment of a fusion power plant based on a low-aspect-ratio
tokamak.  Engineering analyses were performed for two distinct purposes:

\begin{enumerate}

\item
Direct analysis of the in-vessel components and engineering characteristics
of the power plant allows us to evaluate its ability to meet some of its
mission requirements, independent of the overall system performance.  In
addition, characterization of the key engineering issues in some cases requires
more detailed analysis of individual components and systems.

\item
Many of the attributes which affect the attractiveness of a power plant are
computed in a self-consistent manner by the ARIES System Code~\cite{Aries4}.
Inputs to the code require decisions on the design of various components and a
basic level of engineering analysis.

\end{enumerate}

The assessment concentrated on the key components, including the center-post,
first-wall and blanket, shielding, and divertor.  To the extent possible,
design concepts available from the existing literature were used for the
assessments.  In many cases, lack of detail or need for improvement was
identified, suggesting that further design studies will be required in order to
improve upon the assessments and to find innovative solutions to the problem
areas which have been identified.  Our physics understanding of the
low-aspect-ratio tokamak is at an early stage of development; much of the
engineering assessment is based on and sensitive to the reference set of
physics parameters that were selected.

The center-post dominates the assessment.  Minimizing the inboard radial build
is well-known to be important for the overall success of the concept.  However,
an unshielded copper toroidal-field (TF) coil has many critical issues, which
have been assessed.  Activation analysis of the center-post indicates that the
NRC Class-C waste-disposal limit can not be met unless the change-out time is
extremely short (of the order of weeks).  The volume of the waste stream would
be quite large (the center-post alone weighs 680~tonnes, occupying 90~m$^3$).
Given that the copper will embrittle quite rapidly under irradiation (some
estimates are as low as 0.1~dpa), a mechanical design which is ``licensable''
must be developed.  Code rules for brittle structures are unavailable, but it
is likely that, at a minimum, some additional measures will be needed to
isolate the water coolant and provide some mechanical stability to prevent
failure.

Mechanical analysis indicates that the compressive electromagnetic forces in
the center-post are acceptable, but the tensile forces caused by the expansion
of the TF-coils (appearing as vertical forces in the center-post) cannot be
reacted by the copper center-post.  For this reason a sliding connection
between the upper end of the center-post and the outer leg of the TF-coils is
proposed using mercury as the conducting material.  Alternatively, electrical
leads to the center-post can be located at this joint.  The expansive forces
from the TF-coils have to be reacted either by structure included in the shield
around the copper center-post or by external structure.

Our analyses indicate that it is desirable to add 30~cm of shielding to the
center-post.  This level of shielding would help tremendously to reduce the
waste-disposal problem, and reduce the change-out frequency.  The volumetric
heating of the shield ($\sim$150~MW) could be converted into electricity
instead of being dumped at low temperature.  A separate first wall could remove
the surface heat load, protect against plasma erosion, add another level of
safety assurance against center-post failure.  This shield can also be used to
mechanically stabilize the center-post.  Depending on the final physics design
point, it is probably more cost-effective to add a minimal and optimized shield
rather than suffer the many penalties which arise from its absence.

Recirculating power could be a fatal flaw for the low aspect ratio tokamak.
The primary sinks of power are the current-drive and TF-coil systems Estimates
of the TF-coil losses for the cases examined here are extremely large.  The
reference design is a single-turn concept, as described in \cite{Hic91}.  A
single-turn TF coil avoids problems with insulator breakdown in the radiation
environment and simplifies maintenance (although maintenance of the center-post
remains a critical issue).  The singe-turn coil, however requires large
low-voltage, high-current power supplies and large bus-bars and leads. The
ohmic losses are determined primarily from the basic system parameters.  Given
the high currents required, ways to reduce significantly these loads are to
minimize the length, maximize the area (by pushing the post radius to the
largest possible value), maintain low temperature (the conductivity of copper
is sensitive to temperature), and minimize the neutron exposure.  Scoping study
of operation below room temperature to reduce the resistive losses suggests
that cryogenic will not resolve the problem.

Besides concerns over the feasibility and consequences of the center-post,
there are several unique constraints on the design of the blanket and divertor.
These constraints arise due to the potentially high local power density,
including peaking factors, and the additional compatibility concerns with a
water-cooled copper center-post.  The divertor problem is especially severe,
because the area available at low major radius is so restricted, and because
additional sources of plasma heating (from current drive) add to the divertor
heat loads.  Even with significant flux expansion, the local surface heat flux
is likely to exceed 10~MW/m$^2$.

The use of water in the center-post will make the use of liquid lithium in the
power plant difficult or impossible.  Additional barriers to leakage are
necessary, and substantial additional analyses must be performed to satisfy
licensing and regulatory requirements.  The impact of ruling out liquid lithium
is to reduce the bulk heat removal capabilities.  In addition, no other coolant
is known to be compatible with vanadium (compatibility with helium containing
low levels of impurities is still uncertain), such that the design may be
forced to use ferritic steel as the primary structural material and copper as a
heat sink.  These material choices further limit our ability to extract heat at
high efficiency for the purpose of generating electricity, and further degrade
the safety and environmental attractiveness of the concept.


\subsection{Center-Post Assessment}
\label{s-8-3-2}

\subsubsection{Joule losses}
\label{s-8-3-2-1}

The center-post of the low-aspect-ratio tokamak plays a major role in the
design as well as the economics and safety of the machine.  The reference
design considered here utilizes a single-turn center-post; the back legs of all
16 TF coils connect to the center-post at the top and bottom.  Thus, the
center-post is the conduit for the return electrical current for all the TF
coils.  It is made of either pure copper or, more likely,
dispersion-strengthened copper and cooled by water in the temperature range of
40 to 110\SdegC.  Because the electrical current in the center-post is large
and the cross section is small, the ohmic heating is high, despite the
relatively low resistivity of copper.

The resistivity of metals increases with temperature, making it necessary to
operate the center-post at a low temperature.  As a result, the ohmic and
nuclear heating cannot be converted to electricity.  If the outlet temperature
of the water is increased such that it could be used in a power cycle at a low
conversion efficiency, the copper resistivity increases, resulting in drastic
increase in Joule losses in the center-post such the net power consumed by the
TF coils (Joule losses minus the recovered electricity) is increased. Further
increase in resistivity due to transmutations compounds the problem.

Generally, a completely unshielded center-post has been proposed in fusion
power plant studies based on low-aspect ratio tokamaks ({\it e.g.,}
\cite{Hic91}).  Rather high nuclear heating inside the copper center-post as
well as high radiation damage and, therefore, a short lifetime result.  Short
lifetime has a large impact on the availability and waste disposal problems.
For these reasons, a second design option has been investigated, employing a
30-cm thick shield around the copper center-post.  The thermal analysis has
been restricted to the ohmic heat input because this is by far the largest part
of the total heat input and the nuclear heat is substantially different for the
two design options.  Nuclear heating requires either increased flow rate of the
cooling water or an increase in the coolant temperature rise.

For the system code runs, the center-post average temperature was assumed to be
65\SdegC, and the resistivity of pure unirradiated copper was used.  These are
highly optimistic assumptions for several reasons (see Fig.~\ref{sti-8-3-1}):

\begin{figure}[b]
\centerline{\psfig{file=sti-8-3-1.ps,height=3.5truein}}
\caption[sti-8-3-1]
{Resistivity of pure , dispersion-strengthened , and irradiated
copper.}
\label{sti-8-3-1}
\end{figure}

{\bf Coolant temperature.}
Previous design studies have used a water inlet temperature of $\sim$40\SdegC\
and an outlet temperature over 100\SdegC.  The copper temperature will exceed
this value depending on the internal design and heat transfer coefficient
(resistivity can easily be 25\% higher).

{\bf Copper alloying.}
Pure copper is weak at room temperature, and extremely weak at elevated
temperatures.  It is most likely that a dispersion-strengthened copper alloy
would be needed in order to withstand the internal stresses generated by the
high current.  The resistivity of dispersion-strengthened copper ({\it e.g.,}
\hbox{GlidCop Al-15}) is typically about 10\% higher than pure copper.

{\bf Radiation damage.}
Nickel production is unavoidable in any copper alloy.  Without inboard
shielding, the nickel concentration will continue to build up.  Depending on
the lifetime of the center-post, the effect could be very significant.  For
example, analysis in \hbox{Sec.~\ref{s-8-3-5}} indicates that an average
of $\sim$0.3\%~Ni impurity would exist after one full power year (FPY) of
operation, contributing perhaps 20\% to 30\% additional ohmic losses.  The peak
damage is significantly higher, and can increase the resistivity by a factor of
2.  Since the current density also is peaked at the location of peak damage,
the net increase in ohmic losses will be compounded.  Therefore, the
center-post may have to be replaced even before its irradiation lifetime (or
waste-disposal lifetime) is reached in order to limit this increase in the
recirculating power.

\subsubsection{Mechanical design of the center-post}
\label{s-8-3-2-2}

The center-post is a relatively long, skinny column with a large aspect ratio.
In addition to issues associated with its stability as a standing column, the
center-post is subjected to compressive forces due to the current it carries,
as well as tensile forces due to the loads applied by the outboard TF coil
connections at the extremities of the center-post.  Because copper becomes
rapidly embrittled under a very low neutron fluence, the center-post is
designed to carry only the radial compressive forces.  This can be accomplished
by providing a sliding electrical connection at the upper extremity, such that
the expansive forces from the outboard TF coil legs are not transmitted to the
center-post.  Instead, these forces are reacted either by structure going
through the inboard shield, all the way from the upper to the lower extremities
or by external structure.  In this way the expansive forces are reacted without
impacting the center-post.

The pressure applied perpendicular to the direction of current is proportional
to $B^2/2\mu_0$, where $B$ is the magnetic field and $\mu_0$ is the
permeability of air (or vacuum).  The magnetic field at the edge of the
center-post conductor is $\sim$16~T, therefore, the pressure on the center-post
is $\sim$100~MPa. This is larger than the yield strength of pure copper,
indicating that probably the higher strength of dispersion-strengthened copper
is needed.  Table~\ref{t-8-3-1} shows the yield strength of GlidCop~C15720
irradiated to 15~dpa as 343~MPa or a factor of 2--3 higher than the magnetic
pressure~\cite{GlidCop}.

\begin{table}[t]
\caption{}
\centering{\bf Tensile Properties Of Irradiated GlidCop Alloys~\cite{GlidCop}}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{0.5cm}} c c c c c c c c}
\hline
               &        & \multicolumn{2}{c}{Tensile strength} &
                                       & \multicolumn{2}{c}{Yield Strength}
                                                       & & Reduction \\
\cline{3-4} \cline{6-7}
Material & Condition & (MPa) & (KSI) & & (MPa) & (KSI)
                                            & Elongation & in Area  \\
\hline \\
        C15720 & Control & 395  & 57 & & 337 & 49 & 14\% & 48\% \\
               & \N3 dpa & 411  & 60 & & 353 & 51 & 14\% & 51\% \\
               &  15 dpa & 395  & 57 & & 343 & 50 & 20\% & 50\% \\
\\
        C15760 & Control & 466  & 68 & & 397 & 58 & 12\% & 56\% \\
               & \N3 dpa & 468  & 68 & & 397 & 58 & 10\% & 43\% \\
               &  15 dpa & 449  & 65 & & 379 & 55 & 11\% & 50\% \\
\\ \hline
\end{tabular}
\label{t-8-3-1}
\end{table}


The expansive force between the outboard TF horizontal legs can be found by
taking moments about the point where the horizontal leg of the TF coil meets
the vertical center-post.  This moment, $M$, is given by
%
\begin{eqnarray}
\label{e-8-3-1}
M & = & \int^{r_o}{r_{cp}}  \, f(r) \, (r_o - r) \, dr \: ,
\end{eqnarray}
%
where $r_{cp}$ is the radius of the center-post, and $r_o$ is the radius of the
outboard return current.  After integrating, the moment becomes:
%
\begin{eqnarray}
\label{e-8-3-2}
M & = & 10^{-7} I^2 \left[ r_o \,  \ln \left( \frac{r_o}{r_{cp}} \right) +
r_{cp} - r_o \right] \: ,
\end{eqnarray}
%
where $I$ is the total current. For $I=40$~MA, $r_{cp}=0.56$~m and $r_o=17$~m,
we calculate \hbox{$M=6.7\X 10^9$~Nm}.  The tensile force is then $M/r_o$ and
the average tensile stress on the center-post is \hbox{$M/(\pi r_{cp}^2
r_o)$}. In this case the stress is at 415~MPa.

Noting that the tensile stresses would be much larger at the connection point
between the outboard leg and the center-post, a sliding electrical connection
is provided between the outboard TF coil leg and the center-post.  The
expansive force of \hbox{$4\X 10^8$~N} would have to be reacted through
structure placed either within the inner shield or externally.

\paragraph{Sliding electrical connection.}

A sliding electrical connection would only have to be provided on one end of
the center-post with a rigid connection on the other end.  At this point only
one connection has been considered.  This joint may be a natural location to
attach the current leads to the TF coils.  If the leads are located elsewhere,
a conducting liquid such as mercury can be used to bridge the gap between the
conducting members.  In this case, we assume that a 0.5~cm gap exists between
each TF coil leg and the center-post.  Rolling bearings keep the TF coil legs
concentric with the center-post, and the gap is filled with mercury. Mercury
has a resistivity of 1.03~$\mu\Omega$-m at 100\SdegC and the contact area per
TF coil is \hbox{1.1~m $\X$ 0.66~m}.  The electrical resistance of the gap is
\hbox{$7\X 10^{-9}~\Omega$} ohms and the ohmic heating per coil is 50~kW.  The
total ohmic dissipation for the 16 coils is 0.8~MW. Several issues have to be
considered.  Mercury forms amalgams with many metals.  It may be necessary to
electroplate the contacting parts with a metal with which it does not react
with mercury.  The mercury would have to be contained within the gap and may
have to be periodically processed.  \hbox{Figure~\ref{sti-8-3-2}} shows a
possible design of a sliding electrical connection.

\begin{figure}[b]
\centerline{\psfig{file=sti-8-3-2.ps,height=4.in}}
\caption[sti-8-3-2]
{A possible design of a sliding electrical connection on the
upper end of the center-post.}
\label{sti-8-3-2}
\end{figure}


\subsubsection{Fabrication of the center-post}
\label{s-8-3-2-3}

In this design it is assumed that the center-post will be fabricated from the
dispersion strengthened copper C15725.  Present facilities for making
dispersion-strengthened copper are limited to about a length of $\sim$2.5~m
whereas the center-post is about 34~m long.  However, there is no reason why
larger pieces cannot be fabricated if appropriately sized heated pressure
vessels can be provided.

Dispersion strengthened copper is made from a pure copper powder mixed with
small quantities of AlO$_2$.  A form is made of OFHC copper, the powder is then
poured into the form and then heated while being subjected to a pressure of
$\sim$10~MPa.  The exact temperature and pressure values are proprietary and
are not precisely known.  Another method is called the internal oxidation
technique used by SCM Corp. in producing GlidCop~\cite{GlidCop}.  The patented
process begins with melting a dilute solid solution alloy of copper and
aluminum.  The melt is atomized into powder and used in forms as described
above.  During heating, the aluminum in the alloy is selectively oxidized
in-situ.  The powder is then consolidated into fully dense shapes by one of
several conventional metal working processes.

For making the center-post with the tubes in place, it would be necessary to
pre-locate the tubes in the form prior to filling the form with the
powder~\cite{Sol95}.  In this case, the tubes will also be made of
dispersion-strengthened copper.  The tubes can be shaped both in diameter and
configuration, then fixed in position by dispersion-strengthened copper straps.
It is possible that minor shifting of the tubes will occur during
consolidation.  This should not be a serious problem since copper has a very
high thermal conductivity.

It would also be possible to fabricate the center-post in several sections.
For example, the central portion can be fabricated separately, and each of the
end portions separately.  The three sections can then be diffusion bonded into
one whole.  Care will be needed to ensure that the tubes line up and that no
distortion will occur at the interfaces.

The center-post can also be fabricated from other copper alloys, but the design
would have to be different from that described above.  For example, the use of
concentric rings as proposed in Ref.~\cite{Hic91} can be fabricated from other
copper alloys.  Center-Post shaping for minimizing ohmic heating will be more
difficult to accomplish.

\subsubsection{Conclusions}
\label{s-8-3-2-4}

Based on these scoping calculations, the following conclusions have been
reached regarding the calculations of the mechanical behavior of the
center-post:

\begin{enumerate}

\item
The compressive stresses caused by the magnetic field probably require the
higher strength of dispersion-strengthened copper which has a higher electrical
resistivity than pure copper.

\item
The expansive forces on the outboard TF-coil legs cannot be reacted by the
copper center-post.  Either external clamping by the building structure or a
rather strong structure inside the center-post is required to avoid excessive
high tensile stresses in the copper.

\item
Fabrication of the center-post can be envisaged using present technology in
consolidating dispersion strengthened copper, assuming that sufficiently large
ovens will be available.  Alternatively, the center-post can be made in
segments which are diffusion bonded together.

\end{enumerate}


\subsection{Design Implications of Copper and Copper Alloy Embrittlement}
\label{s-8-3-3}

The design implications of a structural material which embrittles during
neutron irradiation are very subtle.  The issue is being addressed within ITER
by the combined efforts of the Interim Structural Design Criteria and Material
Properties Handbook Working Groups.  An elaborate set of design guidelines and
a material properties database have been published in draft form for austenitic
stainless steel [Type 316L(N)] and is currently under review.  With regard to
copper and copper alloys, the current draft of the Handbook contains properties
for Cu-Cr-Zr but not for pure copper or oxide-dispersion-strengthened copper
({\it e.g.,} \hbox{GlidCop Al-15}).  Design rules have not yet been formulated
for the use of copper and copper alloys in ITER. It should be noted, the
``licensing'' agencies (such as NRC) will ultimately develop or adopt the
design codes for fusion power plants.

Given the status of the ITER design and material properties efforts, the
approach taken in this assessment of the impact of copper embrittlement on the
design of the center-post magnet for a low-aspect-ratio tokamak design is to
parallel the rules developed for 316L(N) taking into account the similarities
and differences between steel and copper embrittlement.  In other words, design
criteria will be established for copper alloys which preserve the same design
margins or safety factors ($SF$) as have been established for 316L(N) for
average ($P_m$) and local ($P_l$) primary membrane stresses, for primary
bending stresses ($P_b$), for secondary stresses ($Q$) and for peak local
stresses ($F$) in the region of cracks, notches, {\it etc.}

The operating conditions for both types of materials are restricted in this
work based on the design applications.  For ITER 316L(N), the nominal operating
temperatures are 100--400\SdegC with short time excursions up to 500\SdegC.
For this evaluation it is assumed that in order to maintain high enough
electrical and thermal conductivity, the nominal operating temperature of the
copper is in the range 0--100\SdegC, with short time excursions to higher
temperatures possible.  Also, to focus this section, only the properties of
high purity copper (annealed) and \hbox{GlidCop Al-15}
[hot-isostatically-pressed (HIP)] are considered.

\subsubsection{Effects of embrittlement on design criteria}
\label{s-8-3-3-1}

Pure copper, \hbox{GlidCop Al-15}, and 316L(N) are all ductile materials at the
beginning of life in a fusion environment.  Conventional design rules ({\it
e.g.,} ASME Boiler and Pressure Vessel Code~\cite{ASME95}) may be used to
establish temperature and stress limits to achieve the desired lifetime of a
component.  Although the strengths of these materials increase with
radiation-induced hardening, the allowable loads must be determined based on
the softer unirradiated properties to cover the beginning of life (BOL)
operation.  The design challenge is to demonstrate that these design-allowable
loads lead to acceptable safety factors as the material hardens.  This is
clearly no problem with regard to strength as both the yield (YS) and ultimate
tensile (UTS) strengths increase with neutron fluence.  However, these strength
increases cannot be relied on from a design perspective as short-term
over-heating during off-normal events, as well as cycling during normal
operation, can soften the irradiated material.  Rather, the challenge is to
demonstrate that the decrease of uniform elongation (UE) and local failure
elongation [as measured by reduction in area (RA) in the neck region of a
failed tensile specimen] due to embrittlement do not lead to unacceptable
safety factors.  Because ductility is not as recoverable with over-heating as
the strength properties, the design-conservative approach of loss of ductility
with no increase in strength is adopted for these materials.

For the materials of interest in the established temperature ranges, the
uniform elongation drops to below 1\% at neutron damage levels $>0.01$ dpa for
copper and copper alloys \cite{Fab95,Fab95A} and \hbox{$\geq 7$~dpa} for
316L(N)~\cite{Bil95}.  From a design viewpoint, both materials would be
classified as brittle with such low values of uniform elongation.  Clearly, the
copper and copper alloys embrittle at much lower dpa values than the ITER
316L(N) steel.  Also, 316L(N) has another advantage over copper.  While the
uniform elongation decreases significantly with dpa, the local failure
elongation (as measured by RA) remains high, at least up to 10 dpa.  This is
important with regard to low-cycle fatigue strength and blunting of stress
concentrations at crack tips.  Although no direct RA data were found for
irradiated copper and copper alloys, the decrease in total elongation (TE),
particularly for the dispersion-strengthened alloys (\hbox{${\rm TE} \sim 2\%$}
at \hbox{$\geq 0.01$~dpa}), suggests that the local failure strain also
decreases significantly with neutron exposure.  Thus, copper-alloy
embrittlement in a low-aspect-ratio tokamak presents two challenges above and
beyond those for ITER 316L(N): the rate of embrittlement is much higher and
both the uniform and local failure elongations decrease to very low values.

The approach to incorporating embrittlement into the center-post design
criteria is to establish design acceptable stresses based on unirradiated
properties and to insure that adequate design margins to failure are preserved
as the material embrittles.  If this is not the case, then either the design
lifetime must be limited and/or the allowable stresses must be reduced until
the design margins are satisfied.  For the ITER 316L(N), it has been shown that
the conventional design rules based on unirradiated properties lead to a
minimum safety factor of 3 for primary membrane stresses ($P_m$) and a safety
factor of 2 for local primary membrane plus primary bending stresses
($P_l+P_b$) even with irradiation embrittlement~\cite{Bil95A}. The safety
margins for secondary stresses ($Q$) and peak stresses ($F$) have not yet been
established for irradiated 316L(N), and thus will have to be treated separately
for the copper alloys.

\subsubsection{Design stress intensity}
\label{s-8-3-3-2}

The design stress intensity ($S_m$) is a tabulated parameter in design codes
for code-qualified structural materials.  For ductile materials such as
unirradiated copper and copper alloys, it is generally determined from
%
\begin{eqnarray}
\label{e-8-3-3}
S_m & = & {\rm Min} (2/3 S_{yd}, 1/3 S_{ud})
\end{eqnarray}
%
where $S_{yd}$ and $S_{ud}$ are design yield and ultimate strengths based on
minimum room temperature values and normalized average temperature dependence.
For low temperature operation and simple design rules, $S_m$ is sometimes
listed as the design allowable stress with a limiting temperature of operation.
For the more elaborate design rules, $S_m$ is referred to as the design stress
intensity and the rules include criteria for higher temperature application.

\begin{figure}[b]
\centerline{\psfig{file=sti-8-3-3.ps,height=3.5truein,clip=}}
\caption[sti-8-3-3]
{Design stress intensity ($S_m$) values for pure copper (C10200 in
O60 temper) and \hbox{GlidCop Al-15} (HIP) in the unirradiated condition.}
\label{sti-8-3-3}
\end{figure}

It is important to be specific with regard to material, tempering treatment,
and component size when quoting a design stress intensity.  The Metals
Handbook~\cite{MetalHB} uses the UNS designations C10100 for Oxygen Free
Electronic (OFE) copper and C10200 for Oxygen Free (OF) copper.  The ASME
Boiler and Pressure Code~\cite{ASME95} has qualified C10200 as a
low-temperature structural material.  The $S_m$ values for this material in the
``O60'' temper condition are shown in Fig.~\ref{sti-8-3-3}.  The limiting
temperature of operation is listed as about 200\SdegC, because the tensile and
creep-rupture strengths of this material decrease dramatically above this
temperature.  The material can be hardened ({\it e.g.,} H80 tempering
condition) to produce significantly higher allowable stresses ({\it e.g.,} more
than double up to 180\SdegC).  However, the effects of this type of hardening
are lost above 200\SdegC and it is more suitable for wires and small diameter
tubes than for large complicated structures such as the center-post.  Also, the
ductility of hardened ({\it e.g.,} cold worked) copper is much lower than that
of annealed copper in the irradiated condition.  Thus, this strengthening
mechanism is not considered.  Rather, oxide-dispersion hardening is an
appealing option if higher strength is desired up to higher operating
temperatures.  The design stress intensity of the commercial alloy
\hbox{GlidCop Al-15} (UNS C15715)~\cite{GlidCop} is also shown in
Fig. \ref{sti-8-3-3}.  The particular values are for products produced by
hot-isostatic-pressing (HIP) which may be a practical fabrication technique for
the center-post.  Beyond the 500\SdegC limit of the horizontal axis in
Fig.~\ref{sti-8-3-3}, the GlidCop properties decrease linearly with
temperature, unlike the pure copper properties which decrease significantly
beyond 200\SdegC.  At 100\SdegC, $S_m=37.4$~MPa for pure copper (C10200 in the
O60 temper) and 108~MPa for \hbox{GlidCop Al-15} (HIP).

\subsubsection{Stress limits}
\label{s-8-3-3-3}

The design stress intensity, along with other limiting parameters, is used to
determine acceptable levels of average primary membrane stress ($P_m$), local
primary membrane stress ($P_l$), primary bending stress ($P_b$), secondary
stress ($Q$), and peak stress ($F$).  Safety factors ($SF$) for each loading
combination can be determined by ratioing the stress which causes failure to
the allowable stress.  These safety factors are explored as a function of
decreasing ductility in the following.  The average primary membrane stress
({\it e.g.,} wall-averaged hoop stress in a pressurized coolant tube) is
limited by \hbox{$P_m \leq S_m$}.  The safety factor for this type of loading
is
%
\begin{eqnarray}
\label{e-8-3-4}
SF_m & = & S_{ud}/P_m
\end{eqnarray}
%
and is always \hbox{$\geq 3$} for a material which hardens regardless of
ductility loss.  With the way that $S_m$ is defined for unirradiated 316L(N),
the $SF_m$ varies from 3 to 4 within the temperature range of 100 to 400\SdegC.
For irradiation-hardened 316L(N), the $SF_m$ can be \hbox{$\geq 5$}, but this
is not taken advantage of because of the possible loss of hardening with
cycling and temperature excursions.  A similar situation exists for pure copper
and \hbox{GlidCop Al-15}.

The design limit for primary bending stresses ({\it e.g.,} bending of a beam
due to mechanical or electromagnetic moments) is given in the design codes as
%
\begin{eqnarray}
\label{e-8-3-5}
P_l + P_b & \leq & K \, S_m \: ,
\end{eqnarray}
%
where $K$ is a shape factor for ductile materials ({\it e.g.,} $K=1.5$ for a
beam or plate with rectangular cross-section).

Two approaches have been used to account for the effects of irradiation-induced
hardening on this limit.  The first approach is to increase $S_m$ with strength
hardening and decrease $K$ based on the loss of plastic ductility.  This
approach is not recommended because of the possibility of temperature-induced
softening. The second approach is to define $S_m$ and K based on unirradiated
properties and to explore the effect of decreasing ductility, with no increase
in strength, on the safety factor $SF_{mb}$ for combined primary membrane and
bending loads.  In Ref.~\cite{Bil95A} it is shown that \hbox{$2 \leq SF_{mb}
\leq 4$} for 316L(N) within the temperature range of 100 to 400\SdegC.  As the
ductility drops to zero, the safety factor becomes 2 for all combinations of
primary membrane plus bending loads as long as $P_b>0$.  Sample results are
shown in Fig.~\ref{sti-8-3-4} for 316L(N) at 160\SdegC.  If the confidence
level in the design is based on the minimum safety factor, then the
embrittlement of the copper or copper alloy during irradiation will result in
the same safety factor as the minimum value accepted for 316L(N).  Until some
change is made in the design rules for ITER 316L(N), it is concluded that
designing the center-post based on the unirradiated properties of the material
will lead to acceptable safety factors for primary membrane and bending
stresses as the material loses ductility due to irradiation hardening.

\begin{figure}[t]
\centerline{\psfig{file=sti-8-3-4.ps,height=3.6truein}}
\caption[sti-8-3-4]
{Safety factors for primary membrane plus bending loads as functions of primary
membrane loads based on an elastic/linearly-plastic model.  The assumed design
properties correspond to those of unirradiated 316L(N) at 160\SdegC:
$S_{ud}=440.1$~MPa, $S_{yd}=163$~MPa, $S_m=147$~MPa, and $e_u=30\%$.  A
rectangular cross-section ($K=1.5$) is assumed.  The other two cases
($e_u=0.3\%$ and $e_u=0\%$) are presented to show the effects of decreasing
ductility on the safety factor.}
\label{sti-8-3-4}
\end{figure}

Secondary stresses ($Q$) such as thermal stresses are limited in combination
with the primary stresses.  The limit is designed to protect against local
ratcheting of the material and fatigue damage, and is given by
%
\begin{eqnarray}
\label{e-8-3-6}
P_l + P_b + Q & \leq & 3 S_m \: .
\end{eqnarray}
%
However, as the uniform and total elongation drop below about 1\%, this rule
cannot be used.  While ratcheting is not a problem for such a hardened
material, low cycle fatigue may be.  In particular, the fatigue damage is
assumed to scale with the square of the ratio of the unirradiated failure
ductility to irradiated failure ductility.  While these failure ductilities are
supposed to be determined from RA measurements, very little data were found in
the literature for the reduction in area as a function of neutron fluence.  For
the sake of a numerical example, though, the ratio of the total elongations is
used.  For annealed pure copper irradiated and tested at about 100\SdegC, the
total elongation decreases from about 50\% in the unirradiated condition to
about 10\% at 1 dpa.  Thus, the reduction in low cycle fatigue life is
estimated to be a factor of about 25 due to irradiation embrittlement.  For
\hbox{GlidCop Al-15}, the ratios of total elongations can be as much as a
factor of 10, leading to a reduction in low cycle fatigue life of as much as
100.  These examples both illustrate the potential reduction in low cycle
fatigue life for irradiated copper and copper alloys, as well as emphasize the
need for either fatigue data on irradiated samples and/or reduction-in-area
measurements for irradiated tensile specimens.

The stress limit criterion on the peak stress ($F$) in the area of cracks and
notches is designed to guard against rapid or instantaneous crack growth
leading to failure under static loading conditions.  It is expressed as a sum
of all of the stresses:
%
\begin{eqnarray}
\label{e-8-3-7}
P_l + P_b + Q + F ² S_l \: ,
\end{eqnarray}
%
where $S_l$ is a complicated function of a number of parameters and properties,
including the fracture toughness.  Although no fracture toughness data were
found for irradiated copper in the literature reviewed, based on the decreases
in both uniform and total elongation, it is anticipated that there will be a
significant reduction in fracture toughness and that this reduction may result
in further limitations on $P_m, P_l, P_b$ and $Q$.

In addition to fatigue considerations which represent time-dependent phenomena,
rules are available to protect against creep rupture.  However, if the
operating temperature is maintained to $\leq$100\SdegC for the
center-post design, then this high temperature phenomenon should not limit the
operating stresses beyond the limits already imposed.

In summary, the embrittlement of copper and copper alloys with neutron exposure
may introduce further limitations on design allowable primary and secondary
stresses because of reduction in low-cycle fatigue life and because of
reduction in fracture toughness with ductility loss.  To assess the magnitudes
of these impacts, it is important to have low cycle fatigue data and/or
reduction-in-area data for irradiated samples, as well as fracture toughness
data.  These limitations on allowable design stresses would have to be compared
to anticipated operating stresses.  For very low operating stresses and/or very
short design lifetime, the additional limitations may not pose a problem.

\subsubsection{Deformation limits}
\label{s-8-3-3-4}

In addition to the traditional stress limits, deformation limits may pose
additional constraints to the design-allowable operating stresses.  Such limits
are highly design dependent.  Generic ones which have been developed for
scoping studies are: $\leq$5\% volumetric swelling, $\leq$5\% irradiation
creep, and $\leq$1\% thermal creep.  For operating temperatures
$\leq$100\SdegC, none of these should be design-life-limiting beyond limits
imposed by the effects of transmutation products on the electrical
conductivity.  However, the swelling criterion needs to be closely evaluated in
the light of possible transient overheating events.  With enough transmutation
helium present, 5\% volumetric swelling could be achieved during short-time,
high-temperature transients.  These He bubbles would essentially be frozen into
the material upon return to the low operating temperatures.

\subsubsection{Limits based on high-strain-rate events}
\label{s-8-3-3-5}

Most of the above design limits assume strain-rate-independent properties and
operating conditions.  High strain rate events ({\it e.g.,} disruptions acting
on the first wall or divertor) could cause problems with embrittled irradiated
material.  To assess this issue, fracture toughness data under high strain rate
loading should be gathered for irradiated copper and copper alloys.  For
scoping tests, the Charpy impact test may be used.

\subsubsection{Summary}
\label{s-8-3-3-6}

Developing design criteria for materials which are ductile in the unirradiated
condition, but embrittle significantly with neutron exposure, is an evolving
art.  The criteria which have been developed within the ITER design and
materials groups for 316L(N) stainless steel have been used to guide the
information presented in this section.  While both 316L(N) and copper alloys
increase in strength and decrease in uniform elongation with neutron exposure,
the copper alloys present a particular challenge because of their faster rate
of embrittlement and their decrease in local failure strain, as well as uniform
elongation.  The approach to developing design criteria is based on using the
softer unirradiated properties to set operating stress limits and then to track
the factors of safety with fluence to determine at what level of dpa
embrittlement causes unacceptable decreases in safety factors or design
margins.  This was done explicitly for primary membrane and bending stresses,
for which it was concluded that copper embrittlement would not be a problem.
However, it was established that the embrittlement may result in either a
decrease in lifetime or a decrease in allowable operating stresses based on low
cycle fatigue considerations, rapid crack growth under static loading and crack
growth under high strain rate loading.  To assess the impact of these possible
design restrictions, additional data need to be gathered or generated on low
cycle fatigue and/or local tensile failure strain for irradiated copper, as
well as fracture toughness for a wide range of strain rates for irradiated
copper.

Material properties and design limits alone do not preclude the use of a
material in a fusion application such as the center-post of a low-aspect-ratio
tokamak.  These must be compared to anticipated operating stresses.  If the
operating stresses are low enough and/or the design lifetime is short enough,
then the design margins may be high enough to justify the use of materials such
as copper and copper alloys even with their loss in ductility with neutron
exposure.

\subsection{TF-Coil Power Supply and Busbar}
\label{s-8-3-4}

\subsubsection{Power supply}
\label{s-8-3-4-1}

The use of a single-turn TF coil system raises several concerns regarding power
supplies that can provide high current at low voltage, and regarding the
potentially large losses in lead wires and joints.  The TF coils of
low-aspect-ratio tokamak devices are different from smaller normal conducting
magnets in that they have a large ratio of stored energy to dissipated power.
This is a consequence of the large size of the magnet, since the energy scales
with the cube of the linear size while the dissipated power scales with the
square of the linear size (for comparable fields).  This fact may have a large
effect on the choice of the power supply.  For the low-aspect-ratio tokamak
strawman, the energy stored is about 9~GJ, while the dissipated power is about
1,500~MW.  It is straightforward to determine the time constant for the system,
%
\begin{eqnarray}
\label{e-8-3-8}
\tau & = \frac{L}{R} \: = \: \frac{2 \, E}{P}
\end{eqnarray}
%
where $L$ is the inductance, $R$ is the resistance, $E$ is the energy, and $P$
is the resistive power.  For the low-aspect-ratio tokamak strawman,
\hbox{$\tau \sim 10$~s}.

If fully-rectified 60~Hz is used as the driver, the resulting ripple is
$\lesim$0.1\% (without any filtering or regulation).  Therefore, there is no
need for regulation, and the power supply basically is reduced to a transformer
and one or several diodes, using the line power as the driver.  Typical 25-kA
transformers are large and heavy, but inexpensive. Each power supply module has
a 25-kA capability, driving its own busbar (either normal conducting or
superconducting, as described above).  For the strawman, there are 1,200 power
supplies.  Due to the simplicity of the power supply, the cost of the power
conditioning equipment is not a dominant cost of the device.  The system looks
very similar to those used in large scale electrolytic processes for the
production of aluminum, although with much larger scale.

Alternatively, a switching power supply could be used at higher frequencies,
reducing the size of the transformer but increasing the cost of the driver.
These switching power supplies have been developed recently to deliver very
high specific power, on the order of 3~MW/m$^3$.  Estimated costs for small
units (2~kW) are \$750.  The power supplies include protection, stability and
regulation.  Therefore, if 150,000 of these units are used, the maximum cost of
the power supply would be \$110M.  Larger units are being developed, and
advantages of scale, together with larger units with combined controls, could
reduce the cost of the power supply to less than \$50M.  This power supply
design is much more compact but more expensive than the 60-Hz transformers.
Because of the reduced size, it would be possible to place the power supply
closer to the reactor, decreasing the required busbar length and associated
costs.

\subsubsection{Busbar concerns}
\label{s-8-3-4-2}

There is a tradeoff between the size of the busbar and the dissipated power,
for room temperature busbar.  This tradeoff is shown in Figure~\ref{sti-8-3-5}
as a function of the cross sectional area of a room temperature copper busbar.
Very large costs are associated with the busbar and dissipated power.  It is
assumed that the copper in the busbar cost is \$20/kg and that the busbar is
50~m long.  The cost of the electricity is 90~mills/kWh, the plant operates for
30 years with a 80\% capacity and the interest rate is 5\%.  Means of
decreasing the costs are analyzed in the following sections.

\begin{figure}[tb]
\centerline{\psfig{file=sti-8-3-5.ps,height=3.8in}}
\caption[sti-8-3-5]
{Amortized costs of busbar and power as a function of busbar area.}
\label{sti-8-3-5}
\end{figure}

\paragraph{Cryogenic busbar.}

There are large decreases in the resistivity of materials with decreasing
temperature.  These advantages, however, are lost in the case of magnets due to
the substantial magnetoresistivity of most materials.  However, for busbars
with small magnetic fields, it is possible to obtain large decrements in power
dissipation (ohmic and cryogenic losses) by decreasing the temperature of
operation of the busbar, as long as the busbar is broken into a large number of
current leads and each lead is placed next to other ones with opposite
currents, in order to decrease the value of the magnetic field generated by the
leads.  Concentric leads, of course, satisfy the latter requirement.  The
largest gains are obtained at temperatures in the range of 20 to 90~K.
However, as shown in the next section, it is possible to have superconducting
leads at these temperatures, and therefore we only analyze that case.

\paragraph{High-$T_c$ superconducting busbar.}

Work in progress at MIT with Hoechst, Los Alamos, American Superconductor, and
others indicate that superconducting current leads operating at 77~K (liquid
nitrogen) can be manufactured.  A collaboration between MIT and Hoechst in
Frankfurth, Germany, has tested current leads that, at 77~K, can carry up to
16~kA at a current density of 1.5~kA/cm$^2$ (at self-field).
\hbox{Figure~\ref{sti-8-3-6}} shows the voltage per unit length of a 75-mm
outer diameter (OD) hollow cylinders (5~mm thick), tested both at 77~K and at
66~K (obtained by pumping on liquid nitrogen).  The maximum current of 16~kA is
due to power supply limitations.

\begin{figure}[tb]
\centerline{\psfig{file=sti-8-3-6.ps,height=3.4truein}}
\caption[sti-8-3-6]
{Electric field of high-$T_c$ Superconductor vs current.}
\label{sti-8-3-6}
\end{figure}

The resistive losses of the silver/superconductor joints of the same sample was
measured to be less than 0.4~mW.  The specific, resistive loss of the joints
is about 2~mW/cm$^2$.  For the current leads that are being discussed, a joint
area of about 250~cm$^2$ is feasible by making the conducting region 10~cm
long.

Figure~\ref{sti-8-3-7} shows a schematic diagram of the analyzed joint.
Hoechst together with MIT have developed reliable metallurgical bonds between
the high-temperature superconductor and the silver contacts.  For a
50-m busbar length, the power dissipated in the superconducting busbar is
150~kW, with a room temperature refrigeration of about 1~MW.  The dissipation
in the joints between the superconductor and the copper leads is minimum.

\begin{figure}[tb]
\centerline{\psfig{file=sti-8-3-7.ps,height=4.0truein}}%width=6.25truein}
\caption[sti-8-3-7]
{Schematic diagram of a joint, showing metallurgical and solder bonds.}
\label{sti-8-3-7}
\end{figure}

The present day cost of the high temperature superconductor current leads
(hollow cylinders), offered commercially by Hoechst, is \$500 for a tube that
is 0.4~m long.  The low-aspect-ratio tokamak strawman requires about 30~MA, and
with 25~kA busbar current, 1,200 current lead pairs are needed.  The large
number of current lead pairs is also needed in order to minimize the self field
of the busbar.

In collaboration with Hoechst it has been determined that the hollow cylinders
of 2212~BSCCO can be built with lengths on the order of 8~m (30~ft) for about
twice the unit price of the 0.4-m lead (the bulk of the cost being the
setup/testing time).  Several of these cylinders can be connected in series in
order to span the distance between the power supply and the nuclear
island. Currently, the superconducting leads are being investigated as a means
of carrying the current from the generator to the transformer in present-day
power stations.

The cost of the busbar is on the order of \$1.2~M/m, if 0.4~m long sections are
used, or \$60~M for a 50-m busbar length.  These tubes are commercially
available now.  If 8-m tubes can be manufactured, then the cost would be
\$300~k/m, or \$15M for a 50-m-long busbar.

In order to minimize self-field, conductors with opposite currents need to be
placed near each other.  The case that was studied had concentric hollow
cylinders, with the magnetic fields exclusively in the region between the
tubes.  The space between the tubes is used for coolant.  The evaporation of
liquid nitrogen was determined to be several liters per hour of operation.  The
overall size of the bus bar, even with the gaps for liquid nitrogen, is much
smaller than that for normal conducting busbars.  Also, the power dissipation
is virtually eliminated.


\subsection{Shielding Options, Radiation Effects, and Activation}
\label{s-8-3-5}

This section covers the neutronics, activation, and shielding requirements for
the low-aspect-ratio tokamak power plant.  A scoping analysis was performed for
a preliminary configuration in order to guide the design toward a final
configuration that satisfies all the top level requirements developed by the
Starlite team for U.S. fusion power plants.  Generally, U.S. designs emphasize
safety and environmental aspects of fusion, providing a strong incentive to
generate low rad-waste, not greater than Class C.

The center-post presents important design problems to designers.  Therefore,
most of the neutronics, activation, and shielding activities were devoted to
the center-post and little attention was given to the outboard and divertor
regions.  Due to the severe radiation environment, it is important to identify
potential radiation-induced effects and failure mechanisms so that the
center-post lifetime could be reasonably predicted.  In addition, changes in
the electrical resistivity of the center-post and the volume of waste produced
have dramatic impact on the design.

In order to identify the potential neutronics-related issues, we first
considered a preliminary case with no intervening shield between the first wall
and the center-post.  This center-post is this configuration has a minimum
diameter of 0.874~m composed of 70\% dispersion-strengthened copper alloy with
0.25~wt.\% aluminum (dispersion-strengthened \hbox{GlidCop~Al25}) and cooled
with 30\% water at an average copper temperature of 75\SdegC.  The maximum
temperature of copper is kept at roughly 110\SdegC.  The machine planned
operation is 40~y (30 full power years, FPY) with an average neutron wall
loading of 4.1~MW/m$^2$.  The peak inboard wall loading occurs at the mid-plane
at a value of 3.6~MW/m$^2$.  The total height of the center-post for this
preliminary case is 16 m and no inboard shield was placed between the plasma
and coil, except for a 1.4 cm thick heat removal surface.

The neutronics model employed the 1-D transport code ONEDANT~\cite{Ode82} with
the P3 Legendre expansion for the scattering cross section and the S8 angular
quadrature set.  The associated 46 neutron and 21 gamma group cross section
library was derived from the ENDF/B-V evaluation.  The transport problem is
modeled in toroidal cylindrical geometry around the machine axis and
calculations are carried out to investigate the various responses of interest.
For the center-post, there are concerns with mechanical degradation of copper
due to radiation damage, increase in copper resistivity due to neutron-induced
transmutations, and activation of copper leading to an unacceptable level of
radioactive waste.  The results reported below illustrate how severe the
radiation damage and activation levels are for the unshielded center-post.

\subsubsection{Waste-disposal requirements}
\label{s-8-3-5-1}

The goals and objectives developed by the Starlite team for the fusion power
plants clearly state that all designs must demonstrate that fusion is a safe,
environmentally attractive, and licensable power source that is ready for
commercialization at an economically superior total cost.  This led to a set of
requirements on materials and technology choices and design specifications.
Among these requirements, fusion plants should generate only low-level waste
and all waste can be recycled and/or disposed of at an acceptable level.

The safety and environmental requirements mandates that all in-vessel
components generate no radioactive waste greater than Class-C level.  Vanadium
alloys and low-activation ferritic steels have the potential of meeting these
requirements if employed for first-wall, blanket, and shield operating at a
moderately high fluence of 12~MWy/m$^2$ or less.

Fusion power plants in U.S. will probably be licensed and regulated by the
Nuclear Regulatory Commission (NRC) and other environmental agencies.  Thus
far, the NRC guidelines for Class C waste currently documented in the 10CFR61
report~\cite{10CFR61} were basically developed for nuclear waste generated by
medical and industrial facilities, and fission power plants.  Therefore, NRC
regulations do not contain some materials that may be of interest to fusion,
such as V, W, Si, Sn, ..., {\it etc.}  A complete list of rad-waste limits
specifically for commercial fusion plants will be eventually developed or
adopted by NRC.  Several years ago, Fetter~\cite{Fet90} used and modified the
NRC methodology to develop waste-disposal limits for a wide range of
radionuclides which are of interest to fusion researchers.  He also used
different conversion factors and transfer coefficients and limited the dose to
the whole body rather than to a single organ, as in NRC.  In addition, he used
his work to re-evaluate the isotopes that have already NRC limits and, in some
cases, arrived at different limits than those of NRC.

Development of waste-disposal limits is outside the scope of this research.
Therefore, the philosophy adopted by our Starlite team is that all the
in-vessel components should meet both NRC and Fetter's limits for Class-C
waste.  In fact, all previous ARIES designs have adopted the same philosophy
and we plan to continue imposing both limits on all ARIES/Starlite designs
until NRC develops official guidelines for fusion plants.

For the low-aspect-ratio tokamak design under consideration, the activation
analysis has indicated that the unshielded center-post will not qualify as
Class C waste under neither NRC nor Fetter's waste-disposal limits, unless the
center-post lifetime is limited to 0.12 FPY ($\sim$2 calendar months).  To
prolong the center-post lifetime, a 20--30-cm thick inboard shield is needed
for 4 years or more of operation.  The option of replacing the unshielded
center-post every 2 months is not acceptable for power plants due to the loss
in availability and the generation of large amount of radioactive waste.  The
unshielded center-post option could, however, be acceptable for experimental
devices operating at much lower fluences and not bound by the Class C waste
requirement.  In addition, this shield helps improve the design in many aspects
(Sec.~\ref{s-8-3-1}).  Hence, the shielded center-post is the reference design
for the low-aspect-ratio tokamak machine.

The low-aspect-ratio tokamak machines will generate more rad-waste compared to
conventional-aspect-ratio tokamaks.  The low-aspect-ratio tokamak designs are
highly elongated (30~m for low-aspect-ratio tokamak compared to $<$15~m for
conventional-aspect-ratio tokamaks) resulting in a much larger waste stream for
the low-aspect-ratio tokamak in-vessel components, especially when considering
the disposal of the massive center-post ($\sim$300~tonnes) every $\sim$4~years.
It is true that recycling helps reduce the waste, but this will come at a much
larger cost for the low-aspect-ratio tokamak machines.

\subsubsection{Waste-disposal rating analysis}
\label{s-8-3-5-2}

The activation calculations were conducted using the DKR-ICF computer
code~\cite{Hen87} with activation cross sections taken from the
USACT93~\cite{Man91} library and radial neutron flux distribution generated by
the transport code ONEDANT.  The neutron transmutation data used is in a
46-group structure format.  The center-post activation results were utilized in
the rad-waste classification.  The rad-waste of all alloys were evaluated
according to both NRC 10CFR61~\cite{10CFR61} and Fetter~\cite{Fet90} waste
disposal concentration limits.  The copper alloy used in the activation
analysis is the dispersion-strengthened alloy GlidCop Al25 and the poloidal
average neutron wall loading over the center-post is 2.5~MW/m$^2$.  The Class-C
waste-disposal ratings of the center-post as a function of operation time are
as shown in \hbox{Table~\ref{t-8-3-2}}.

\begin{table}[tb]
\caption{}
\centering{\bf Class-C Waste-Disposal Rating$^{(a)}$ for \\
the Dispersion-Strengthened Glidcop Al25 Center-Post \\
(2.5~MW/m$^2$ and 1 y Cooling-Off Period)}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c @{\hspace{1.cm}} c @{\hspace{1.cm}} c}
\hline
Time of Operation & Fetter Limits & NRC Limits \\
\hline \\
 6 months         & 2.64$^{(b)}$  & 4.11 \\
 1 year           & 4.94$^{(c)}$  & 8.53 \\
 3 years          & 12.2$^{(d)}$  & 29.1 \\
\\ \hline
\multicolumn{3}{l}{${(a)}$Dominant isotope is given in parenthesis.} \\
\multicolumn{3}{l}{${(b)}$Breakdown: 2.56 ($^{108m}$Ag) and 0.081 ($^{26}$Al).}
\\
\multicolumn{3}{l}{${(c)}$Breakdown: 4.76 ($^{108m}$Ag) and 0.16 ($^{26}$Al).}
\\
\multicolumn{3}{l}{${(d)}$Breakdown: 11.6 ($^{108m}$Ag) and 0.48 ($^{26}$Al).}
\end{tabular}
\label{t-8-3-2}
\end{table}

In order to be qualified for Class-C waste disposal, the waste-disposal rating
(WDR) should be below unity. As shown in the table, the center-post will not
qualify for disposal as Class-C waste for operating times longer than a few
weeks to a few months, regardless of the limits used.  Note that $^{63}$Ni
(half-life of 100~y) dominates the WDR considering the NRC limits.  Longer
cooling off periods of 10~y or more will not change the WDR for such long lived
radionuclides.  The isotope $^{63}$Ni is produced directly from copper while Ag
is an impurity and Al is an additive to the GlidCop alloy.  Since $^{63}$Ni is
generated from copper, using different copper alloys would not make a
difference in the WDR.  Reducing the silver content in the copper alloy
(currently at 0.002~w/o) by an order of magnitude would allow for the disposal
of the center-post after 1~FPY as Class-C waste according to Fetter's limits
only.  Table~\ref{t-8-3-2} indicates that the NRC limits are more stringent
and will thus be used to determine the center-post lifetime and shielding
requirements.  In order to dispose of the center-post as Class C waste, the
center-post should be subjected to a fluence $\lesim$0.3~MWy/m$^2$.  Since the
economics mandate operating at high neutron wall loading, it is therefore
necessary to either limit the center-post lifetime to 0.12 full power year
(FPY), or shield the center-post for longer life.  It was estimated that in
order to replace the center-post every 3~FPY (same as blanket) or more, the WDR
should drop by a factor of $>$30.  This could be achieved by placing 20 to
30~cm of shield between the plasma and center-post.  The shielded center-post
will thus have an end-of-life fluence of 0.3~MWy/m$^2$ or less.

To conclude, only two options seem to satisfy the Class-C waste requirement.
The first option is to limit the lifetime of the unshielded center-post to
0.12~FPY.  The second option is to shield the center-post with 20 to 30~cm
thick shield to prolong the lifetime to 3 FPY or more.  Both options will
alleviate most of the radiation damage problems encountered in the highly
irradiated center-post.  However, the unshielded case is penalized for
generating a lot more rad-waste and having longer machine down time ({\it
i.e.,} lower availability).

\subsubsection{Irradiation Effects}
\label{s-8-3-5-3}

The lifetime of the various components is determined by the design criteria and
requirements developed by the Starlite team for U.S. commercial power plants.
These requirements include structural integrity, Class C waste, and radiation
damage.  The high radiation environment in the low-aspect-ratio tokamak machine
mandates that some of the in-vessel components need frequent replacement during
the 40-y plant life due to radiation damage considerations.  These components
are the blanket, divertor, and center-post.  The blanket (and divertor) useful
lifetime is determined by the integrity of the main structural material.  The
criterion adopted in this study is that no more than 150~dpa is desirable for
ferritic steel and vanadium structures.  This corresponds to an end-of-life
fluence at $\sim$12~MWy/m$^2$ for a typical first wall neutron spectrum.  This
means that for an outboard neutron wall loading of 4--6~MW/m$^2$ the blanket
(and divertor) lifetime is in the order of 2--3 operational years at full
power.

Four radiation effects are likely to degrade the performance and thus limit the
life of the center-post, namely the embrittlement of copper, changes in copper
resistivity due to transmutations, neutron-induced swelling, and activation of
copper leading to high level waste.  At present, the dominant life limiting
factor for the center-post is found to be the activation of copper.  The highly
irradiated unshielded center-post appears to have a lifetime of 2 months,
limited by the Class C waste disposal requirement.  To prolong the coil life
and thus increase the plant availability, the center-post should be shielded.
It is estimated that a 20--30 cm shield would allow the center-post to function
for 4 years or more.

\paragraph{Radiation-induced changes in copper resistivity}

Neutron irradiation will increase the copper resistivity due to the production
of neutron-induced transmutations, defects, and dislocations.  The resistivity
effect due to radiation damage is small compared to that due to
transmutation~\cite{Elg83} because of the fact that a large fraction of the
defect-induced resistivity is expected to anneal out at the operating
temperature of the center-post ($\sim$100\SdegC).  Moreover, the
defect-induced resistivity saturates at low fluence while the
transmutation-induced resistivity scales somewhat linearly with fluence.
Increasing resistivity with time means increasing power dissipation in the
coil, leading to higher recirculation power, lower net efficiency, and a larger
machine required to offset the effects of higher recirculating power and lower
net thermal efficiency.

At any instant of time, the two stable isotopes of copper (69.2\% $^{63}$Cu and
30.8\% $^{65}$Cu) interact with neutrons and give rise to transmutation
products.  The reactions of interest are Cu(n,H)Ni, Cu(n,$\alpha$)Co,
Cu(n,$\gamma$), and Cu(n,2n).  The last two reactions produce unstable copper
isotopes that $\beta$ decays to Zn and Ni stable isotopes.  The time-dependent
inventory of both stable and unstable Ni, Co, and Zn isotopes is computed using
the activation code DKR-ICF.  After 0.12~FPY (the maximum lifetime allowed for
the unshielded center-post to meet Class-C waste-disposal limits), the peak and
average nuclide productions are listed in \hbox{Table~\ref{t-8-3-3}}.  The peak
values represent the maximum values obtained at the center-post plasma facing
surface having a peak wall loading of 3.6~MW/m$^2$.  The average values are
calculated for an average wall loading of 2.5~MW/m$^2$ and averaged radially
over the center-post assuming a radius of 0.51~m.

\begin{table}[b]
\caption{}
\centering{\bf Amount of Transmutation Products (appm) Produced \\
in the Unshielded Center-Post after 0.12 FPY}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c @{\hspace{1in}} c @{\hspace{1in}} c}
\hline
Nuclide & Peak     & Average \\
\hline \\
Ni      &  2,108\D &   360\D \\
Co      & \D\N23.6 & \N\N2.6 \\
Zn      &  1,258   &   259\D \\
\\ \hline
\end{tabular}
\label{t-8-3-3}
\end{table}

The inventory of transmutations that is used to calculate the change in the
center-post copper resistivity should be based on the average value, not on the
peak.  The average value is quite lower (factor of $\sim$6) than the peak which
occurs at the mid-plane outermost layer of the center-post.  The difference is
due to the exponential attenuation of the neutron flux with radial distance
through the coil and the vertical drop of the neutron wall loading over the
height of the center-post.  As shown later, the outer-most layer of the
center-post exhibits a large inventory of $^{62}$Ni, $^{64}$Ni and Co produced
by (n,2n) and (n,$\alpha$) reactions which have threshold energies of 9 and
5~MeV, respectively.  These reactions decrease rapidly within the coil.  The
resulting resistivity change in the copper conductor can then be computed from
the transmutations using a linear combination of the induced resistivity from
the different elements as given by the relation:
%
\begin{eqnarray}
\label{e-8-3-9}
\rho & = & \rho_{Cu} + \Sigma_i \rho_i A_i \: ,
\end{eqnarray}
%
where $\rho_i$ and $A_i$ are the specific resistivity per atom percent and the
atom percentage of the transmutation product $i$, respectively.  The room
temperature resistivity of copper and specific resistivities of Ni, Co, and Zn
are \hbox{$1.71\X 10^{-8}$}, \hbox{$1.12\X 10^{-8}$}, \hbox{$6.4\X 10^{-8}$},
\hbox{$3\X 10^{-9}$~$\Omega$-m}, respectively~\cite{Zin95}.

The percentage increases in copper resistivity due to Ni, Co and Zn
transmutations are listed in \hbox{Table~\ref{t-8-3-4}}.  Nickel has the
highest impact on copper resistivity followed by Zn and then Co.  The average
change in copper resistivity amounts to $\sim$3\% implying that the power
dissipation in the unshielded center-post due to transmutation-induced
resistivity is small compared to the intrinsic ohmic losses in the center-post.
The same conclusion holds true for the case of shielded center-post which is
subjected to $\sim$0.3~MWy/m$^2$ fluence with a much softer neutron spectrum.
The softer spectrum will result in a significant reduction in the high energy
neutron-induced transmutations generated by the (n,2n) and (n,$\alpha$)
reactions.  Thus, the shielded center-post will experience even lower change
($<3\%$) in resistivity due to transmutations.

\begin{table}[b]
\caption{}
\centering{\bf Increase in Copper Resistivity of Unshielded Center-Post
At 0.12 FPY}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c @{\hspace{1.in}} c @{\hspace{1.in}} c}
\hline
Nuclide & Peak      & Average \\
\hline \\
Ni      &  13.8\%\Z & 2.36\%  \\
Co      & \Z0.88\%  & 0.1\%\Z \\
Zn      & \Z2.21\%  & 0.45\%  \\
\\ \hline
\end{tabular}
\label{t-8-3-4}
\end{table}

Table~\ref{t-8-3-5} gives the detailed nuclide (impurity) production for the
unshielded center-post.  Since the nuclide production is given per unit
fluence, the peak values are only representative of surface values (local
values) and the average values are representative of values averaged over the
total center-post thickness.

\begin{table}[p]
\caption{}
\centering{\bf Nuclide Production (appm per MW-y/m$^2$)
For Unshielded Center-Post}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{1.in}} c @{\hspace{1.in}} c}
\hline
Nuclide             & Peak          & Average \\
\hline \\
$^{60}$Ni           &    \D\Z\Z2.7  &      \D\Z\Z0.4  \\
$^{61}$Ni           &    \D\Z\Z6.6  &      \D\Z\Z0.6  \\
$^{62}$Ni           &      1,312\D  &        \D203\D  \\
$^{63}$Ni           &      \D262\D  &       \D\Z47\D  \\
$^{64}$Ni           &      3,297\D  &        \D937\D  \\
{\bf{\it Total} Ni} & {\bf 4,880\D} &   {\bf 1,190\D} \\
\\
$^{59}$Co           &      19.6     &      3.0  \\
$^{60}$Co           &      35.1     &      5.6  \\
{\bf{\it Total} Co} & {\bf 54.7}    & {\bf 8.6} \\
\\
$^{64}$Zn           &        1,929  &        550\D  \\
$^{65}$Zn           &      \D\Z\Z7  &      \Z\Z1.0  \\
$^{66}$Zn           &        \D974  &        306.5  \\
$^{67}$Zn           &      \D\Z\Z3  &      \Z\Z0.6  \\
{\bf{\it Total} Zn} & {\bf 2,913\D} & {\bf 858\D\Z} \\
\\ \hline
\end{tabular}
\label{t-8-3-5}
\end{table}

\paragraph{Atomic displacement in copper.}

There is serious concern over ductility degradation of the copper conductor
under irradiation.  The dispersion-strengthened GlidCop Al25 alloy proposed for
the center-post undergoes rapid radiation hardening and loss of ductility at
irradiation temperatures below 200\SdegC.  Recent irradiation data showed rapid
embrittlement of all copper alloys and complete loss of work hardening by 0.1
dpa~\cite{Mat95}.  The 0.1~dpa is reached for the unshielded center-post after
one day of operation.  For longer times of operation, the copper will be
extremely brittle.  Thus, the copper will have no structural integrity and
design rules for brittle material should be used in designing the center-post.
The radiation damage to the unshielded center-post is given in
\hbox{Table~\ref{t-8-3-6}} for an arbitrary operation time of 1~FPY.  The peak
values are evaluated at the mid-plane for a peak inboard wall loading of
3.6~MW/m$^2$.  The average values are 4 to 8 times lower than the peak.  This
reduction is due to the radial attenuation of the neutron flux over the 50~cm
thick center-post and the vertical drop in wall loading over the inboard side.
At 0.12~FPY, the peak value is 6 dpa for an unshielded center-post.  Using a
20--30-cm-thick shield to protect the center-post will result in a peak value
slightly lower than 6~dpa for 3~FPY.  Other designs comprising different
materials and using different data libraries and geometric models may produce
different neutron spectra and different levels of radiation damage than the
values reported in \hbox{Table~\ref{t-8-3-5}}.

\begin{table}[b]
\caption{}
\centering{\bf Radiation Damage To Unshielded Center-Post}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{1.in}} c c}
\hline
                                & Peak     & Average$^{(a)}$ \\
\hline \\
Neutron wall loading (MW/m$^2$) & 3.6      & 2.5  \\
Atomic displacement (dpa/FPY)   & 48       &  8   \\
Helium production (appm/FPY)    & 540      &  130 \\
Hydrogen production (appm/FPY)  & 2,840    &  330  \\
Nuclear heating                 & 100~MW/m$^3$ & 160 MW \\
\\ \hline
\multicolumn{3}{l}{$^{(a)}$For a homogeneous mixture of copper and water
coolant.}
\end{tabular}
\label{t-8-3-6}
\end{table}

\clearpage
Present knowledge of the annealing process indicates that a significant
fraction of the radiation damage in the copper alloys could be removed by
periodic annealing at 200 to 300\SdegC for a few hours.  However, this might
not be feasible for power plants.  Periodic annealing for a few hours after
each day of operation (upon reaching 0.1~dpa) will certainly lower the
availability of the system.  Continuous annealing is also unacceptable due to
the increase in copper resistivity, neutron-induced swelling, and coolant
corrosion at high operation temperature (200 to 300\SdegC).

\paragraph{Swelling of copper.}

Experimental data shows a rapid increase in the GlidCop swelling rate in the
100 to 250\SdegC temperature range of interest (Sec.~\ref{s-8-3-3}).  Higher
temperature swelling behavior is not relevant to this low-aspect-ratio tokamak
design.  Table~\ref{t-8-3-7} lists the swelling rates at various temperatures
and the peak/average swelling for an arbitrary operation time of 1~FPY.  The
peak swelling occurs at the mid-plane where the wall loading peaks at
3.6~MW/m$^2$.  The average value is a factor of 6 lower than the peak.  At
0.12~FPY, the peak swelling is $<$2\%.  The shielded center-post should have
lower swelling rate.  It is recommended to keep the maximum temperature below
180\SdegC to limit the swelling of copper.

\begin{table}[b]
\caption{}
\centering{\bf Neutron-Induced Swelling In Dispersion-Strengthened GlidCop \\
For Unshielded Center-Post}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c}
\hline
Operating   & Swelling & Peak     & Average \\
Temperature & Rate     & Swelling & Swelling \\
\hline
\hspace{1.25in} &  \hspace{1.25in} & \hspace{1.25in} & \hspace{1.25in} \\
          $<$180\SdegC &       \Z\Z$<$0.019\%/dpa & \Z1\% & \Z0.16\% \\
\phantom{$<$}220\SdegC & \phantom{$<$}0.3\%/dpa   &  14\% &   2.5\%  \\
\phantom{$<$}250\SdegC & \phantom{$<$}0.5\%/dpa   &  24\% &   4.2\% \\
\\ \hline
\end{tabular}
\label{t-8-3-7}
\end{table}

\subsubsection{Outboard shield}
\label{s-8-3-5-4}

The need for water coolant for the center-post will most likely exclude the use
of liquid lithium as a breeder for the outboard blanket of the low-aspect-ratio
tokamak machine.  Serious safety concerns arise when large amounts of Li and
water coexist in a single machine.  While a multiple-barrier approach could be
applied to lessen the safety concerns, detailed assessments will be needed to
demonstrate the ability to meet the requirement for no evacuation at the site
boundary following any credible accident.

Other candidate breeders are lead-lithium, FLiBe, or solid breeders.  The later
will most likely requires beryllium multiplier to achieve tritium
self-sufficiency.  A 50-cm thick space is reserved on the outboard side for a
blanket which is followed by a shield, and then a vacuum vessel .  The shield
should be sized to mainly protect the vacuum vessel and ensure that the
neutron-induced helium production does not exceed 0.1 appm at the vacuum
vessel.  For costing evaluation, the systems code employed the ARIES-II
outboard Li/V blanket (although it may be incompatible with the water cooled
center-post) and the modified shield and vacuum vessel designs.  The system has
an overall neutron energy multiplication of 1.36 and a gross thermal efficiency
of 46\%.

\subsubsection{Conclusions}
\label{s-8-3-5-5}

Radiation-induced effects have a major impact on the center-post design of the
low-aspect-ratio tokamak power plant.  At present, the lifetime of the
center-post appears to be limited by the Class-C waste-disposal requirements.
As a consequence, the lifetime of the unshielded center-post is limited to
0.12~FPY (for a fluence of 0.3~MWy/m$^2$) or the center-post must be protected
with 20--30~cm of shield.  Since it is not feasible to replace 100~tonnes of
copper every 2~months, the center-post should be shielded to prolong the
lifetime to 4~years or more, reduce the cumulative rad-waste and replacement
cost, increase the system availability and alleviate most of the center-post
radiation damage problems.

We have assessed the effects of neutron fluence on conductor resistivity,
swelling, and atomic displacement.  The Ni and Zn transmutations will continue
to build up and the effect on copper resistivity could be significant.  For
example, after one FPY operation without an inboard shield, 0.3 at.\% Ni and
0.1 at.\% Zn would contribute 25\% additional ohmic losses.  Even though the
radiation-induced swelling and changes to copper resistivity due to
transmutations are small at 0.3~MWy/m$^2$, there is a serious concern on the
ductility degradation and the consequences of the high atomic displacement on
the conductor strength.  The copper will embrittle quite rapidly at 0.1~dpa
which is reached after one day of operation for the unshielded center-post and
after 1--2 months for the shielded center-post.  Designing a center-post with
extremely brittle copper alloy will be a challenging engineering task.

The shielded center-post is our reference design for the low-aspect-ratio
tokamak power plant.  The inboard shield helps improve the design in many
aspects.  The shield can be used to add mechanical support for the center-post
(Sec.~\ref{s-8-3-2}).  The nuclear heating ($\sim$150 MW) can be removed as
useful heat from the shield (instead of being dumped at low temperature) and
converted into electricity, reducing the center-post thermal hydraulic
problems.  The shield alleviates most of the center-post radiation damage
problems and helps tremendously to reduce the rad-waste stream, replacement
cost and change-out frequency, and increase the availability of the machine.
Overall, the center-post shield improves the low-aspect-ratio tokamak design
significantly.

\subsection{Blanket and Power Conversion Assessment}
\label{s-8-3-6}

\subsubsection{First wall and blanket design options}
\label{s-8-3-6-1}

The selection of the blanket option for the low-aspect-ratio tokamak design is
even more complicated than the process for a conventional tokamak for the
following reasons:

\begin{enumerate}

\item
The center-post is cooled by water.  This places severe design constraints on
design options with lithium as coolant and/or breeder.  Lithium has the best
potential for a high temperature system and has the capability of handling high
neutron wall loading.

\item
To reduce the capital cost, it will be attractive for a fusion power plant to
operate with a high neutron wall loading ($>5$~MW/m$^2$).  Solid breeder
blanket concepts have limited potential to handle neutron wall loading greater
than 3~MW/m$^2$.

\item
For a low-aspect-ratio tokamak design, the recirculating power can be high, due
to the current-drive power and the high Joule losses in the normal coil system,
including the center-post.  Therefore, an efficient power conversion system is
needed.  This recirculation power can be much higher than a conventional
tokamak.

\end{enumerate}

\begin{sidewaystable}
\caption{}
\centering{\bf Assessment of Blanket Options for a
Low-Aspect-Ratio Tokamak Power Plant$^{(a)}$}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{0.5cm}} c c c c c c}
\hline \\
Blanket concept     \\
\quad Structure &  SiC Composite
                           & V-alloy & V-alloy
                                         & Ferritic steel & Ferritic steel
                                                            & Ferritic steel \\
\quad Coolant   & He       & Li      & He       & LiPb    & He     & H$_2$O \\
\quad Breeder   & Li$_2$O  & Li      & Li       & LiPb    & Li$_2$O  & LiPb \\
\\
Blanket pressure&5--20~MPa & few atm & 5-20~MPa & few MPa &5--20~MPa & 20~MPa\\
Thermal Efficiency$^{(b)}$
                & 49\%    & 46\%     & {\bf 33\%} &  {\bf 33\%}   & 46\%
                                                            & {\bf 33\%} \\
\\
%\multicolumn{3}{l}{Issues Generic to low-aspect-ratio tokamak  Designs:} \\
\multicolumn{3}{l}{Neutron wall loading capacity} \\
\quad at 3~MW/m$^2$ & OK  & OK       & OK      & OK       & OK       & OK \\
\quad at 6~MW/m$^2$ & {\bf No} & OK  & OK      & {\bf No} & {\bf No} & OK \\
Safety concerns & -- & {\bf Li/water}
                               & {\bf Li/water}
                                     & --      &  --        & -- \\
\\ \hline
\multicolumn{7}{l}{$^{(a)}$\parbox[t]{7.5truein}{Issues specific to
low-aspect-ratio tokamak designs with water-cooled center-posts are listed.
Each blanket concept also has ``generic'' critical issues (see text). }} \\
\multicolumn{7}{l}{$^{(b)}$The thermal efficiency is the net thermal efficiency
of the power cycle.}
\end{tabular}
\label{t-8-3-8}
\end{sidewaystable}

With limited resource available for the low-aspect-ratio tokamak assessment, an
attractive blanket for the low-aspect-ratio tokamak design that can satisfy
both performance and safety requirements has not been identified.
Table~\ref{t-8-3-8} lists all the blanket options considered, each with some
critical issues specific to low-aspect-ratio tokamak tokamak power plants. It
should be noted that each blanket concept has some ``generic'' critical issues
independent of the plasma operating mode: (1)~development of low-activation
structural material (noting small data base for SiC composites and DBTT issues
for ferritic steels); (2)~considerably higher afterheat in designs with
ferritic steels; (3)~Polonium production in designs with LiPb as coolant or
breeder (a waste-disposal issue); (4)~tritium contamination and clean-up of the
coolant in water-cooled design; and (5)~MHD pressure drop in lithium
self-cooled designs (may require insulating coating).

It appears that the only blanket concepts which can avoid the safety concerns
over the water-lithium combination will have modest thermal performance and
wall loading limits, and will employ ferritic steel as the structural material.
There are critical issues yet to be resolved for these blankets, such as the
low irradiation temperature of the structural material which could cause DBTT
issues and the modest cycle thermal efficiency (33\%).  If the recirculation
power in the low-aspect-ratio tokamak design remains high, then the net thermal
efficiency will be very low.

While we could not identify a high-performance blanket for the low-aspect-ratio
tokamak power plant, in order assess the potential advantage of this concept,
it is decided to use a high-performance blanket (Li/V blanket of ARIES-II) as
the reference blanket and to determine how attractive the low-aspect-ratio
tokamak concept can be. It is clear that this blanket would not be compatible
to the water-cooled center-post, due to the potential lithium-water reaction.

\subsubsection{Thermal efficiency}
\label{s-8-3-6-2}

The large recirculating power for the low-aspect-ratio tokamak design reduces
the net efficiency of the power plant. The effect of the lower thermal
efficiency is reflected by the higher COE from the system evaluation.  In
addition, a low net efficiency has some other effects which are not assessed by
the system evaluation.  One of the possible impact is the thermal pollution
which is caused by the large amount of rejected waste heat.  Also, the large
thermal power outlet will limit the possible choice of site selection.  For the
Starlite design goal, a net efficiency of $\sim$40\% was selected.

Since we could not identify a high-performance blanket for the low-aspect-ratio
tokamak power plant, we parametrically assess the impact of the net thermal
efficiency of the power cycle.  Figure~\ref{sti-8-3-8} shows the net efficiency
of the power plant $\eta_{net}$ as a function of the recirculating power for
two different net thermal cycle efficiencies.  For the reference Starlite
low-aspect-ratio tokamak (1000~MWe net electric output with $\sim$1,500~MW
recirculating power) which ``assumes'' a high-performance blanket
($\eta_{th}=46\%$), the net plant efficiency is very low (17\%).  If the net
thermal efficiency of the system is only 33\%, the net plant efficiency will be
lower than 13\%!

\begin{figure}[tb]
%\vspace*{4.2in}
\centerline{\psfig{file=sti-8-3-8.ps,height=4.2truein}}
\caption[sti-8-3-8]
{The net plant efficiency, $\eta_{net}$ as a function of recirculating power
fraction for different net thermal cycle efficiencies. The Starlite goal is a
net plant efficiency of $\sim$40\%.}
\label{sti-8-3-8}
\end{figure}

This example clearly indicates that: (1)~it is essential to reduce the
recirculating power in the low-aspect-ratio tokamak to $\lesim$500~MWe, and
(2)~it is essential to use a high-performance blanket option, {\it i.e.,} the
most important parameter for the selection of the low-aspect-ratio tokamak
blanket is the thermal conversion efficiency because of the potentially high
recirculating power.  It should be noted that blanket designs which are
consistent with the low-aspect-ratio tokamak requirement tends to have low net
thermal efficiency.  More efficient blanket systems probably are not compatible
with the low-aspect-ratio tokamak design, either with higher wall loading, or
with a water-cooled center-post.

\subsection{Divertor Assessment}
\label{s-8-3-7}

Power and particle handling are two of the most difficult design problems for
magnetic confinement devices.  The high peak surface heat flux and target plate
erosion rate are primary concerns of the divertor design, followed by the
control of impurities migration and helium ash removal. During the normal
operation of a fusion power plant, the goal of peak surface heat flux at the
divertor plate of $\sim$5~MW/m$^2$ has been selected for robustness of the
engineering design, while maintaining a reasonably high coolant outlet
temperature for efficient thermal power conversion.  Presently, the divertor
design approach is to find options for distributing the plasma transport power
to a large enough surface area in order to reduce the peak surface heat flux to
meet the engineering goal.

For comparison, the peak surface heat flux at the outboard divertor plate for
the Starlite low-aspect-ratio reference was estimated and compared with the
single null ITER-EDA~\cite{Die94} and Starlite reversed-shear reference design.
It is assumed that the transport power from the plasma core is transferred to
the inboard and outboard scrape-off-layers, and then to the collector plates at
the divertor by following the separatrix geometry.  A relatively deep outboard
slot ($\sim$1~m) is also assumed to help the trapping of neutrals and reduce
electron temperature and sputtering at the target plate by allowing radial
transport into the private flux region below the $\X$-point.  The device
geometry and transport power parameters are given in
\hbox{Table~\ref{t-8-3-9}}.

The peak heat flux depends on the power balance leading to the power deposited
at the scrape-off-layer ($P_{SOL}$), the magnetic flux expansion between the
plasma mid-plane and divertor target, and the width of the scrape-off layer,
and is given by:
%
\begin{eqnarray}
\label{e-8-3-10}
Q_{peak} & = & \frac{0.7 \, P_{SOL}}{2\pi R_s \lambda_d} \: ,
\end{eqnarray}
%
where $R_s$ is the major radius at the strike point, $\lambda_d$ the width of
the heat flux profile on the target, (taking into the inclination of the target
plate), and the factor of 0.7 represents the power distribution between the
regions outside and inside of the separatrix. In these estimates, we have
accounted for the expected Bremsstrahlung and line-radiation losses in the
core, up/down and in/out distributions of inboard and outboard scrape-off layer
power flow that is consistent with the experimental database from ASDEX, PDX
and DIII-D double-null operation~\cite{Hil95}. The in/out power balance for the
ITER single-null design is consistent with the single-null ELMing H-mode
results from DIII-D~\cite{Leo95}.

\begin{table}[p]
\caption{}
\centering{\bf Simple Estimate of Peak Divertor Heat Fluxes}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{1.cm}} c c c}
\hline
                              &  & \multicolumn{2}{c}{Starlite Reference} \\
\cline{3-4}
                              & ITER-EDA  & Reversed-shear  & Low-$A$    \\
\hline \\
Major radius (m)              & 8.14        & 5.12         & 4.4          \\
Aspect ratio                  & 2.9         & 4            & 1.25         \\
Configuration                 & single-null & double-null  & double-null  \\
Fusion power (MW)             & 1,500       &  1,881       & 3,952        \\
$\alpha$-particle power (MW)  & \D300       &  \D376       & \D790        \\
Current-drive power (MW)      & \D100       & \D\Z67       & \D458        \\
Transport power (MW)          & \D400       & \D443        & 1,248        \\
Core radiation fraction       & 0.40        & 0.35         &   0.35       \\
Up and down split             & NA          & 1:1          &   1:1        \\
Outboard and inboard split    & 1.3:1       & 4:1          &   4:1        \\
Lower/outboard $P_{SOL}$ (MW) & 134         & 125          &   354        \\
Divertor radiation fraction   & 0.3         & 0.3          &   0.3        \\
Mid-plane $\lambda_{mp}$ (cm)$^{(a)}$
                              & 1           & 1            &   1/2        \\
SOL flux expansion            & 4           & 4            &   4          \\
Target plate angle (\degC)    & 20          & 20           &   20         \\
Divertor $\lambda_d$ (cm)$^{(a)}$
                              & 11.7        & 11.7         &   11.7/23.4  \\
$R_s$ (m)                     & 8           & 4.7          &   4.4        \\
$Q_{peak}$ (MW/m$^2$)         & 17          & 18.4         &   53/26.5    \\
\\ \hline
\multicolumn{4}{l}{$(a)$$\lambda$ denotes the radial energy e-folding length.}
\end{tabular}
\label{t-8-3-9}
\end{table}

Respective peak heat flux estimates are presented in Table~\ref{t-8-3-9}.
Based on Table~\ref{t-8-3-9}, it is obvious that a higher peak heat flux
reduction is necessary for the low-aspect-ratio tokamak design even assuming
that certain processese may increase the mid-plane scrape-off layer thickness
to 2 cm~\cite{Pen95p}.  Enhanced radiation by seeding the edge-plasma by
impurities ({\it e.g.,} Ne, Ar, Fe, Kr) and the addition of neutral gas puffing
({\it e.g.,} D$_2$ ) in the divertor have been shown to be effective in
reducing the peak divertor heat flux and target plate plasma temperature by
increasing the divertor density and radiation losses.  These can possibly be
utilized to reach the design goal of peak heat flux of $\sim$5~MW/m$^2$ for the
ITER design.  We would expect similar technique can also be applied to the
Starlite designs.

\section{SYSTEMS ASSESSMENT}
\label{s-8-4}

The low-aspect-ratio (LAR) tokamak is not as developed theoretically
nor experimentally as the other tokamak regimes discussed in this report. The
role of systems analysis for the LAR tokamak, therefore, is to identify
high-leverage parameters affecting power-plant design, rather then to suggest
specific values for these parameters that minimize the cost-of-electricity
(COE). The design parameters examined in this analysis are: the plasma minor
radius, $a_p$; the factor $f_{TF}$ by which the toroidal dimension of the cross
section of the outer log of the toroidal-field (TF) leg is increased; the
plasma ion temperature, $T_i$; the inboard shield thickness, $\delta_{S,I}$;
the factor $f_\Omega$ by which the ohmic dissipation in the center-post is
reduced due to shaping of the center-post; and the plasma toroidal $\beta$. An
additional parameter relevant to the center-post shaping is the factor $f_V$ by
which the center-post volume is increased due to shaping. The plasma aspect
ratio, $A$, is also an important design parameter, but resource limitations
only permitted parametric system examination of designs at $A=1.25$, suggested
in \hbox{Ref.~\cite{Pen95}}. Nominal values for the above variables are listed
in \hbox{Table~\ref{t-8-4-1}}.

\begin{table}[t]
\caption{}
\centering{\bf Nominal Values used in Parametric Analysis}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{l @{\hspace{1cm}} c }
\hline \\
Plasma aspect ratio		                 & 1.25 \\
Equilibrium                                      & STREAC19$^{(a)}$ \\
Plasma $\beta$	                                 & 36.3\%  \\
Plasma minor radius (m)                          & 4.0  \\
Center-Post dissipation scale factor, $f_\Omega$ & 0.61 \\
Center-Post volume scale factor, $f_V$           & 2.03 \\
TF-Coil cross-sectional scale factor, $f_{TF}$   &  5.0 \\
Plasma ion temperature (keV)                     & 12.0 \\
Inboard shield thickness, $\delta_{S,I}$ (m)     & 0.3 \\
Net electrical power (GWe)                       & 1.0 \\
\\ \hline
\multicolumn{2}{l}{$(a)$See Sec.~\ref{s-8-2}.}\\
\end{tabular}
\label{t-8-4-1}
\end{table}

\subsection{Low-Aspect-Ratio-Tokamak Systems Models}
\label{s-8-4-1}

Most of the models used in the ARIES Systems Code (ASC) for the
low-aspect-ratio tokamak are described in Ref.~\cite{Aries4,PULSAR}. Those
models that are specific to the low-aspect-ratio tokamaks ({\it e.g.}, a
scaling for current-drive efficiency, $\gamma_B$) are discussed as needed in
the text.  The plasma equilibrium ({\it e.g.}, magnetic topology, $\beta$, {\it
etc.}), current-drive, and other physics results are based upon the analyses of
the STREAC19 equilibrium presented in \hbox{Sec.~\ref{s-8-2}}.  All analysis is
done for a 1-GWe, commercial power plant with a vanadium blanket and shield
design that is a refinement of the \AII blanket and shield design and is
described in~\hbox{Sec.~6}. Modifications to the blanket and shield design
specific to LAR are described in Sec.~\ref{s-8-3}. These modifications include
the elimination of the inboard blanket and a reduction in the thickness of the
inboard shield to 0.3~m (the minimum thickness that ensures Class-C
waste-disposal rating for the center-post after its approximately
two-full-power-year lifetime).  The poloidal-field (PF) coils are
superconducting coils; whereas the TF coils are normal conducting coils.  A
Level of Safety Assurance (LSA) rating of LSA=2 has been assumed without regard
for possible incompatibility of blanket and center-post coolants, and
appropriate cost credits have been taken.  All costs are reported in 1992
dollars.

\begin{figure}[t]
\centerline{\psfig{file=sti-8-4-1.ps,height=4.2truein,clip=}}
\caption[Fig-8-4-1]
{Elevation view of an upper-half quadrant of the LAR fusion power core.}
\label{sti-8-4-1}
\end{figure}

The base configuration of the LAR fusion power core (FPC) is shown in
Fig.~\ref{sti-8-4-1}. The inboard first wall conforms to a flux surface and
serves as a limiter to the plasma surface inboard of the null.  Vertical space
is provided above the null for a divertor chamber that captures the plasma
exhaust from the outboard plasma surface.  The center-post cross section
expands above the equatorial plane as allowed by the first wall and inboard
shield to provide maximum shaping of the center-post conductor in order to
reduce Joule losses.

\subsection{Sensitivity Studies}
\label{s-8-4-2}

\subsubsection{Plasma minor radius}
\label{s-8-4-2-1}

\begin{figure}[p]
\centerline{\psfig{file=sti-8-4-2.ps,height=3.4truein,clip=}}
\caption[Fig-8-4-2]
{Cost of electricity as a function of plasma minor radius
calculated internally to the ASC for the given LAR parameters.}
\label{sti-8-4-2}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{file=sti-8-4-3.ps,height=3.4truein,clip=}}
\caption[Fig-8-4-3]
{The mass power density (left scale) and the engineering gain (right scale)
for the minor-radius variation of \hbox{Fig.~\ref{sti-8-4-2}}.}
\label{sti-8-4-3}
\end{figure}

The most basic design parameter is the physical size of the tokamak.
Typically, the ASC determines the plasma minor radius that minimizes the cost
of electricity (COE) for a specified plasma aspect ratio and ion temperature
with an internal scan of minor radius. The COE for such a scan in $a_p$ is
shown in \hbox{Fig.~\ref{sti-8-4-2}} for the LAR design.  The COE for the LAR
generally displays a minimum in minor radius resulting from the tradeoff of
maximizing the Fusion Power Core (FPC) mass power density (MPD) and of
maximizing the engineering gain ($1/\epsilon$, where $\epsilon$ is the
recirculating power fraction), as is shown in \hbox{Fig.~\ref{sti-8-4-3}}.
Decreasing the plasma minor radius decreases the blanket, shield, and coil
masses and increases the MPD. For an LAR-based power plant, the center-post
dissipation dominates the recirculating-power losses that determine the
engineering gain. Decreasing the plasma minor radius increases the ohmic
dissipation in the resistive TF coils, because the center-post cross-sectional
area decreases with decreasing plasma minor radius, and the center-post
resistance and ohmic dissipation then increase.

The minimum of COE in \hbox{Fig.~\ref{sti-8-4-2}} is also affected by the
discrete nature of turbine capacity. The ASC costing algorithms limit the
capacity of a single turbine to 1,360 MWe which gives rise to the discontinuity
in the COE {\it versus} minor radius shown in \hbox{Fig.~\ref{sti-8-4-2}}.
Because the ASC models for the LAR project large (1,185 MW) center-post
dissipations, the gross electric power is always near the transition point
where the number of turbines required changes from two to three.  Minor changes
in the assumed parameters listed in \hbox{Fig.~\ref{sti-8-4-2}} can move the
location of this transition point to larger plasma minor radius and eventually
cause the minor radius that minimizes the COE to jump in a discontinuous
fashion to a local minimum where $\partial\rm{COE}/\partial a_p=0$, as opposed
to the extremum shown in \hbox{Fig.~\ref{sti-8-4-2}}. Although this jumping
effect does not significantly affect the COE, discontinuities in various
variables displayed in subsequent parametric curves are possible.

\subsubsection{Size of the outboard TF-coil cross section}
\label{s-8-4-2-2}

Because the LAR tokamak has a large recirculating power fraction, methods to
reduce the recirculating power should be exercised.  One such method is to
increase the cross sectional area of the outboard TF-coil leg. In the ASC the
``base'' cross section of the outboard TF-coil leg is a rectangle with a
toroidal dimension which is equal to the length of one side of the 16-sided
polyhedronal cross section of the center-post in the equatorial plane.  The
radial leg of this ``base'' cross section is always assumed to be increased by
a factor $f_r=2.5$. In order to ass the impact of the change in the size of the
outboard TF-coil cross section, the toroidal leg of this ``base'' cross section
is increased by a variable factor $f_{TF}$.  The cost-of-electricity
sensitivity to $f_{TF}$ is shown in \hbox{Fig.~\ref{sti-8-4-4}}.  The minimum
in the COE with $f_{TF}$ represents the classic tradeoff between maximizing
mass power density and maximizing the engineering gain, as is shown in
\hbox{Fig.~\ref{sti-8-4-5}}.  The engineering gain increases with increasing
$f_{TF}$, because increasing $f_{TF}$ increases the cross sectional area for
the outboard TF coil that, in turn, decreases the coil resistance (resistance =
resistivity $\times$ conductor length $/$ conductor area) and the power
dissipated in the outboard TF-coil leg. Increasing $f_{TF}$ also increases the
mass of the outboard TF-coil leg, thus decreasing the mass power density.

\begin{figure}[p]
\centerline{\psfig{file=sti-8-4-4.ps,height=3.4truein,clip=}}
\caption[sti-8-4-4]
{The cost of electricity for LAR parameters
as a function of the outboard TF-coil cross-sectional scale factor.}
\label{sti-8-4-4}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{file=sti-8-4-5.ps,height=3.4truein,clip=}}
\caption[sti-8-4-5]
{The mass power density (left scale) and the engineering gain (right scale)
for the minor-radius variation of \hbox{Fig.~\ref{sti-8-4-4}}.}
\label{sti-8-4-5}
\end{figure}

\subsubsection{Plasma ion temperature}
\label{s-8-4-2-3}

\begin{figure}[b]
\centerline{\psfig{file=sti-8-4-6.ps,height=3.6truein,clip=}}
\caption[sti-8-4-6]
{The cost of electricity for LAR parameters as a function of the plasma ion
temperature.  Each point along the curve represents the COE optimum in plasma
minor radius, $a_p$.}
\label{sti-8-4-6}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{file=sti-8-4-7A.ps,height=3.8truein,clip=}}
\vspace*{-1.cm}
\centerline{\psfig{file=sti-8-4-7B.ps,height=3.8truein,clip=}}
\caption[Fig-8-4-7]
{Variation of the mass power density (left scale, top frame) and engineering
gain (right scale, top frame) and the fuel $\beta$ (bottom frame) for the
ion-temperature variation of \hbox{Fig.~\ref{sti-8-4-6}}.}
\label{sti-8-4-7}
\end{figure}

The cost-of-electricity sensitivity to variations in the plasma ion temperature
is shown in \hbox{Fig.~\ref{sti-8-4-6}}.  The temperature that minimizes the
COE also maximizes the mass power density, as is shown in
\hbox{Fig.~\ref{sti-8-4-7}}. The engineering gain does not play a role in the
optimization of the plasma ion temperature, because the only component of the
engineering gain that is plasma-temperature dependent is the current-drive
efficiency, and the current-drive power is negligible ($\lesim$7~MW) for this
equilibrium (Sec.~\ref{s-8-2-3}). At low temperatures ({\it i.e.},
$\leq$12~keV), the mass power density increases slightly with increasing
temperature because $\langle\sigma v\rangle/T^2_i$ is increasing with
increasing temperature until its maximum at $\approx$15~keV.  The MPD decreases
with increasing temperature above the optimum temperature because of decreased
reactivity ({\it i.e.}, \hbox{$n^2\langle\sigma v\rangle$}). However, the
plasma density decreases (at constant $\beta$) and the fusion products are
thermalized more slowly which increases the hot alpha-particle pressure and
decreases the fuel pressure (see \hbox{Fig.~\ref{sti-8-4-7}}), and requires a
higher toroidal-field strength, which increases TF-coil costs for the same net
electrical power.

As mentioned above, current drive does not play a role in the determination of
the optimum temperature, because the STREAC19 equilibrium maximizes the
bootstrap-current fraction at a value of $f_{BC}=$ 0.997.  Attaining a
bootstrap-current fraction approaching unity is essential for a low-aspect-ratio
tokamak power plant because even at a high bootstrap-current fraction of
$f_{BC}=0.9$, the current-drive power (including system efficiency) exceeds
600~MWe.

\subsubsection{Inboard shield thickness}
\label{s-8-4-2-4}

\begin{table}[b]
\caption{}
\centering{\bf Effect of Removing the Inboard Shield$^{(a)}$}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{1.cm}} c c c c}
\hline
Case                                     & Base  &  A    &   B    &   C    \\
\hline \\
Inboard shield thickness (m)             &  0.30 &  0.   &   0.   &  0.    \\
Plant capacity factor                    &  0.76 &  0.76 &   0.76 &  0.59  \\
First-Wall fluence lifetime (MWy/m$^2$)  & 12.0  & 12.0  &  12.0  & 12.0   \\
Center-Post fluence lifetime (MWy/m$^2$) & 12.0  & 12.0  &   0.3  &  0.3   \\
Plasma minor radius (m)                  &  4.00 &  2.40 &   2.55 &   2.46 \\
Engineering gain $Q_E$                   &  1.58 &  1.83 &   1.92 &   1.87 \\
Mass power density (kWe/tonne)           &  26.2 & 53.9  &  47.1  &  55.6  \\
Cost of electricity (mill/kWeh)          & 116.0 & 88.7  & 163.5  & 182.0  \\
\\ \hline
\multicolumn{5}{l}{$(a)$ For the fixed parameters of Table~\ref{t-8-4-1}.}\\
\end{tabular}
\label{t-8-4-2}
\end{table}

The impact of an 0.3-m thick inboard shield surrounding the center-post is
indicated in \hbox{Table~\ref{t-8-4-2}}.  The first column of
\hbox{Table~\ref{t-8-4-2}} represent the ``base'' case with an 0.3-m thick
inboard shield.  For Case~A, the inboard shield has been removed. Without an
inboard shield the space that would have been allocated for shielding can be
used as additional center-post conductor to reduce the center-post dissipation
and associated recirculating power and to increase the engineering gain.
Because the engineering gain is raised for all plasma minor radii, the COE
minimizes at a lower plasma minor radius, thus increasing the mass power
density.  The COE is 20~mill/kWeh lower for Case~A relative to the ``base''
case with an inboard shield.  However, this second case does not account for
the consequences of not having an inboard shield, and the conter-post does not
qualify for Class-C low level waste.

The fluence lifetime of the center-post should be reduced to 0.3~MWy/m$^2$ to
meet the requirements for Class C waste disposal (Sec.~\ref{s-8-3}), affecting
just the replacement cost for the center-post (Case~B of
\hbox{Table~\ref{t-8-4-2}}). No penalty has been assessed for an expected 3\%
increase in the center-post resistivity. For this case the optimum COE occurs
at a larger plasma minor radius than for Case~A because the center-post
replacement cost has risen relative to the cost associated with recirculating
power.  The COE of Case~B is 48 mill/kWeh higher than for the base case.

Lastly, the plant availability is reduced from 0.76 to 0.59, to simulate a
center-post lifetime of $\approx$1.44~months and a replacement time of
$\approx1$~month (Case~C of Table~\ref{t-8-4-2}).  Although the cost of
electricity is explicitly dependent upon $1/p_f$, the economics governing the
optimization is also influenced by the plant availability factor. Reducing the
plant factor places a greater emphasis on reducing production costs ({\it
i.e.}, mass power density and associated costs). This case is 66~mill/kWeh
higher than for the base case.

It should be noted that for the four cases of \hbox{Table~\ref{t-8-4-2}}, the
current-drive power is negligible ({\it i.e.}, $\sim$10~MW).  If a more careful
analysis of MHD equilibrium and stability, current drive, and profile control
indicates the current-drive power is not effectively zero, then eliminating the
inboard shield does not have nearly as significant an effect on the COE as is
indicated herein, because of the added cost of current drive.

\subsubsection{Center-post shaping}
\label{s-8-4-2-5}

\begin{table}[b]
\caption{}
\centering{\bf Effect of Shaping the Center-Post$^{(a)}$}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l  c c}
\hline
                                      &    With & Without \\
				      & shaping & shaping  \\
\hline \\
Center-Post dissipation scale factor, $f_\Omega$ & 0.609 & 1.0\\
Center-Post volume scale factor, $f_V$  & 2.031 & 1.0  \\
Center-Post dissipation (MW)            & 1,185 & 1,654\\
Gross electrical power (GWe)            & 2,708 & 3,143\\
Plasma minor radius (m)                 &  4.00 & 4.50 \\
Engineering gain                        &  1.58 & 1.47 \\
Mass power density (kWe/tonne)          &  26.2 & 21.7 \\
Cost of electricity (mill/kWeh)         & 116.0 & 139.0 \\
\\ \hline
\multicolumn{3}{l}{$(a)$ For the fixed parameters of Table~\ref{t-8-4-1}.}\\
\end{tabular}
\label{t-8-4-3}
\end{table}

For the purposes of analyzing the effect of shaping the center-post, the
unshaped center-post is characterized by: (1)~a conductor filling fraction
$\lambda=0.7$, (2)~a Cu resistivity $\eta=2\times10^{-8}$ ohm-m, (3)~a height
$H_{CP}$ and a radius $R_{CP}$, (4)~a current density \hbox{$j_{CP}=I_{CP}/\pi
\, R_{CP}^2$}, (5)~a volume \hbox{$V_{CP}=\pi \, R_{CP}^2\lambda \, H_{CP}$},
and (6)~a ohmic dissipation \hbox{$P_{CP}=\eta I_{CP} \, j_{CP} \,
H_{CP}/\lambda$.}  The volume of the shaped center-post can be expressed as
$V_{CP}^s=V_{CP}f_V$, where
%
\begin{eqnarray}
\label{e-8-4-1}
f_V & = &
\left[\pi\int_0^{H_{CP}} R^2 dz - (1-\lambda) V_{CP}\right]/\lambda V_{CP}
 \: .
\end{eqnarray}
%
Similarly, the ohmic dissipation in the shaped center-post can be expressed as
$P_{CP}^s=P_{CP}f_\Omega$, where
%
\begin{eqnarray}
\label{e-8-4-2}
 f_\Omega & = & \frac{\lambda R_{CP}^2}
{H_{CP}}\int_0^{H_{CP}}\frac{dz}{R^2-(1-\lambda)R_{CP}^2} \: .
\end{eqnarray}

The effect of shaping the center-post on the COE is indicated in
\hbox{Table~\ref{t-8-4-3}} for the LAR parameters.  A significant savings of
25~mill/kWeh is obtained by shaping the center-post, suggesting that all
LAR-tokamak power-plant designs should invoke center-post shaping. This
reduction in COE is possible because the additional center-post conductor
reduces the center-post dissipation and the recirculating power by 469 and
435~MW, respectively.  The reduced recirculating power permits a reduction in
machine size ({\it i.e.}, $a_p$) to increase MPD and further decrease COE.

\subsubsection{Plasma $\beta$}
\label{s-8-4-2-6}

\begin{figure}[b]
\centerline{\psfig{file=sti-8-4-8.ps,height=3.7truein,clip=}}
\caption[sti-8-4-8]
{The cost of electricity for LAR parameters as a function of the plasma
$\beta$.  Note that this variation is not based on self-consistent plasma
equilibria.  Rather the achievable $\beta$ of the base equilibrium (36\%) is
arbitrarily increased.}
\label{sti-8-4-8}
\end{figure}

The LAR tokamak offers high beta, but also carries along the disadvantages of
high recirculating power that have been evident in the previous parametric
variations. One question that arises immediately for the LAR tokamak is how
high must the plasma $\beta$ be to suppress the deleterious effects of the
recirculating power? The variation in the COE with plasma $\beta$ is shown in
\hbox{Fig.~\ref{sti-8-4-8}} for LAR parameters.  Note that this variation is
not based on self-consistent plasma equilibria.  Rather the achievable $\beta$
of the base equilibrium (36\%) is arbitrarily increased.

The benefits of increasing the $\beta$ are a reduction in plasma size ({\it
i.e.}, $a_p$) and an increase in mass power density, while simultaneously
increasing the engineering gain.  The Starlite requirement and goal for the COE
are 65 and 80 mill/kWeh, respectively, as indicated in \hbox{Sec.~3} and
\hbox{Ref.~\cite{Wag95}}.  Extrapolation of the results of
\hbox{Fig.~\ref{sti-8-4-8}} indicate that \hbox{$\beta \approx 100\%$} is
required just to meet the Starlite COE requirement with the current-drive power
effectively set to zero.

\subsection{Design Point Determination}
\label{s-8-4-3}

Although the LAR tokamak analysis is not as complete nor as self-consistent as
the analysis of the other tokamaks in this report, a list of parameters for the
reference design used in the systems analysis are given in Tables~\ref{t-8-4-4}
through \ref{t-8-4-6}.

\begin{table}[b]
\caption{}
\centering {\bf Low-Aspect-Ratio (LAR) Power-Plant Physics Parameters}
\vspace{0.5cm}
{\small
\renewcommand{\arraystretch}{0.83}
\begin{tabular}{l @{\hspace{1.0 truecm}} c}
\hline \\
Plasma major toroidal radius, $R_T$ (m)            &  5.00 \\
Plasma minor radius, $a_p$ (m)                      &  4.00 \\
Plasma vertical elongation, $\kappa_\X$             &  3.00 \\
Plasma vertical elongation, $\kappa_{95}$           &  3.40 \\
Plasma triangularity, $\delta_\X$                   &  0.45 \\
Plasma triangularity, $\delta_{95}$                 &  0.55 \\
Plasma aspect ratio, $A=R_T/a_p$                    &  1.25 \\
On-axis safety factor, $q_0$                        &  2.94 \\
Circularized safety factor, $q_\ast$                &  3.54 \\
Plasma-edge safety factor, $q$                      & 28.87 \\
Troyon coefficient, $C_T$ (Tm/MA)                   &  0.06 \\
Plasma $\beta$                                      &  0.36 \\
Plasma poloidal $\beta$                             &  1.42 \\
Stability parameter, $\epsilon\beta_p$              &  1.13 \\
Peak-to-average density, $n_0/n$                    &  1.57 \\
Peak-to-average temperature, $T_0/T$                &  1.06 \\
Normalized edge density, $n_E/n$                    &  0.29 \\
Ion temperature, $T_i$ (keV)                        & 12.00 \\
Electron temperature, $T_e$ (keV)                   & 12.19 \\
Ion density, $n_i$ (10$^{20}$~m$^{-3}$)             &  1.00 \\
Electron density, $n_e$ (10$^{20}$~m$^{-3}$)        &  1.24 \\
Particle-to-energy confinement time ratio, $\tau_p/\tau_E$          &  4.00 \\
Ion-to-electron energy confinement time ratio, $\tau_{Ei}/\tau_{Ee}$&  1.00 \\
Lawson parameter, $n_i\tau_E$ (10$^{20}$~m$^{-3}$s) &  4.89 \\
ITER-89P scaling~\cite{Pos90} multiplier, $H_{89P}$ &  3.02 \\
On-axis toroidal field, $B_{\phi 0}$ (T)            &  1.77 \\
Radiation fraction, $f_{RAD}$                       &  0.32 \\
Plasma current, $I_p$ (MA)                          & 40.06 \\
Bootstrap-current fraction, $f_{BC}$                &  1.00 \\
Current-drive power to plasma, $P_{CD}$ (MW)        &  7.27 \\
Current-drive efficiencies: &     \\
\quad $I_{EX}/P_{CD}$ (mA/W)                        & 16.52 \\
\quad $\gamma$ (10$^{20}$ A/W-m$^2$)                & 34.23 \\
 \\ \hline
\end{tabular}
}
\label{t-8-4-4}
\end{table}

\begin{table}[p]
\caption{}
\centering {\bf Low-Aspect-Ratio (LAR) Power-Plant Engineering Parameters}
\vspace{0.5cm}
{\small
\renewcommand{\arraystretch}{0.90}
\begin{tabular}{l @{\hspace{2.0 truecm}} c }
\hline \\
Field at TF coil, $B_{\phi c}$ (T)                        & 15.95 \\
Center-post resistive losses (MW)			  & 1,185  \\
Total TF resistive losses (MW)				  & 1,565  \\
TF-coil current density, $j_{TF}$ (MA/m$^2$)              & 27.49 \\
Magnetic-field energy, $W_B$ (GJ)                         & 25.24 \\
Total specific energy, $W_B/M_c$ (MJ/tonne)               &  2.63 \\
Fusion power, $P_F$ (GW)                                  &  4.53 \\
Neutron power, $P_N$ (GW)                                 &  3.62 \\
Average neutron wall loading, $I_w$ (MW/m$^2$)            &  2.23 \\
Peak neutron wall loading, $\hat I_w$ (MW/m$^2$)          &  3.14 \\
Neutron fluence life, $I_w\tau$ (MWy/m$^2$)               & 12.00 \\
Average blanket energy multiplication                     &  1.37 \\
Average first-wall heat flux, $q_w$ (MW/m$^2$)            &  0.27 \\
Blanket power density, $P_{TH}/V_{BLK}$ (MW/m$^3$)        & 13.87 \\
Current-drive system efficiency, $\eta_{CD}$              &  0.95 \\
Thermal conversion efficiency, $\eta_{TH}$                &  0.46 \\
Thermal power (GWth):   \\
\quad peak, $\hat P_{TH}$                                 &  5.89 \\
\quad average, $P_{TH}$                                   &  5.89 \\
Auxiliary site power, $P_{AUX}$ (MW)                      &108.33 \\
Primary loop pumping power (MW)                           & 27.08 \\
Gross electrical power, $P_{ET}$ (GWe)                    &  2.71 \\
Net electrical power, $P_E$ (GWe)                         &  1.00 \\
Recirculating-power fraction, $\epsilon = 1/Q_E$          &  0.63 \\
Plasma gain, $Q_p=P_F/P_{CD}$                             &622.66 \\
Engineering gain, $Q_E$                                   &  1.59 \\
Net plant efficiency, $\eta_p = \eta_{TH} (1 - \epsilon)$ &  0.17 \\
Plant capacity factor, $p_f$                              &  0.76 \\
Masses (tonne): \\
\quad First wall                                          & \D\Z\Z17 \\
\quad Blanket and reflector                               &  \D\Z259 \\
\quad Shield                                              &  \Z8,036 \\
\quad TF coils                                            &  \Z8,167 \\
\quad PF coils                                            &  \Z1,447 \\
\quad Fusion power core                                   &   2,4863 \\
Mass power density, MPD (kWe/tonne)                       &  26.25 \\
Level of Safety Assurance, LSA                            &     2  \\
\\ \hline
\end{tabular}
}
\label{t-8-4-5}
\end{table}

\begin{table}[p]
\caption{}
\centering {\bf Low-Aspect-Ratio (LAR) Power-Plant Economic Parameters
(1992 \$)}
\vspace{0.5cm}
{\small
\renewcommand{\arraystretch}{0.8}
\begin{tabular}{l l @{\hspace{1.cm}} c}
\hline
Acct.\ No. & Account Title                         & Million Dollars \\
\hline \\
20.     & Land and land rights                                      &  \D\Z10.4 \\
21.     & Structures and site facilities                            &   \D433.5 \\
22.     & Reactor plant equipment                                   &   2,209.9 \\
22.1.1  & \ \ First wall, Blanket, and Reflector                    &  \D\Z78.7 \\
22.1.2  & \ \ Shield                                                &   \D356.2 \\
22.1.3  & \ \ Magnets                                               &   \D619.7 \\
22.1.3.1& \ \ \ TF coils                                            &   \D507.4 \\
22.1.3.3& \ \ \ PF coils                                            &   \D112.4 \\
22.1.4  & \ \ Supplemental heating systems                          &   \D127.6 \\
22.1.5  & \ \ Primary structure and support                         &   \D109.6 \\
22.1.6  & \ \ Reactor vacuum systems                                &   \D140.9 \\
22.1.7  & \ \ Power supplies                                        &   \D137.3 \\
22.1.8  & \ \ Impurity control                                      &  \D\Z16.7 \\
22.1.9  & \ \ Direct energy conversion system                       & \D\Z\Z0.0 \\
22.1.10 & \ \ ECRH breakdown system                                 & \D\Z\Z4.3 \\
22.1    & \ \ Reactor equipment                                     &   1,591.0 \\
22.2    & Main Heat Transfer and Transport                          &   \D365.8 \\
23.     & Turbine plant equipment                                   &   \D628.4 \\
24.     & Electric plant equipment                                  &   \D164.5 \\
25.     & Miscellaneous plant equipment                             &  \D\Z90.6 \\
26.     & Special materials                                         &  \D\Z27.0 \\
90.     & Total direct cost                                         &   3,564.4 \\
91.     & Construction services and equipment                       &   \D427.7 \\
92.     & Home office engineering and services                      &   \D185.3 \\
93.     & Field office engineering and services                     &   \D213.9 \\
94.     & Owner's costs                                             &   \D658.7 \\
96.     & Project contingency                                       &   \D852.2 \\
97.     & Interest during construction (IDC)                        &   \D975.1 \\
98.     & Escalation during construction (EDC)                      & \D\Z\Z0.\D \\
99.     & Total capital cost                                        &   6,877.3 \\
 \\
  & &                 \us{Constant dollars}  \\
% keep [90] etc in {} or LATEX will screw up.
{[90]}    & Unit direct cost, UDC (\$/kWe)                          & 3,564.4 \\
{[94]}    & Unit base cost, UBC (\$/kWe)                            & 5,902.2 \\
{[99]}    & Unit total cost, UTC (\$/kWe)                           & 6,877.3 \\
 \\
          & Capital return (mill/kWeh)                              &  \Z99.86 \\
{[40-47,51]} & O\&M (1.4\%) (mill/kWeh)                             & \Z\Z9.16 \\
{[50]}    & Blanket replacement (mill/kWeh)                         & \Z\Z6.38 \\
          & Decommissioning (mill/kWeh)                             & \Z\Z0.50 \\
{[02]}    & Fuel (mill/kWeh)                                        & \Z\Z0.07 \\
          & Cost of electricity, COE (mill/kWeh)                    &   115.97 \\
\\ \hline
\end{tabular}
}
\label{t-8-4-6}
\end{table}

\section{SUMMARY \& CONCLUSIONS}
\label{s-8-5}

The assessment of low-aspect-ratio tokamak power plants included
detailed physics calculations, system studies, and preliminary engineering
design considerations.  The physics analyses yielded equilibria that are stable
to all MHD modes (with a wall present) and require very little current drive
power ($\lesim$10~MW) due to bootstrap fractions approaching unity.  These
equilibria had the following characteristics: aspect ratio of 1.25, elongations
of 3 and $\beta$ as high as 40\%.  It is noted that these cases would require a
vertical feedback control system and a conducting wall to stabilize the
external kink mode.  The sensitivity of the current drive power requirements to
variations in plasma profiles is an important issue and must be addressed in
future studies.

Using the cases noted above, system studies and engineering analyses were
conducted.  It appears that even for these cases with low current-drive power
requirements, there still was significant recirculating power ({\it i.e.,}
recirculating power fraction $\sim$0.6) associated with the resistive losses in
the copper center post and return legs of the toroidal field coils.  In this
context, minimizing the amount of inboard shielding for the low-aspect-ratio
tokamak is well known to be important for the power efficiency of the concept.
However, an unshielded copper center post leads to activation levels which
exceed the NRC Class-C waste-disposal limit, unless change-out times of the
order of weeks are employed.  Such short change-out times lead to unacceptable
costs of electricity.  Our systems and engineering analyses indicated that it
is desirable to add about 30~cm of shielding to the center post.  This level of
shielding helps significantly to reduce the waste disposal problem and the
associated deleterious effect on cost of electricity.  At the same time,
however, the addition of this shielding increases the resistive losses in the
center post, however, the net effect is beneficial with respect to the
unshielded case.

In addition to concerns about the design of the center post, preliminary
engineering evaluations also identified constraints with regard to the blanket
and divertor.  In the context of blanket design, the use of water in the
center-post makes the use of liquid lithium in the blanket problematic due to
safety issues.  Additional barriers to leakage would be necessary and
substantial additional analysis would have to be performed to satisfy licensing
and regulatory requirements if liquid lithium were to be used in the blanket.
The impact of ruling out liquid lithium in the blanket is to reduce the bulk
heat removal capabilities of the concept and, thus, further reduce the overall
plant efficiency which is already low ($\sim$17\%).  Finally, the divertor
problem appears especially severe for the low-aspect-ratio tokamak concept
because the divertor surface area available at low major radius is so
restricted.


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\end{thebibliography}
\end{document}
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