% % FINAL VERSION % \documentstyle[12pt,twoside,aries]{report} \setcounter{chapter}{1} \begin{document} \newcommand{\AI}{\hbox{ARIES-I}} \newcommand{\AII}{\hbox{ARIES-II}} \newcommand{\AIII}{\hbox{ARIES-III}} \newcommand{\TI}{\hbox{TITAN-I}} \newcommand{\TII}{\hbox{TITAN-II}} \newcommand{\VTS}{\hbox{V-3Ti-1Si}} \newcommand{\VCT}{\hbox{V-15Cr-5Ti}} \newcommand{\HT}{\hbox{HT-9}} \newcommand{\degC}{$^\circ$C} \newcommand{\SdegC}{$\,^\circ$C} \newcommand{\SPL}{MgAl$_2$O$_4$} \newcommand{\AlO}{Al$_2$O$_3$} \newcommand{\LNO}{LiNO$_3$} \newcommand{\Z}{\phantom{0}} \newcommand{\D}{\phantom{.0}} \newcommand{\W}{\phantom{.}} \newcommand{\Y}{\phantom{$\sim$ }} \newcommand{\X}{\times} \newcommand{\Dtf}{\hbox{$\Delta T_f$}} \newcommand{\Dtw}{\hbox{$\Delta T_w$}} \newcommand{\WR}{\hbox{W-26Re}} \newcommand{\NB}{Nb$_3$Sn} \newcommand{\NA}{Nb$_3$A$\ell$} \newcommand{\NAG}{Nb$_3$(A$\ell$,Ge)} \newcommand{\LZ}{Li$_2$ZrO$_3$} \newcommand{\LS}{Li$_4$SiO$_4$} \newcommand{\LO}{Li$_2$O} \title{\bf 2. SYSTEMS STUDIES} \author{\hbox to \hsize { \hbox to 2.truein{ Ronald L.~Miller \hfil } \hfil \hbox to 2.truein{\hfil Robert A.~Krakowski \hfil } \hfil \hbox to 2.truein{\hfil David A.~Ehst } } \\ } \date{ } \maketitle \begin{titlepage} {%\baselineskip=1.8ex \tableofcontents } \end{titlepage} \setcounter{page}{1} \chapter{SYSTEMS STUDIES} \section{INTRODUCTION} \label{s-2-1} The objective of the systems analysis activity is the systematic study and determination of plant operating parameters through economic analysis and optimization of the power station, emphasizing the performance of the fusion power core. The fusion power core (FPC) includes the tokamak plasma chamber, first wall, blanket, shield, coils, and associated structure. The reference design points are chosen to meet overall design goals of the \AI\ study, such as minimal cost of electricity (COE, mill/kWh) and high mass power density (MPD, kWe/tonne). In addition, trade-off and sensitivity studies were performed to establish and characterize the design window for attractive tokamak reactors. Results and constraints from detailed modeling and engineering design effort are fed back and integrated into the systems model. The systems code, therefore, is used as a tool in the iterative conceptual engineering-design process. The ARIES systems code evolved from a systems code used to model the spherical-torus tokamak~\cite{Mil86} and from the reversed-field-pinch systems code used in the TITAN study~\cite{Naj90}. The ARIES systems code includes a physics model comparable to that of the MUMAK code~\cite{Fen87,Fen89}, which is used for U.S.\ ITER~\cite{Ite89} modeling. Informal benchmarking with the TRAC-II code~\cite{Sta75,Ehs79} (physics and engineering), the European SUPERCOIL/SCAN codes~\cite{Coo89b} (physics, engineering, and costing), and the GENEROMAK code~\cite{She86,Del88a} (physics, engineering, and costing) has been performed with generally satisfactory results~\cite{Mil90}. \begin{figure}[p] \vspace*{7.1in} \caption[Fig-2-1-1] {ARIES systems-code logic and flow diagram, combining plasma engineering, technology, and costing modules. The MacNTAU~\cite{Wer89} power-balance solver is incorporated in the systems code directly. An input file for the separate COLIF code~\protect[\citelist{Ker87}\,-\,\citelist{Ker89b}] (modeling aspects which emphasize the kinetics) is generated for physics modeling.} \label{f-2-1-1} \end{figure} The overall features and logic of the ARIES systems code are schematically given in \hbox{Fig.~\ref{f-2-1-1}}. The systems code incorporates a series of computational search loops ({\it e.g.,} fixed aspect ratio and variable minor radius), which solve for the FPC physics and engineering characteristics. Subject to certain constraints and design goals, cost dependencies are identified, and the key interactions between system variables are determined. Design points identified by this procedure are subjected to detailed analysis and subsystem design, with conceptual design results being fed back to the systems design code throughout the project for further optimization and refinement. Ongoing calibrations with separate subsystem models used in the project are made. Results of plasma engineering, blanket and shield, impurity control, magnet, power conversion, maintenance, tritium handling, safety, and other analyses are integrated, sometimes in approximate form, into the systems optimization model, which typically uses COE as the object function to be minimized. Section~\ref{s-2-2} summarizes the physics and engineering models used in the ARIES systems code. \hbox{Section~\ref{s-2-3}} describes the costing and economics assumptions and algorithms used to project COE estimates. The results of parametric studies leading to the \AI\ reference design point are discussed in \hbox{Sec.~\ref{s-2-4}}. Summary and conclusions are given in \hbox{Sec.~\ref{s-2-5}}. \section{MODELS AND METHODS} \label{s-2-2} \subsection{Plasma Engineering Model} \label{s-2-2-1} The ARIES systems-code model for plasma engineering begins with a steady-state, point-plasma model that is corrected for profile effects. The toroidal plasma geometry is described by the major toroidal radius, $R_T$, and minor radius, $a$ ({\it i.e.}, half width at the mid-plane), and vertical half height, $b$. The plasma aspect ratio is \hbox{$A \equiv R_T/a = 1/\epsilon$}. When needed, parameters at the separatrix surface are denoted by subscript $\X$ ({\it e.g.}, $\kappa_\X$) and at the 95\% flux surface by subscript 95 ({\it e.g.}, $\kappa_{95}$). In addition to the vertical elongation, \hbox{$\kappa \equiv b/a$}, the plasma poloidal boundary is also characterized by the triangularity parameter, $\delta$. The plasma shape factor, S, which is the ratio of the plasma poloidal perimeter to the perimeter of a circular plasma (\hbox{$\kappa = 1$}) with the same value of minor radius, $a$, can be approximated by~\cite{Uck88} % \begin{eqnarray} \label{e-2-2-1} S & = & \oint \frac{\phi{dl}}{2\pi{a}} \left|_{\psi_{edge}} \right. \: \simeq \: \left[\frac{1 + \kappa^2\left(1 + 2\delta^2 - 1.2\delta^3\right)}{2} \right]^{1/2} \: , \end{eqnarray} % as representative of the more refined model~\cite{Gal89} actually used in the ARIES systems code. The plasma cross-sectional area is given by~\cite{Ehs79} % \begin{eqnarray} \label{e-2-2-2} A_p & = & \pi a^2 \kappa \left[\frac{2J_1(\delta)}{\delta}\right] \: \equiv \: \pi r_p^2 \: , \end{eqnarray} where $J_1(x)$ is the first-order Bessel function of the first kind and $r_p$ is the circularized (average) plasma radius. The corresponding major toroidal radius of the centroid of the plasma cross section is approximated by~\cite{Ehs79} % \begin{eqnarray} \label{e-2-2-3} R^\prime_T & = & R_T \left(1 - \frac{0.233 \, \delta}{A}\right) \; , \end{eqnarray} % such that the plasma volume is \hbox{$V_p \equiv 2\pi R^{\prime}_T A_p$}. Then, % \begin{eqnarray} \label{e-2-2-4} V_p & = & 2\pi^2 \, R_T \, a^2 \, \kappa \, \left(1 - \frac{0.233 \, \delta}{A}\right) \left[\frac{2J_1(\delta)}{\delta}\right] \: . \end{eqnarray} The plasma ion and electron densities are denoted by $n_i$ and $n_e$, respectively. Local charge neutrality requires that % \begin{eqnarray} \label{e-2-2-5} n_e(r) = {\displaystyle\sum_j} \, n_j(r) Z_j \:, \end{eqnarray} % where the summation is over all ion species. The effective plasma charge is % \begin{eqnarray} \label{e-2-2-6} Z_{eff} & = & \frac{\displaystyle\sum_j n_jZ_j^2}{\displaystyle\sum_j n_jZ_j} \: . \end{eqnarray} % The ion population comprises deuterium (D) and tritium (T) as the primary fuel species, as well as fusion-product $\alpha$-particles, proton (p), and $^3$He (from the DD reactions), and impurities ({\it e.g.,} 1\%~O$_2$). The concentration of each ion species is \hbox{$f_j \equiv n_j/n_i$}, and inputs to the systems code are iterated with results of the COLIF code\hbox{~[\citelist{Ker87}\,-\,\citelist{Ker89b}]} for self-consistency. The central (peak) plasma temperature is \hbox{$T(0) \equiv T_o$}, and the central (peak) plasma density is \hbox{$n(0) \equiv n_o$}. The radial profiles of electric and ion temperatures and densities are approximated, respectively, by % \begin{eqnarray} \label{e-2-2-7} T(r) & = & T_o \left[ 1 - \left(\frac{r}{r_p}\right)^2 \right]^{\alpha_T} \: , \\ \label{e-2-2-8} n(r) & = & n_s + \left(n_o - n_s\right) \left[1 - \left(\frac{r}{r_p}\right)^2\right]^{\alpha^*_n} \: , \end{eqnarray} % where $n_s$ is the separatrix (edge) density representative of a pedestal in the density profile. The value of the temperature profile exponent is \hbox{$\alpha_T = 1.1$} and the value of the density profile exponent is \hbox{$\alpha^*_n = 1.3$}, yielding an $n/n_o$ ratio similar to that of a density profile without a pedestal ({\it i.e.}, \hbox{$n_s = 0$}) with \hbox{$\alpha_n = 0.3$}. The plasma internal energy is \begin{eqnarray} \label{e-2-2-9} W_p & = & 1.5 \, k_B \, \left(n_e \, T_e + \displaystyle\sum_j \, n_j \, T_j \right) \: , \end{eqnarray} where $k_B$ is the Boltzmann constant. The volume-averaged current density, plasma density, temperature, and density-weighted volume-averaged temperature are defined, respectively, as follows: % \begin{eqnarray} j_\phi & \equiv & \frac{I_\phi}{A_p} \: , \nonumber \\ n & \equiv & \frac{2 \pi}{A_p} {\,} \int^{r_p}_0 n(r) {\,} rdr \: , \nonumber \\ \bar{T} & \equiv & \frac{2 \pi}{A_p} {\,} \int^{r_p}_0 T(r) {\,} rdr \: , \nonumber \\ \label{e-2-2-10} T & \equiv & \frac{2 \pi}{n {\,} A_p} {\,} \int^{r_p}_0 T(r) {\,} n(r) {\,} rdr \: , \end{eqnarray} % where $I_\phi$ is the toroidal plasma current. For a density profile with no pedestal \hbox{($n_s = 0$)}, these integrations yield: % \begin{eqnarray} \frac{n}{n_o} & = & \frac{1}{1+\alpha_n} \: , \nonumber \\ \frac{\bar{T}}{T_o} & = & \frac{1}{1+\alpha_T} \: , \nonumber \\ \label{e-2-2-11} \frac{T}{T_o} & = & \frac{1+\alpha_n}{1+\alpha_n+\alpha_T} \: . \end{eqnarray} % Also, the plasma pressure would have the same radial profile as given by \hbox{Eq.~(\ref{e-2-2-7})} but with the exponent \hbox{$\alpha_p \equiv \alpha_n + \alpha_T$}. The plasma surface-averaged poloidal magnetic field, \hbox{$B_{\theta} \equiv \bar B_{\theta}(r_{p})$}, is produced by the toroidal current, $I_\phi$, and the external poloidal-field coils and is given by % \begin{eqnarray} \label{e-2-2-12} B_{\theta} & = & \frac{\oint B_{\theta} \, d\ell}{\oint \, d\ell} \: = \: \frac{\mu_{o} I_{\phi}}{2\pi a S} \: , \end{eqnarray} % where $\mu_o$ is the permeability of the vacuum. The toroidal magnetic field, produced by external TF coils, provides the primary confining field in a tokamak. The on-axis toroidal magnetic-field strength is \hbox{$B_\phi(R_T) \equiv B_{\phi o}$}, the toroidal beta is \hbox{$\beta \equiv 2\mu_o p/B^2_{\phi o}$}, and the poloidal beta is \hbox{$\beta_{\theta} \equiv 2\mu_o p/B^2_{\theta}$}. The edge safety factor, $q$, is approximated by\hbox{~\cite{Uck88,Fla81}} % \begin{eqnarray} \label{e-2-2-13} q & = & 5 C_q \, \frac{B_{\phi o} \, a}{I_\phi} \, \frac{\epsilon}{\left(1 - \epsilon^2\right)^2} \, S^2 \: , \end{eqnarray} % where the coefficient \hbox{$C_q \equiv C_q(\epsilon\beta_\theta, q) \approx 1$} and $I_\phi$ is in MA. The toroidal and poloidal beta values are related by~\cite{Fla81} % \begin{eqnarray} \label{e-2-2-14} \beta & = & \beta_\theta \, \left[\frac{C_q}{q} \, \frac{\epsilon}{\left(1 - \epsilon^2 \right)^2}\right]^2 \, S^2 \: . \end{eqnarray} % The \AI\ plasma operates in the first stability regime. Therefore, the toroidal beta is taken to be limited by the Troyon relation~\cite{Tro84}, % \begin{eqnarray} \label{e-2-2-15} \beta & = & C_T \, \frac{I_\phi}{a \, B_{\phi o}} \: . \end{eqnarray} The \AI\ reference MHD equilibrium is computed based on pressure and toroidal current-density profiles that are consistent with current-drive, transport, and impurity-control/particle-exhaust analyses. Values of plasma vertical elongation, $\kappa_\X$, approaching 2 are desirable in increasing plasma volume and fusion power. A base-case value of \hbox{$\kappa_\X = 1.8$}, however, is chosen based on the vertical stability analysis for \AI\ and on the desire to place the passive stabilization elements behind the blanket. The values of on-axis safety factor, \hbox{$q_o \sim 1.3$}, and average plasma-edge safety factor, \hbox{$q \sim 4.75$}, reflect the trade-offs between equilibrium and stability analyses and the effort to maximize the bootstrap-current fraction. The base-case toroidal beta is 1.9\% (corresponding to \hbox{$C_T = 0.032$~T$\,$m/MA}) and is found from \hbox{high-$n$} ballooning and \hbox{$n = 1$} kink stability analyses of the \AI\ equilibrium. Details of the plasma equilibrium and stability modeling are discussed in Sec.~3. The steady-state plasma power balance equates the input power ({\it i.e.}, the charged-particle fusion power, $P_{\alpha}$, absorbed current-drive power, $P_{CD}$, and ohmic power, $P_{\Omega}$) with the radiative and transport power losses to give \begin{eqnarray} \label{e-2-2-16} P_\alpha + P_{CD} + P_\Omega & = & P_{BR} + P_{CY} + P_{TR} \: . \end{eqnarray} The two primary radiation-loss channels are bremsstrahlung and cyclotron radiation. The electron-ion and electron-electron bremsstrahlung local power density (W/m$^3$) is generally given by~\cite{Mcn82} \begin{eqnarray} \label{e-2-2-17} P_{BR} & = & 5.35\X 10^{-37} n^2_e T_e^{1/2} \, \left[Z_{eff} \, \left(1 + 1.55\X 10^{-3} \, T_e + 7.15\X 10^{-6} \, T^2_e \right)\right. \nonumber \\ & & + \left. 4.14\X 10^{-3} \, T_e \right] \: , \end{eqnarray} % which includes correction for relativistic effects that tend to be significant only at temperatures above the typical \AI\ operating point at \hbox{$T \simeq 20$ keV}. The cyclotron- (synchrotron) radiation global power density (W/m$^3$) is given by~\cite{Mcn82} % \begin{eqnarray} \label{e-2-2-18} P_{CY} & = & 6.214\X 10^{-17} \, n_e \, T_e \, B^2_{\phi} \, \Phi \: , \end{eqnarray} where the Trubnikov absorption correction is given by~\cite{Mcn82}, \begin{eqnarray} \label{e-2-2-19} \Phi & = & \frac{5.198\X 10^{-3}}{\Lambda^{1/2}} \, T^{1.5}_e \left( 1 + \frac{22.61}{A \, T^{1/2}_e}\right)^{1/2} (1 - R_{CY})^{1/2} \: , \end{eqnarray} \begin{eqnarray} \label{e-2-2-20} \Lambda^{1/2} & = & 7.78\X 10^{-9} \left(\frac{n_e \, a}{B_\phi}\right)^{1/2} \:. \end{eqnarray} In the above expressions, the usual substitution \hbox{($B_{\phi} = B_{\phi o} \sqrt{1-\beta}$)} is made to correct for plasma diamagnetism. The reflectivity of the first wall, without holes, was estimated by Krajcik~\cite{Kra73} to be % \begin{eqnarray} \label{e-2-2-21} R^K_{CY} & = & 1 - \sqrt{2 \, \epsilon_o \, Z^* \, \omega_c \, \rho} \: , \end{eqnarray} % where $\epsilon_o$ is the permittivity of the vacuum, $\omega_c$ is the electron cyclotron frequency, \hbox{$Z^* \simeq 0.32 T_e + 11.6$} $\{${\it cf.,} Fig.~5 of Ref.~\cite{Kra73}$\}$ represents the effective harmonic, and $\rho$ is the electrical resistivity of the first-wall material. A correction to this expression to account for holes in the first wall was suggested by Schaffer~\cite{Sch89a} % \begin{eqnarray} \label{e-2-2-22} R^S_{CY} & = & \left(1-f_h\right) \, R^K_{CY} \:, \end{eqnarray} % where $f_h$ is the fraction of the first-wall surface area devoted to holes. Krajcik~\cite{Kra73} also provides a more accurate expression for synchrotron radiation (without holes). Denoting Krajcik's accurate expression by $K_c$, defining $K_\ell$ from Trubnikov \hbox{Eq.~(\ref{e-2-2-21})} through \hbox{$\Phi =$} \hbox{$K_\ell\sqrt{1-R_{CY}}$}, and limiting \hbox{$K_c < K_\ell$}, Werley~\cite{Wer89b} has proposed the hole-corrected effective reflectivity, % \begin{eqnarray} \label{e-2-2-23} R^W_{CY} & = & \left(1-f_h\right)- \left(1-f_h\right)\left(\frac{K_c}{K_\ell}\right)^2 \: . \end{eqnarray} % This expression is found to be more suitable for the poorly reflecting SiC-composite first wall used in \AI. It should be noted that a number of similar models exist and that they produce various results~\cite{Mcn82}, depending, for example, on the treatment of plasma profiles. Therefore, the simplified cyclotron-radiation model of \hbox{Eq.~(\ref{e-2-2-23})} must be viewed as an approximation to future calculations using more refined codes~\cite{Tam83}. The plasma-core radiation fraction is % \begin{eqnarray} \label{e-2-2-24} f_{RAD} & \equiv & \frac{P_{BR} + P_{CY}}{P_{\alpha} + P_{CD} + P_{\Omega}} \: . \end{eqnarray} Generally, the volume-averaged power densities are given by % \begin{eqnarray} \label{e-2-2-25} P_k & \equiv & \langle P_k (r) \rangle \, = \, \langle f_k [B(r),\ j(r),\ n(r),\ T(r)] \rangle \: , \end{eqnarray} % where the subscript $k$ denotes fusion, radiation, or ohmic-heating power density. The systems model calculates volume-averaged power densities, $P_k$, using averaged parameters; all profile information is contained in the profile-enhancement factors, $g_k$, where % \begin{eqnarray} \label{e-2-2-26} P_k & = & g_k \, f_k (\langle B \rangle, \langle j \rangle, \langle n \rangle, \langle T \rangle) \: . \end{eqnarray} % Henceforth, these quantities are volume-averaged and the notation \hbox{$\langle . \rangle$} is dropped, except as otherwise specified, for simplicity. The profile factors are then defined as % \begin{eqnarray} \label{e-2-2-27} g_k & \equiv & \frac{2 \pi}{P_k \, A_p} \, \int^{r_p}_0 P_k (r) \, rdr \: . \end{eqnarray} % For example, the average plasma DT fusion-power density is given by \begin{eqnarray} \label{e-2-2-28} \frac{P_F}{V_p} & = & 2.186\X 10^{-12} \, g_{_{DT}} \, n_{_D} n_{_T} \, \langle \sigma v \rangle_{_{DT}} \: , \end{eqnarray} % where \hbox{$\langle \sigma v \rangle_{DT}$} is the fusion reactivity evaluated at the density-weighted volume-averaged temperature, and $g_{_{DT}}$ is the fusion-power profile-correction factor. The values \hbox{$\langle \sigma v \rangle_{DT} (T_i)$} are obtained using standard fits~\cite{Hiv77} that have been extended to higher temperature with additional data~\cite{Gre67} and have been augmented with Los Alamos experimental measurements and a temperature-dependent fitting function~\cite{Jar86} in the range \hbox{$0 < T_i < 20$ keV}. The ohmic-heating profile-correction factor is % \begin{eqnarray} \label{e-2-20-29} g_\Omega & = & \frac{2 \pi}{\eta_\| (n,T) \, I^2_\phi \, A_p} \, \int^{r_p}_0 \eta_\| \left[ n(r), \, T(r)\right] \, \left[ j^2_\phi (r) + j^2_\theta (r) \right] \, rdr \: , \end{eqnarray} % where $\eta_\|$ is the classical Spitzer resistivity, \hbox{$\propto Z_{eff} T_e^{-3/2}$}. For the \AI\ base case with \hbox{$T \simeq 20$ keV}, the profile enhancement factors are: \hbox{$g_{DT} = 1.081$}, \hbox{$g_{DD} = 1.327$}, \hbox{$g_{BR} = 1.050$}, \hbox{$g_{CY} = 1.45$}, and \hbox{$g_\Omega = 1.346.$} A 0.10-m scrape-off layer (SOL) at the equatorial plane provides the standoff between the plasma separatrix and the first wall. For \AI,\ power and particle control is provided by two poloidal-field divertors. A simplified analytic scaling model~\cite{Coo89a}, based on detailed Braams code~\cite{Bra87} and other~\cite{Har84} results, has been developed and incorporated into the systems code for a rapid and flexible characterization of the \AI\ edge-plasma/divertor performance. This model estimates the peak plasma temperature at the divertor plate from % \begin{eqnarray} \label{e-2-2-30} T_{PL} & = & \frac{2m_j \, (L_s \, f_d \, \chi \, f_r)^2} {k_B \, (w \, \gamma \, W)^2 \, \lambda^4} \: , \end{eqnarray} where, \itemitem{$m_j \equiv$} plasma ion mass ($2.5 \X 1.66 \X 10^{-27}$ kg), \itemitem{$L_s \equiv$} is the edge field-line connection length between the watershed and null points, \itemitem{$f_d \equiv$} fraction of the total thermal power that is deposited on the plate being considered (typically $\sim$0.5 for two divertors), \itemitem{$\chi \equiv$} cross-field thermal diffusivity in the scape-off layer (typically $\sim$4~m$^2/s$), \itemitem{$f_r \equiv$} fraction of the thermal power in the divertor plasma that reaches the plate through the sheath (typically $\sim$0.71), \itemitem{$w \equiv$} ratio of parallel power flow at the divertor target to that in the scrape-off layer (typically $\sim$0.6), \itemitem{$\gamma \equiv$} sheath power-flow transmission coefficient (typically $\sim$7.2), \itemitem{$W \equiv$} ratio of plasma pressure at the divertor target to the pressure at the watershed (typically $\sim$0.6), and \itemitem{$\lambda \equiv$} radial scale length for the decay of power flow in the SOL appropriate for high-recycle (\hbox{$T_{PL} \ll T_{SOL}$}) or low-recycle limiting cases (\hbox{$T_{PL} \simeq T_{SOL}$}). \noindent The model also estimates the peak divertor-plate heat flux, % \begin{eqnarray} \label{e-2-2-31} \hat q_{PL} & = & \frac{P_H \, (1 - f_{RAD}) \, f_c \, f_d \, f_f \, \sin\alpha} {2\pi \, R_T \, f_x \, w \, \lambda} \: , \end{eqnarray} % where, \itemitem{$P_H \equiv$} plasma (input) heating power, \hbox{$ P_H = P_\alpha + P_{CD} + P_\Omega$}, \itemitem{$f_c \equiv$} fraction of the thermal power transported into the channel (inboard or outboard) being considered, \itemitem{$f_f \equiv$} fraction of the power in the divertor plasma that is absorbed by the divertor target (typically $\sim$0.9), \itemitem{$\alpha \equiv$} angle of inclination between the divertor plate and the magnetic surfaces in the poloidal cross section (typically \hbox{$10^\circ$}), and \itemitem{$f_x \equiv$} average flux expansion factor between the SOL and divertor plate (typically 2). The value of $f_{RAD}$ for a DT plasma at \hbox{$T \simeq 20$ keV} with \hbox{$Z_{ eff} \simeq 1.5$} and a highly reflective \hbox{($P_{CY} \simeq 0$)} first wall is \hbox{$\simeq 0.12$}. For \AI\ with an SiC-composite first wall \hbox{($R_{CY} \simeq 0.45$)}, values of \hbox{$f_{RAD} \simeq 0.5$} are predicted. It should be noted that the above provides only an estimate of $T_{PL}$ and $\hat q_{PL}$. Detailed edge-plasma transport calculations have been performed for \AI\ (Sec. 5). The estimates given above and detailed calculations both show that $T_{PL}$ and $\hat q_{PL}$ decrease as aspect ratio increases, which provides an incentive for high aspect-ratio operation. The plasma-energy confinement time, $\tau_E$, expressed in terms of the net heating power, \hbox{$P_{TR} \simeq P_H(1-f_{RAD})$}, is % \begin{eqnarray} \label{e-2-2-32} \tau_E & = & \frac{W_p}{P_H \left(1 - f_{RAD}\right)} \: . \end{eqnarray} % The ARIES systems code begins with an input value of the Lawson parameter, $n_i \tau_E$, to establish the physics design point. An internal MacNTAU power-balance calculation~\cite{Wer89b} and an auxiliary COLIF-code~\cite{Ker87} calculation are used to establish the appropriate ion constituent fractions and the corresponding values of $n_i \tau_E$ and $\tau_E$. Once $\tau_E$ is known, a comparison can be made to any of a number of proposed empirical scaling relations~\cite{Ite89,Uck88,Uck89}. It should be emphasized that no particular scaling law is assumed in advance to derive the \AI\ design point. The value of $\tau_E$ for a design point is compared to the various scaling alternatives ({\it e.g.,} Goldston) by means of a confinement multiplier, \hbox{$H_j \equiv \tau_E/\tau^j_E$}, where the superscript $j$ denotes the particular relation of interest. For example, the Goldston L-mode confinement time is given by % \begin{eqnarray} \label{e-2-2-33} \tau^G_E & = & {3.7\X 10^{-5} \, I_\phi \: R_T^{1.75}\, a^{-0.37} \, \kappa_\X^{0.5} \: \left(\frac{A_i}{1.5}\right)^{0.5}} \: {[P_H \, (1-f_{RAD})]^{-0.5} } \: , \end{eqnarray} % where $A_i$ is the atomic mass (2.5 for a nominal 50:50 DT fuel mixture). The current-drive cost and recirculating power have a direct impact on the economics of a steady-state reactor. Therefore, the amount of current drive by external means should be minimized. This is achieved in \AI\ by maximizing the bootstrap-current contribution and by decreasing the magnitude of I$_\phi$ ({\it e.g.,} increased $A$ and $q$, and decreased $\beta$). A solution of the Grad-Shafranov equation that is made self-consistent with both RF-driven and neoclassical current densities~\cite{Hsi89} provides the following expression % \begin{eqnarray} \label{e-2-2-34} \Gamma & \equiv & \frac{\Delta P_{CD}}{P_{CD}} \: \simeq \: \frac{f \, C \, \beta_{Io}}{A^{1/2}} \: , \end{eqnarray} % where $P_{CD}$ is the current-drive power if all the plasma current has to be driven by external means and $\Delta P_{CD}$ is the predicted current-drive power when neoclassical bootstrap effects are included. The on-axis Shafranov poloidal beta is given by % \begin{eqnarray} \label{e-2-2-35} \beta_{Io} & = & \frac{p_o}{(B^2_\theta/2\mu_o)} \: , \end{eqnarray} % where $B_{\theta}$ is given by \hbox{Eq.~(\ref{e-2-2-12})}. The coefficient $C$ is given by % \begin{eqnarray} \label{e-2-2-36} C & = & C_1 - \frac{\alpha_T}{\alpha_p} \left(C_2 + C_3 \, n_{eo} \, \frac{T_{io}}{Z_{eff} \, p_o} \right) \: . \end{eqnarray} % Typically \hbox{$C_1 = 2.1$}, \hbox{$C_2 = 1.3$}, \hbox{$C_3 = 1.2$}, and \hbox{$f \simeq$ 0.23}, which in actuality is a weak function of geometry and profiles. Although the expression for $\Gamma$ was derived for \hbox{A$\gg 1$}, \hbox{$\kappa = 1$}, and \hbox{$I_{BC}/I_\phi \ll 1$}, it agrees well with the numerical result given in Ref.~\cite{Hsi89}. This model is applied as stated to all ARIES systems calculations and is described in \hbox{Secs.~2.4.1} and~2.4.2. Both neutral-beam current drive (NBCD) and ion-cyclotron range-of-frequencies (ICRF) fast-wave current drive (FWCD) were considered for \AI, with lower-hybrid waves being used in both cases to drive current in the edge plasma. The NBCD is a prime candidate for ITER~\cite{Ite89} and promises a higher normalized current-drive efficiency, % \begin{eqnarray} \label{e-2-2-37} \gamma & = & \frac{(n_{e}/10^{20})R_{T}I_{\phi}}{P_{CD}} \: . \end{eqnarray} % The FWCD has a lower unit cost (\$/W), requires less technology extrapolation for RF sources, and is more compatible with the fusion power core (superior maintenance and safety characteristics). Because of these advantages and since no overriding total cost differences were found between NBCD and FWCD, FWCD is selected as the reference current-drive system for the \AI\ design. The following expression, corrected for electron trapping effects, is used in the ARIES systems code to scale the normalized efficiency of the FWCD: % \begin{eqnarray} \label{e-2-2-38} \gamma & = & 0.72 \: \bar T_e^{0.77} \, (0.041 + 0.235 \, \beta) \: , \end{eqnarray} % where $\bar T_e$ is the volume-averaged electron temperature. This expression is based on a fit to 19 self-consistent equilibrium/current-drive calculations with \hbox{$\alpha_p = 1.4$}, \hbox{$\alpha_n = 1.1$}, \hbox{$Z_{eff} = 1.5$}, and volume-averaged ion temperature of \hbox{$\bar T = 12$~keV}; limits on \hbox{$\bar T_e > 10$~keV} and \hbox{$\beta \leq 0.08$} are also noted~\cite{Hsi89}. The tritium-fuel source rate, $S_T$, in steady state balances the sum of the tritium burnup and loss rate, \hbox{$L_T = f_tn_i/\tau_p$}. The tritium burnup fraction, $f_B$, is % \begin{eqnarray} \label{e-2-2-39} f_B & \equiv & \frac{S_{T}-L_{T}}{S_{T}} = \left[ 1 + \frac{1} {g_{DT} \, f_D \, \langle\sigma v \rangle_{DT} \, n_i \, \tau_E \, \left( \tau_p / \tau_E \right)} \right]^{-1} \:\: , \end{eqnarray} % where $\tau_p$, the particle confinement time, is taken to be \hbox{$\tau_p \approx 4\tau_E$}. For the reference \AI\ design, \hbox{$f_B \simeq 0.19$}. The value of $\tau_p$ is assumed to be the same for both fuel ions and the fusion $\alpha$-particles, resulting in a steady-state concentration, \hbox{$n_\alpha/n_i \sim 10\%$}. The partial pressure, \hbox{$n_\alpha T_\alpha / (n_iT_i + n_e T_e)$}, of the thermalizing $\alpha$-particles reduces the ``useful beta'' of the device by $\sim$20\%, calculated consistent with Ref.~\cite{Den87}. \subsection{Reactor Engineering Model} \label{s-2-2-2} The systems-code model for reactor engineering characterizes the first wall, blanket, shield, coils, current-drive, and impurity control hardware in order to provide the basis for the cost estimate of the fusion power core (FPC). Additionally, the reactor power balance is modeled in order to size the balance-of-plant requirements of the \AI\ electric generating station. The inboard and outboard thicknesses of the first wall, blanket, and shield are determined by detailed calculations, discussed in \hbox{Sec.~8}, and held fixed by the systems code. The SiC-composite first-wall thickness is 0.012~m on both the inboard and outboard sides of the plasma. The inboard blanket thickness, $\delta_{Bi}$, is 0.67~m, and the outboard blanket thickness, $\delta_{Bo}$, is 0.97~m. The inboard shield thickness, $\delta_{Si}$, is 0.71~m and the outboard shield thickness, $\delta _{So}$, is 0.81~m. Thus, the inboard standoff between the inner plasma edge and the inside face of a TF coil, where the peak toroidal field, $B_{\phi c}$, is found is $\sim$1.5~m for all values of $a$ and $R_T$ searched by the systems code. The toroidal magnetic field is produced by a set of \hbox{$N_{TF} = 16$} D-shaped TF coils, carrying the coil current, $I_c$, and producing the on-axis field strength, \hbox{$B_{\phi}(R_T) \equiv B_{\phi o}$}, given by % \begin{eqnarray} \label{e-2-2-40} B_{\phi o} & = & \frac{\mu_o \, N_{TF} \, I_c}{2\pi \, R_T} \: . \end{eqnarray} % The peak toroidal magnetic-field strength on the inboard leg of the TF coil is given by % \begin{eqnarray} \label{e-2-2-41} B_{\phi c} & = & \frac{\mu_o \, N_{TF} \, I_c}{2\pi \, R_c} \: , \end{eqnarray} % where $R_c$ is the major radius of the inboard face of the inboard \hbox{TF-coil} leg. Using a convenient model~\cite{Tur77} to characterize the poloidal shape of the \hbox{TF coil}, the \AI\ \hbox{TF coil} is illustrated schematically in \hbox{Fig.~\ref{f-2-2-1}}. \begin{figure}[tb] \vspace*{4.6in} \caption[Fig-2-2-1] {Schematic elevation view of the \AI\ plasma and TF coil, showing the bucking cylinder (BC) and the notation used for various major toroidal radii. The view shown is a figure of revolution about the vertical axis (\hbox{$R = 0$}).} \label{f-2-2-1} \end{figure} The \hbox{TF-coil} set dominates the size and cost of a tokamak reactor, particularly for the \AI\ design with \hbox{$B_{\phi c}$ = 21 T}. The centrum of the \hbox{TF-coil} model is a scaling relationship between the overall current density, $j_{\phi c}$, and $B_{\phi c}$. This \hbox{TF-coil} scaling is given by~\cite{Sch88} % \begin{eqnarray} \label{e-2-2-42} j_{\phi c} & = & \frac{0.8 \, \sigma_{all} - B^2_{\phi c}/2\mu_o}{\sigma_{all} \, (1/j_{SC} + 1/j_{ST}) + (B_{\phi o} R_T/4)\, ln (R_2/R_1) - \sigma_{ST}/j_{ST}} \: , \end{eqnarray} % where $R_1$ and $R_2$ are, respectively, the major radii of the TF-coil inboard and outboard legs, and are in m. The factor \hbox{0.8 = 1/1.25} reflects the typical 25\% contribution of insulation and helium-coolant channels to the coil cross section. The allowable stress in the coil support structure is $\sigma_{all}$ and $\sigma_{_{ST}}$ is the stabilizer (tensile) yield stress (both in GPa). The stabilizer provides both the usual quench protection and (unlike for ITER) contributes to the overall load-carrying capability. Note that the structural term, (\hbox{$B_{\phi o} R_T/4)ln(R_2/R_1$}), dominates except for small values of $j_{SC}$. This \hbox{TF-coil} scaling relationship is utilized in the ARIES systems code, using the following formula for $j_{SC}$ % \begin{eqnarray} \label{e-2-2-43} j_{SC} & = & 10^4 \: \frac{1- (B_{\phi c}/46)^2}{B^{0.5}_{\phi c}} \:. \end{eqnarray} % Typical for the \AI\ design is \hbox{$j_{\phi c} = 28$ MA/m$^2$} at \hbox{$B_{\phi c} = 21$ T}, compared to the ITER design with \hbox{$j_c \simeq 14$ MA/m$^2$} at \hbox{$B_{\phi c} \simeq 11$ T}. \hbox{Figure~\ref{f-2-2-2}} gives a typical dependence of $j_{\phi c}$ on $B_{\phi c}$ for a range of coil scaling assumptions (\hbox{$R_0 = 7.5$ m}, \hbox{$R_1 = 3.5$ m}, \hbox{$R_2= 11$ m}, \hbox{$j_{ST} = 200$ MA/m$^2$}, and \hbox{$\sigma_{ST} = 0.8$~GPa} for options III and IV). \begin{figure}[tb] \vspace*{4.0in} \caption[Fig-2-2-2] {Scaling relationships for overall \hbox{TF-coil} current density and peak (inboard leg) \hbox{TF-coil} magnetic-field strength. Options I~and~II (independent of D-coil geometry) were used in the GENEROMAK/ESECOM studies~\protect\cite{She86,Del88a,Hol89,Kra88}. Options III~and~IV were developed~\protect\cite{Sch88} for advanced high-field applications, consistent with \AI. The ITER design point~\protect\cite{Ite89}, developed independently of these scaling laws, is shown for comparison.} \label{f-2-2-2} \end{figure} With the $j_c$ {\it versus} $B_{\phi c}$ relationship available and $N_{TF}$ specified, the inboard \hbox{TF-coil} cross section can be determined for a given reactor design ({\it e.g.,} $R_T$, $R_1$, $R_2$, $B_{\phi c}$, {\it etc.}). Specifications of the shield and standoffs determine the vertical half height of the (near-constant-tension) D-shaped TF coil. This D-shaped coil is extended to the outboard leg using the standard shape factor, which, when established by the vertical shield/divertor standoff relative to the elongated plasma, is more than adequate to meet the radial build requirements at the outboard equatorial-plane location. The toroidal-field ripple is monitored at the inboard plasma edge, the plasma magnetic axis, and the outboard plasma edge, with $N_{TF}$ being chosen to hold the outboard plasma-edge ripple to below $\sim$1\%. The outboard \hbox{TF-coil} ``transparency'' or coverage factor is also monitored to aid in defining the FPC layout and maintenance. The net centering forces generated by the \hbox{TF-coil} set are reacted by the hollow bucking cylinder, as is illustrated in \hbox{Fig.~\ref{f-2-2-1}}. The critical compressive pressure, $P_c$, at which the elastic buckling of the (thin) cylinder occurs is approximated by~\cite{Roa82} % \begin{eqnarray} \label{e-2-2-44} P_c & = & 0.807 \: \frac{E \, \delta^2_{BC} }{h_{BC} \, R_{BCi}} \, \left[ \left( \frac{1}{1-\nu^2} \right)^3 \, \frac{\delta_{BC}^2}{R_{BCi}^2} \right]^{1/4} \: , \end{eqnarray} where \hbox{$E = 2.9\X 10^{11}$~Pa} is Young's modulus for the bucking cylinder material (steel), \hbox{$\nu \simeq 0.27$} is Poisson's ratio, $h_{BC}$ is the vertical height of the bucking cylinder, and $R_{BCi}$ is the major toroidal inner (bore) radius of the bucking cylinder with thickness, \hbox{$\delta_{BC} \equiv R_{BCo} - R_{BCi}$}. The outer radius of the bucking-cylinder (actually a polygon with \hbox{$N_{TF} = 16$} sides) coincides with the major radius of the faces of the TF coils at major radius \hbox{$R_{BCo} = R_1 - \delta_{TF}/2$}. The external pressure on the bucking annular cylinder is \hbox{$P= (R_{BCo}/R_1)\,(B_{\phi C}^2/2\mu_o) = f_{ BC} P_c$}, where the factor \hbox{$f_{BC} = 0.20$} provides a margin of safety. \hbox{Equation~(\ref{e-2-2-44})} is solved for the bucking cylinder thickness, $\delta_{BC}.$ The bucking cylinder is also monitored for the compressive hoop stress, $\sigma_\theta$, at $R_{BCi}$, which is approximated by~\cite{Tim51} % \begin{eqnarray} \label{e-2-2-45} \sigma_\theta & = & - \, \frac{2R^2_{BCo} \, P}{R^2_{BCo} - R^2_{BCi}} \: , \end{eqnarray} % which is not allowed to exceed 1.0~GPa. The bucking cylinder is made sufficiently thick to meet the more restrictive of the constraints provided by \hbox{Eqs.~(\ref{e-2-2-44})} and \hbox{(\ref{e-2-2-45})}. This approximate procedure has been found to be comparable to, but more conservative than, the detailed coil-design calculation reported in Sec.~7. For example, finite-element analysis shows that the peak stress in the \AI\ TF coil is $\sim$700~MPa and, for the same design, the systems code predicts \hbox{1.0-GPa} stress levels. Additional coil structural support beyond the bucking cylinder must also be provided. Following \hbox{Ref.~\cite{Del88a}}, the steel structural volume is estimated to be 50\% of the \hbox{TF-coil} volume, pending a detailed mechanical design. The poloidal-field (PF) coils are modeled in the ARIES systems code as two divertor-field (DF) and two equilibrium-field coils. The ratio of DF-coil current to plasma current and the ratio of DF-coil major radius to plasma major radius is calibrated from detailed equilibrium calculations using the NEQ code~\cite{Str84}. These ratios, along with the specification of engineering current density, $j_c$, in the DF coil allow a coil sizing to be made. Similar PF-coil current and major radius ratios are provided by the NEQ code equilibrium calculations, but a second option using a dipole model to provide the correct vertical field at the plasma outboard edge is adequate for purposes of the systems study. The ratio of PF-coil to \hbox{TF-coil} mass is monitored to ensure agreement with detailed coil design activities, this ratio typically being in the range of 0.4 to 0.6. An important design choice in \AI\ is to locate all PF coils outside the \hbox{TF-coil} set. This specification, along with the requirement that a separatrix be formed within the vacuum chamber in a minimum-energy configuration, gives the dependence of triangularity, $\delta$, on elongation, $\kappa$, depicted in \hbox{Fig.~\ref{f-2-2-3}(A)}; lower $\kappa$ requires higher $\delta$ for a given aspect ratio in order to maintain an {\it in vacuo} separatrix. The dependence of $\kappa$ and $\delta$ on the inverse aspect ratio, as determined by the NEQ code, is weak and is modeled by~\cite{Gal89} % \begin{eqnarray} \label{e-2-2-46} \kappa_{\X} & = & \kappa_o \, (1 + 0.44 \: \epsilon^{2.1}) \: , \\ \label{e-2-2-47} \delta_{\X} & = & \delta_o \, (1 + 0.77 \: \epsilon^3) \: , \end{eqnarray} where the calibration coefficients are \hbox{$\kappa_o = 2.1$} and \hbox{$\delta_o = 0.53$} ({\it cf.,} Ref.~\cite{Gal89}). Maintaining this weak dependence on $\epsilon$ but including the $\delta$ {\it versus} $\kappa$ dependence shown in \hbox{Fig.~\ref{f-2-2-3}} gives~\cite{Bat89} % \begin{eqnarray} \label{e-2-2-48} \delta_o = - 0.5 \kappa_o + 1.56 \: . \end{eqnarray} % Since the coupling of PF coils located outside the \hbox{TF-coil} set is poor, and decreases, this $\delta$ {\it versus} $\kappa$ requirement translates into increased PF-coil currents and mass at the lower values of $\kappa$. The required PF-coil mass increase has been estimated by utilizing the multipolar field expansion~\cite{Bat89} used to generate \hbox{Fig.~\ref{f-2-2-3}(A)} and is expressed in \hbox{Fig.~\ref{f-2-2-3}(B)} as follows: % \begin{eqnarray} \label{e-2-2-49} \frac{\Delta M}{M_{PF}} & = & \left( \frac{188}{\kappa_{95}} \right)^{3.0} - 1 \: , \end{eqnarray} % where the reference point has been taken at \hbox{$\kappa = 1.88$}. The dependence on $A$ is weak and, therefore, is not modeled. \begin{figure}[p] \vspace*{7.1in} \caption[Fig-2-2-3] {(A)~Plasma triangularity as a function of vertical elongation to ensure a minimum-energy magnetic design for a range of plasma aspect ratios; and (B)~Added PF-coil mass required to maintain an {\it in vacuo} separatrix as the plasma vertical elongation increases.} \label{f-2-2-3} \end{figure} The thermal conversion efficiency of the blanket power, $P^B$, to electricity is denoted by $\eta^B_{TH}$. We denote the thermal-conversion efficiency for the divertor-coolant loop by $\eta^D_{TH}$, in order to model the case of a compound system (typically \hbox{$\eta^D_{TH} < \eta^B_{TH}$}) or the extreme case where the divertor power, $P^D$, is dumped to the environment as low-grade (waste) heat (\hbox{$\eta^D_{TH} \simeq 0$}). The power-to-wall area weighted-average thermal-conversion efficiency becomes % \begin{eqnarray} \label{e-2-2-50} \eta_{TH} & = & \frac{P^B \, \eta^B_{TH} + P^D \, \eta^D_{TH}}{P^B + P^D} \: , \end{eqnarray} % where % \begin{eqnarray} \label{e-2-2-51} P^B & = & f_{RAD} \, (1-f_D) \, (P_{CP}+P_{CD} +P_\Omega) + P_N \, M_N \: , \\ \label{e-2-2-52} P^D & = & (1 - f_{RAD} + f_{RAD} \, f_D ) \, (P_{CP}+P_{CD}+P_\Omega) \: , \end{eqnarray} % the core plasma-radiation fraction is \hbox{$f_{RAD} \simeq 0.5$}, \hbox{$f_D = A_D/A_{FW} \simeq 0.25$} is the fraction of the first-wall surface area devoted to the divertor plate surfaces, and \hbox{$P_{CP} \simeq P_\alpha$} is the charged-particle fusion power. The blanket neutron-energy multiplication is \hbox{$M_N = 1.3$}, as found from neutronics calculations reported in Sec.~8. In order to avoid the design complication of two separate coolant circuits, the \AI\ blanket and divertor use the same helium-coolant inlet/outlet temperatures and pressures to yield \hbox{$\eta^D_{TH} = \eta^B_{T H} = \eta _{TH} = 0.49$} (Secs.~5, 8, and 9). \begin{figure}[b] \vspace*{4.0in} \caption[Fig-2-2-4] {ARIES-I overall power-flow model as used in the systems code. Typically net electrical power is fixed. Other parameters adjust as plasma radius is varied for fixed plasma aspect ratio.} \label{f-2-2-4} \end{figure} The \AI\ power flow, as modeled in the systems code, is illustrated in \hbox{Fig.~\ref{f-2-2-4}}. Given a stipulated target for the net electric-power output, $P_E$, the thermal-power output, $P_{TH}$, is determined for a nominal value of the thermal conversion efficiency, $\eta_{TH}$, such that \hbox{$P_E = \eta_{TH} (1-\epsilon) P_{TH}$}, where \hbox{$\epsilon = 1/Q_E$} is the recirculating power fraction and \hbox{$Q_E \equiv P_{ET}/P_C$} is the engineering $Q$-value. The gross electric-power output is \hbox{$P_{ET} = \eta_{TH} P_{TH}$}. A fraction $f_{AUX}$ of $P_{ET}$ ($P_{AUX} = f_{AUX} P_{ET}$) is allocated for auxiliary functions. A fraction $f_{pump}$ of $P_{ET}$ (\hbox{$P_{pump} = f_{pump} P_{ET}$}) is allocated for primary-loop pumping power. Of this latter contribution to the recirculating power, it is assumed that \hbox{$0.90\, P_{pump}$} is recoverable as useful thermal power in the primary coolant loop. The engineering $Q$-value figure of merit, $Q_E$, can be written as % \begin{eqnarray} \label{e-2-2-53} Q_E & = & \frac{1}{\epsilon} \: = \: \eta_{TH} \: \frac{M_N \, P_N + P_{\alpha} + P_{\Omega} + 0.90 \, P_{pump}} {P_{AUX} + P_{pump} + \left(P_{CD}/ \eta_{CD}\right)} \: , \end{eqnarray} % The ratio of fusion power to absorbed current-drive power is the plasma $Q$-value, or gain, \hbox{$Q_p \equiv P_F/P_{CD} \simeq 20$} for \hbox{$Q_E \simeq 5$}. The average 14.1-MeV-neutron first-wall load, \hbox{$P_N/A_w = I_w$}, is given by % \begin{eqnarray} \label{e-2-2-54} I_w & = & \frac{14.06}{17.58} \: \frac{P_F \, x}{4\pi^2 \, A \, r_p^2} \: , \end{eqnarray} % where $P_F$ is the DT fusion power and \hbox{$x \equiv r_p/r_w$} is the plasma filling fraction, which is calculated based on the assumed scrape-off layer thickness, \hbox{$\delta_s = r_w - r_p = 0.10$~m}. Subsequent calculations of the poloidal distribution of the neutron wall load, including the effects of the source distribution, are performed using the NEWLIT code~\cite{Att85}. The peak-neutron wall load as computed from NEWLIT is 3.8~MW/m$^2$ at the outboard first wall at the equatorial plane, compared with the average value of 2.5~MW/m$^2$. The total FPC mass, $M_{FPC}$, is the sum of coil, blanket, shield, and structure component masses. The ratio $P_E/M_{FPC}$ in units of kWe/tonne is used as the FPC mass power density (MPD) figure of merit. The threshold of marginal economic interest, met by \AI, has been identified at $\sim$100 kWe/tonne~\cite{Con85}. Higher power-density systems lead to higher values of MPD, with the MPD value that minimizes COE being highly dependent on the confinement concept. \section{COSTING AND ECONOMICS} \label{s-2-3} \begin{table}[p] \caption{} \centering{\bf Fusion-Power-Core Unit Costs (1988 \$)$^{\bf (a)}$} \vspace{1.cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l c c}\hline Material/Component & Mass Density (kg/m$^3$) & Unit Cost (\$/kg) \\ \hline \hspace{1.75truein} & \hspace{1.75truein} & \hspace{1.75truein} \\ Breeder materials \\ \quad Li (natural) & \D500 & \Z47.5 \\ \quad Li$_2$O & 2,000 & \Z47.5 \\ \quad LiA$\ell$O$_2$ & 2,550 & \Z47.5 \\ \quad Li$_4$SiO$_4$ & 2,400 & \Z47.5 \\ \quad Li$_2$ZrO$_3$ & 4,150 & \phantom{$^{(b)}$}\Z51.3$^{(b)}$ \\ \quad FLiBe & 2,000 & \phantom{$^{(c)}$}\Z74.3$^{(c)}$ \\ \quad PbLi & 9,400 & \Z\Z4.3 \\ Structural materials \\ \quad PCA & 7,800 & \Z52.8 \\ \quad RAF & 7,800 & \Z52.8 \\ \quad Fe-1422 & 7,800 & \Z21.1 \\ \quad HT-9/Cu & 7,800 & \Z58.1 \\ \quad SiC (composite) & 3,200 & 400.0 \\ \quad A$\ell$ & 2,700 & \Z14.2 \\ Other materials \\ \quad SiC (bulk) & 3,200 & \Z50.0 \\ \quad Be & 1,850 & 530.5 \\ \quad BeO & 3,000 & 212.0 \\ \quad Cu & 8,900 & \Z58.5 \\ \quad C & 2,000 & \Z10.7 \\ \quad B$_4$C & 2,500 & \Z21.2 \\ Coils \\ \quad Ternary \NB\ & 7,300 & \Z90.0 \\ \quad \NB & 7,300 & \phantom{$^{(d)}$} \Z80.0$^{(d)}$ \\ \quad NbTi & 7,300 & \Z70.0 \\ \quad Cu & 7,300 & \Z57.0 \\ \quad Divertor system & -- & \phantom{~k\$/m$^2$}\Z63.0~k\$/m$^2$ \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Reported 1986-\$ unit costs (Ref.~\cite{Del88a}, primary source) are updated to 1988 \$.} \\ \multicolumn{3}{l}{$^{(b)}$Ref.~\cite{Abd87}.}\\ \multicolumn{3}{l}{$^{(c)}$$\sim$\$50/kg suggested~\cite{Smi84}.}\\ \multicolumn{3}{l}{$^{(d)}$Ref.~\cite{Tho88}.}\\ \end{tabular} \label{t-2-3-1} \end{table} The plasma physics and reactor engineering models of the ARIES systems code together provide a characterization of the \AI\ FPC that is adequate for conceptual cost projections. The systems-code unit-cost data base is summarized in \hbox{Table~\ref{t-2-3-1}}. Unit costs for NBCD and FWCD options are summarized in \hbox{Table~\ref{t-2-3-2}}. \begin{table}[tb] \caption{} \centering{\bf Current-Drive Unit Costs (1988 \$)} \vspace{1.0cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l c c} \hline & System Efficiency, $\eta_{CD}$ & Unit Cost (\$/W)\\ \hline \hspace{1.75truein} & \hspace{1.75truein} & \hspace{1.75truein} \\ 2-MeV neutral beam & 0.68 & 3.0 \\ Lower hybrid (80 MHz) & 0.68 & \Z1.25 \\ ICRF fast-wave$^{(a)}$\\ \quad \Z\Z80~MHz & 0.84 & 1.0 \\ \quad \Z158~MHz & 0.72 & 1.0 \\ \quad \Z250~MHz & 0.65 & 2.0 \\ \quad \Z800~MHz & 0.63 & 2.0 \\ \quad 2500~MHz & 0.48 & 2.0 \\ \quad 8000~MHz & 0.44 & 2.5 \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Ref.~\cite{Rei89}}\\ \end{tabular} \label{t-2-3-2} \end{table} The estimated cost of electricity (COE, mill/kWh) is the most important evaluation tool for optimizing and comparing with alternative energy sources. The estimated COE is the object function of the parametric systems analysis (PSA) code. Both constant-1988 and then-current-1994 dollar analyses (consistent with the assumed \hbox{$\tau_c = 6$-y} construction time) are used to evaluate the \AI\ costs. The bus-bar energy cost is given by % \begin{eqnarray} \label{e-2-3-1} \mbox{COE} & = & \frac{C_{AC} + \left(C_{O\&M} + C_{SCR}+ C_F\right) (1 + y)^Y} {8760 \, P_E \, p_f} \: , \end{eqnarray} % where \itemitem{COE} = Cost of electricity in constant or then-current dollars (mill/kWh); \itemitem{$C_{AC}$} = Annual capital cost charge, equals the ``overnight'' total capital cost multiplied \phantom{ =} by the fixed charge rate; \itemitem{$C_i$} = Cost of account $i$; \itemitem{$C_{O\&M}$} = Annual operations and maintenance cost, \hbox{$C_{40} + C_{41} + \dots + C_{47}$}; \itemitem{$C_{SCR}$} = Annual scheduled component-replacement cost, \hbox{$C_{50} +C_{51}$}; \itemitem{$C_F$} = Annual fuel costs, $C_{02}$ and $C_{03}$; \itemitem{$y$} = Annual escalation rate; \itemitem{$Y$} = Construction period (year); \itemitem{$P_E$} = Net plant electric-power output (MWe); \itemitem{$p_f$} = Plant availability factor; \itemitem{TDC} = Total direct cost = \hbox{$\displaystyle \sum^{26}_{i = 20} C_i$}; \itemitem{$C_{IDC}$} = Interest during construction, \hbox{$C_{97} = f_{IDC} \,$TDC}; \itemitem{$C_{EDC}$} = Escalation during construction, \hbox{$C_{98} = f_{EDC} \,$TDC}; \itemitem{TCC} = Total capital cost \hbox{= $\displaystyle \sum^{98}_{i = 20} C_i$}. \begin{table}[tb] \caption{} \centering {\bf Effective Cost of Money (COM) and\\ Levelized Annual Fixed-Charge Rate (LAFCR)$^{\bf (a)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{c c l}\hline Escalation Rate, $y$ (\%/y) & COM, $x$ (\%/y) & \multicolumn{1}{c}{LAFCR (\%/y)} \\ \hline \hspace{5.0truecm} & \hspace{3.0truecm} & \hspace{4.0truecm} \\ \Z0 & \Z4.2 & \Z8.0$^{(b)}$(10.0)$^{(c,d)}$ \\ \Z2 & \Z5.8 & \Z9.7 \\ \Z4 & \Z7.4 & 11.6 \\ \Z5 & NA & NA~~(15.0)$^{(d)}$ \\ \phantom{$^{(b,e)}$}\Z6$^{(b,e)}$ & \phantom{$^{(b,e)}$}\Z9.0$^{(b,e)}$ & 13.6$^{(e)}$ (16.5)$^{(c)}$ \\ \Z8 & 10.6 & 15.8 \\ 10 & 12.2 & 18.1 \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Ref.~\cite{Ham85}.}\\ \multicolumn{3}{l}{$^{(b)}$TITAN study~\cite{Naj90} reference case for constant-dollar mode.}\\ \multicolumn{3}{l}{$^{(c)}$Ref.~\cite{She86}.}\\ \multicolumn{3}{l}{$^{(d)}$Ref.~\cite{Sch78}.}\\ \multicolumn{3}{l}{$^{(e)}$TITAN study~\cite{Naj90} reference case for then-current-dollar mode.} \end{tabular} \label{t-2-3-3} \end{table} The detailed methodology for calculating the time-related cost factors is described in \hbox{Refs.~\cite{She86}} and \cite{Ham85} and will not be repeated here. This description differs from the U.S.\ fusion-reactor-community standards~\cite{Sch78,Phu79} used in the period 1980 to 1985 because of a slightly different ``s-shaped'' spending profile assumed, but represents the pending standard for U.S.\ fusion-reactor studies for the foreseeable future~\cite{Tho86}. Levelized annual fixed-charge-rate (LAFCR) values are summarized in \hbox{Table~\ref{t-2-3-3}}. The reference TITAN study~\cite{Naj90} assumed an annual inflation/escalation rate, \hbox{$y = 0.06$/y}, to give, under the standard assumptions~\cite{Ham85}, an annual utility cost of money (COM) of \hbox{$x = 0.09$/y}, a constant-dollar-mode LAFCR of 0.08/y, and a corresponding then-current-dollar-mode LAFCR of 0.136/y. Factors used to obtain interest-during-construction and escalation-during-construction costs are summarized in \hbox{Table~\ref{t-2-3-4}}. The TITAN study used the upper set of factors (\hbox{$y = 0.06$/y} and \hbox{$x = 0.09$/y}). The lower set, assuming \hbox{$y = 0.05$/y} and \hbox{$x = 0.10$/y}, is characteristic of the older U.S.\ fusion-community standard~\cite{Ham85,Sch79}. Differences between the standards, however, are not large. The economic parameters assumed in the costing model used in the \AI\ study are summarized in \hbox{Table~\ref{t-2-3-5}}. \begin{table}[p] \caption{} \centering {\bf Time-Related Cost Factors$^{\bf (a)}$\\ as Functions of Construction Lead Time} \vspace{0.5cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{c @{\hspace{1.0cm}} c c @{\hspace{1.0cm}} c}\hline Lead Time, $Y$ & \multicolumn{2}{c}{Capitalization Factors,} & Interest Factor,\\ (y) & $f^\prime_{cap}$ (nominal dollar) & $f^\prime_{cap0}$ (constant dollar) & $f^\prime_{IDC}$ \\ \hline \\ \multicolumn{4}{c}{\us{For $y = 0.06$/y and $x = 0.09$/y$^{(b)}$}} \\ \\ \Z1 & 0.0788 & 0.0177 & 0.0556 \\ \Z2 & 0.1558 & 0.0287 & 0.0908 \\ \Z3 & 0.2386 & 0.0399 & 0.1274 \\ \Z4 & 0.3273 & 0.0513 & 0.1652 \\ \Z5 & 0.4224 & 0.0629 & 0.2042 \\ \phantom{$^{(c)}$}\Z6$^{(c)}$ & 0.5244 & 0.0747 & 0.2444 \\ \Z7 & 0.6338 & 0.0866 & 0.2858 \\ \Z8 & 0.7511 & 0.0986 & 0.3284 \\ \Z9 & 0.8768 & 0.1109 & 0.3723 \\ 10 & 1.0117 & 0.1233 & 0.4176 \\ \\ \multicolumn{4}{c}{\us{For $y = 0.05$/y and $x = 0.10$/y}}\\ \\ \Z1 & 0.0812 & 0.0297 & 0.0617 \\ \Z2 & 0.1558 & 0.0484 & 0.1011 \\ \Z3 & 0.2359 & 0.0676 & 0.1424 \\ \Z4 & 0.3217 & 0.0873 & 0.1853 \\ \Z5 & 0.4135 & 0.1076 & 0.2299 \\ \Z6 & \phantom{$^{(d)}$}0.5120$^{(d)}$ & \phantom{$^{(d)}$}0.1282$^{(d)}$ & \phantom{$^{(d)}$}0.2761$^{(d)}$ \\ \Z7 & 0.6174 & 0.1494 & 0.3240 \\ \Z8 & 0.7303 & 0.1711 & 0.3738 \\ \Z9 & 0.8512 & 0.1933 & 0.4254 \\ 10 & 0.9808 & 0.2161 & 0.4790 \\ \\ \hline \multicolumn{4}{l}{$^{(a)}$$f_j \equiv 1+f^\prime_j$.}\\ \multicolumn{4}{l}{$^{(b)}${\it cf.}, Ref.~\cite{She86}, Table D.I, p.~237.}\\ \multicolumn{4}{l}{$^{(c)}$TITAN study~\cite{Naj90} base case for $Y = 6$~y construction time.}\\ \multicolumn{4}{l}{$^{(d)}${\it cf.}, Ref.~\cite{Sch78}; (IDC and EDC are, respectively, interest and escalation during construction.)}\\ \multicolumn{4}{l}{\phantom{$^{(a)}$}$f^\prime_{cap} \equiv (f_{IDC} + f_{EDC}) = (0.316 + 0.190) = 0.506,\ f^\prime_{cap0} = 0.129,\ f^\prime_{IDC} = 0.316$.}\\ \end{tabular} \label{t-2-3-4} \end{table} \begin{table}[p] \caption{} \centering{\bf Reference ARIES-I Economic Parameters$^{\bf (a)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{2.truecm}} c}\hline \\ Plant life (y) & 30 \\ Plant lead time (y) & \Z6 \\ Indirect cost factor & 0.35 \\ Contingency factor & 0.10 \\ Factor for escalation and interest during construction & 1.1652 \\ Nominal capacity factor & 76\% \\ Spare parts multipliers \\ \quad Blanket & 1.0 \\ \quad Coil & 1.0 \\ \quad Limiter & 1.0 \\ Effective cost of money \\ \quad Nominal dollars & 0.0957 \\ \quad Constant dollars & 0.0435 \\ Inflation rate (y$^{-1}$) & 5\% \\ Effective tax rate & 0.3664 \\ Tax depreciation life (y) \\ \quad Overall plant & 15 \\ \quad Replaceable blankets, {\it etc.} & \Z5 \\ Fixed charge rate \\ \quad Nominal dollars & 0.1638 \\ \quad Constant dollars & 0.0966 \\ \\ \hline \multicolumn{2}{l}{$^{(a)}${\it cf.}, Ref.~\cite{Del88a}, Table 3.1.}\\ \end{tabular} \label{t-2-3-5} \end{table} The direct-cost account entries, $C_i$, are obtained by applying relevant (installed) unit-cost estimates ({\it e.g.,} \$/W, \$/kg, \$/m$^3$), where known, to the calculated usage of these items in the conceptual design, such that \hbox{$C_i (\$) = u_i (\$/{\rm unit}) X_i$ (unit)}. A learning curve or mass production credit is taken for a ``tenth-of-a-kind'' commercial reactor installation, consistent with U.S.\ fusion-reactor design-community practice. The \AI\ study, like STARFIRE~\cite{Bak80} and most other U.S.\ fusion-reactor designs reported in the last decade, assumes unit costs consistent with these learning curve~\cite{Arg90} credits, rather than first-of-a-kind unit costs (including R\&D) appropriate for ITER~\cite{Ite89} or some other reported designs~\cite{Coo89b}. A ``75\% learning curve'' ({\it i.e.,} 0.75 progress ratio, $p$), as used for ARIES~\cite{Tho88}, represents the expectation that each doubling of production represents a (\hbox{$1 - p$}) reduction in unit costs. A ``tenth-of-a-kind'' reactor represents, nominally, $\sim$3.3 doublings of production or $\sim$50\% cost reduction relative to initial costs. Of course, actual production experience varies~\cite{Arg90}. \clearpage Often, the cost data base consists of cost scaling relationships of the form % \begin{eqnarray} \label{e-2-3-2} C_j \, (\$) & = & c_j \, (X_j)^{e_j} \: , \end{eqnarray} % where $X$ can be either a descriptive variable ({\it e.g.,} power, mass, volume) or a scaled variable, $X_j/X_{REF}$, related to a reference value, $X_{REF}$ of $X_j$, and $e_j$ is an appropriate scaling exponent (usually \hbox{$0 < e_j < 1$}). \hbox{Equation~(\ref{e-2-3-2})} can be rewritten in the form % \begin{eqnarray} \label{e-2-3-3} C_j & = & \frac{c_j \, (X_j)^{1-e_j}\ (X_j)^{e_j}}{(X_j)^{1 - e_j}} \: = \: \left[ \frac{c_j}{(X_j)^{1-e_j}}\right] \, X_j \: , \end{eqnarray} % which allows the definition of a (variable) unit cost % \begin{eqnarray} \label{e-2-3-4} u_j & = & \frac{c_j}{(X_j)^{1-e_j}} \: , \end{eqnarray} % and is a dependent function of the descriptive variable, $X_j$, itself. While the cost accounting scheme allows for detailed cost breakdowns (to four levels), only a relatively sparse, but comprehensive, subset of items are estimated and reported explicitly. The cost scaling exponents, $e_j$, used in the \AI\ study are typically consistent with those of the U.S.\ fission nuclear-reactor experience~\cite{Del88b} and represent the inclusion of quality-control costs associated with nuclear-grade (N-stamped) components. Potential cost savings derived from the substitution of conventional (non-nuclear) components under the condition of demonstrable inherent safety are significant but controversial~\cite{Lee86}. Two sets of such cost factors are summarized in \hbox{Table~\ref{t-2-3-6}}. These savings can amount up to 25\% on selected items, but have not been included in the \AI\ data base. The essential elements of the \AI\ cost data base are summarized in \hbox{Table~\ref{t-2-3-7}}. Costs that date from sources using a 1980-dollar reporting base are scaled to a 1988 reporting base by using the multiplicative factor 1.424. \begin{table}[tb] \caption{} \centering {\bf Safety-Assurance Cost-Credit Factors$^{\bf (a)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.2} \begin{tabular}{l c c}\hline Cost Area & Perkins' Factor~\cite{Del88a,Lee86} & ICF Factor~\cite{Del88a,May85} \\ \hline & \hspace{3.5truecm} & \hspace{3.5truecm} \\ Blankets & 2.0\Z & 1.11 \\ Shield & 2.0\Z & 1.11 \\ Coils & 1.44 & 1.11 \\ Reactor building and hot cells & 1.47 & 4.0\Z \\ Other structures and improvements & 1.47 & 1.15 \\ Heat transfer and transport & 2.5\Z & 1.11 \\ Other reactor plant equipment & 1.0\Z & 1.11 \\ Turbine plant equipment & 1.0\Z & 1.18 \\ Electrical plant equipment & 1.75 & 1.54 \\ Miscellaneous plant equipment & 1.3\Z & 1.67 \\ Heat reject system & 1.25 & 1.11 \\ Land & 1.0\Z & 1.18 \\ Indirect costs & 1.25 & 1.32 \\ O\&M costs & 1.0\Z & 1.32 \\ All other cost areas & 1.0\Z & 1.0\Z\\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Divisor factor applied to cost model accounts.}\\ \end{tabular} \label{t-2-3-6} \end{table} \begin{table}[p] \caption{} \centering {\bf Summary of ARIES-I/TITAN Cost Data Base$^{\bf (a,b)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l l l }\hline Acct. No. & \multicolumn{1}{c}{Account Title} & \multicolumn{1}{c}{Cost (M\$, 1980$^{(c)}$)} \\ \hline & \hbox to 3.00truein{\hfil} & \hbox to 2.25truein{\hfil} \\ {\bf 20.} & {\bf Land and Land Rights} & 3.3 \\ \\ {\bf 21.} & {\bf Structures and Site Facilities} \\ 21.1 & Site improvements and facilities & 11.28 \\ 21.2 & Reactor building & $3.0\X 10^{-4} V_{RB} + 39.5$ \\ 21.3 & Turbine building & 33.5 \\ 21.4 & Cooling structures & $7.135 \, (P_{ET}/1000)^{0.3}$ \\ 21.5 & Power supply and energy storage bldg. & 9.16 \\ 21.6 & Miscellaneous buildings & 76.5 \\ 21.7 & Ventilation stack & 1.81 \\ \\ {\bf 22.} & {\bf Reactor Plant Equipment (RPE)} \\ 22.1 & Reactor equipment \\ 22.1.1.1 & \ \ Breeding material: & Table 2.3-I \\ & \ \ \ \ Liquid metal (LM): PbLi$^{(d,e)}$ & (see Acct.~No.\ 26.1) \\ & \ \ \ \ \phantom{Liquid metal (LM):} Li$^{(d,f)}$ & (see Acct.~No.\ 26.1) \\ & \ \ \ \ Water solution: LiNO$_3$$^{(d,g)}$ & ($7.83 f_{6_{Li}} + 2.46) \, \X 10^{-3} \, M$ \\ 22.1.1.2 & \ \ Blanket and first-wall structure & $0.0533 \, M_{BL}$ \\ 22.1.1.3 & \ \ Be multiplier$^{(g)}$ & $0.3338 \, M_{Be}$ \\ 22.1.2 & \ \ Shield:\\ & \ \ \ \ V alloy$^{(f)}$ & $0.1855 \, M_{SHD}$ \\ & \ \ \ \ Ferritic steel$^{(g)}$ & $0.0157 \, M_{SHD}$ \\ 22.1.3 & \ \ Magnet coils: \\ & \ \ \ \ Normal conducting & Table 2.3-I \\ & \ \ \ \ Superconducting & Table 2.3-I \\ 22.1.4 & \ \ Supplemental (RF) heating systems & Table 2.3-II \\ 22.1.5 & \ \ Primary structure and support & $0.1125 \, V_{STR}$ \\ 22.1.6 & \ \ Reactor vacuum system & $0.015 \, M_{VAC}^{(f)} + 2.5$ (kg/d) \\ \\ \end{tabular} \label{t-2-3-7} \end{table} \begin{table}[p] \centering {\bf Table~2.3-VII (Cont'd)} \vspace{1.0cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l l l }\hline Acct. No. & \multicolumn{1}{c}{Account Title} & \multicolumn{1}{c}{Cost (M\$, 1980$^{(c)}$)} \\ \hline & \hbox to 3.00truein{\hfil} & \hbox to 2.25truein{\hfil} \\ 22.1.7 & \multicolumn{2}{l}{\ \ Power supply (switching, energy storage):} \\ & \ \ \ \ Normal-conducting coils & 18.55~\$/kVA \\ & \ \ \ \ Superconducting coils & 296.7~\$/(kVA)$^{0.8}$ \\ & \ \ \ \ IBC$^{(f)}$ & 37.09~\$/kVA \\ & \ \ \ \ OFCD$^{(f,g)}$ & 37.09~\$/kVA \\ & \ \ \ \ Other & 1.0 \\ & \ \ TF IBC busbars$^{(f)}$ & 3.034 \\ & \ \ DF IBC busbars$^{(f)}$ & 1.625 \\ 22.1.8 & \ \ Impurity control system & $0.66 \, A_{D}$ \\ 22.1.9 & \ \ Direct energy conversion$^{(h)}$ & 0.0 \\ 22.1.10 & \ \ ECRH breakdown system & 1.589 \\ 22.2 & Main heat-transfer system \\ 22.2.1 & \ \ Primary coolant: \\ & \ \ \ \ Li$^{(f)}$ & $u_j = 0.2013 \, [P_{TH} (1 - f_w)]^{-0.2}\ ^{(i)}$ \\ & & $X_j = P_{TH} (1 - f_w)$ \\ & \ \ \ \ He, H$_2$O$^{(g)}$ & $u_j = 0.1030 \, [P_{TH} (1 - f_w)]^{-0.2}\ ^{(i)}$ \\ & & $X_j = P_{TH} (1 - f_w)$ \\ 22.2.2 & \ \ Intermediate coolant system & $u_j = 0.1030 \, P_{TH}^{-0.2}\ ^{(i)}$ \\ & & $X_j = P_{TH}$ \\ 22.2.3 & \ \ Secondary coolant system & $u_j = 0.1030 \, (P_{TH} \, f_w)^{-0.2}\ ^{(j,e)}$ \\ & & $X_j = P_{TH} \, f_w$ \\ 22.3 & Auxiliary cooling systems & $6.7\X 10^{-4} \, P_{TH}$ \\ 22.4 & Radioactive waste treatment & $1.2\X 10^{-3} \, P_{TH}$ \\ 22.5 & Fuel handling and storage \\ 22.5.1 & \ \ Pellet injectors & 3.709 M\$~each $\X 2$ $^{(f,g,i)}$ \\ 22.5.2 & \ \ Fuel processing system & 0.5~(g/d)$^{0.7}$ \\ 22.5.3 & \ \ Fuel storage & 3.709 \\ 22.5.4 & \ \ Atmospheric tritium recovery & 0.2~(m$^3$/h)$^{0.6}$ \\ 22.5.5 & \ \ Water detritiation system: \\ & \ \ \ \ TITAN-I, ARIES-I & 5 \\ & \ \ \ \ TITAN-II & 140 \\ \\ \end{tabular} \end{table} \begin{table}[p] \centering {\bf Table~2.3-VII (Cont'd)} \vspace{1.0cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l l l }\hline Acct. No. & \multicolumn{1}{c}{Account Title} & \multicolumn{1}{c}{Cost (M\$, 1980$^{(c)}$)} \\ \hline & \hbox to 3.00truein{\hfil} & \hbox to 2.25truein{\hfil} \\ 22.6 & Other reactor plant equipment & $1.09\X 10^{-3} \, P_{TH}$ \\ 22.7 & Instrumentation and control & 23.41 \\ \\ {\bf 23.} & {\bf Turbine Plant Equipment} \\ 23.1 & Turbine generators & $59.9 \, (P_{ET}/1000)^{0.7}$ \\ 23.2 & Main steam system & $4.80 \, (P_{TH}/2860)$ \\ 23.3 & Heat rejection systems & $0.0632 \, (P_{TH}-P_{ET})^{0.8}$ \\ 23.4 & Condensing system & $13.8 \, (P_{ET}/1000)^{0.9}$ \\ 23.5 & Feed heating system & $7.55 \, (P_{TH}/2860)$ \\ 23.6 & Other turbine plant equipment & $40.9 \, (P_{ET}/1000)^{0.6}$ \\ 23.7 & Instrumentation and control & $7.8 \, (P_{ET}/1000)^{0.3}$ \\ \\ {\bf 24.} & {\bf Electric Plant Equipment} \\ 24.1 & Switchgear & $8.6 \, (P_{ET}/1000)$ \\ 24.2 & Station service equipment & $14.2 \, (P_{ET}/1000)$ \\ 24.3 & Switchboards & $5.4 \, (P_{ET}/1000)$ \\ 24.4 & Protective equipment & 2.11 \\ 24.5 & Electrical structures and wiring containers & $11.12 + 6.28 \, (P_{ET}/1440)$ \\ 24.6 & Power and control wiring & $23.0 + 13.0 \, (P_{ET}/1440)$ \\ 24.7 & Electrical lighting & 8.2 \\ \\ {\bf 25.} & {\bf Miscellaneous Plant Equipment} \\ 25.1 & Transportation and lifting equipment & 15.68 \\ 25.2 & Air and water service systems & 12.35 \\ 25.3 & Communications equipment & 6.22 \\ 25.4 & Furnishings and fixtures & 1.20 \\ \\ {\bf 26.} & {\bf Special Materials} \\ 26.1 & Reactor LM coolant/breeder:$^{(d)}$ \\ & \ \ PbLi$^{(e)}$ & $(7.83 f_{^6Li} + 2.46) \X 10^{-3} M_{LM}$ \\ & \ \ Li$^{(f)}$ & $(1169 f_{^6Li} + 58.0) \X 10^{-3} M_{LM}$ \\ 26.4 & Other & 0.25 \\ 26.5 & Reactor-building cover gas (Argon)$^{(f,h)}$ & 0.13 \\ \end{tabular} \end{table} \begin{table}[p] \centering {\bf Table~2.3-VII (Cont'd)} \vspace{1.0cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l l l }\hline Acct. No. & \multicolumn{1}{c}{Account Title} & \multicolumn{1}{c}{Cost (M\$, 1980$^{(c)}$)} \\ \hline & \hbox to 3.00truein{\hfil} & \hbox to 2.25truein{\hfil} \\ {\bf 90.} & {\bf Total Direct Cost (TDC)} \\ \\ {\bf 91.} & \multicolumn{2}{l}{\bf Construction Services and Equipment (10\% of TDC)} \\ \\ {\bf 92.} & \multicolumn{2}{l}{\bf Home Office Engineering and Services (10\% of TDC)} \\ \\ {\bf 93.} & \multicolumn{2}{l}{\bf Field Office Engineering and Services (10\% of TDC)} \\ \\ {\bf 94.} & {\bf Owner's Cost (5\% of TDC)} \\ \\ {\bf 95.} & {\bf Process Contingency (5\% of TDC)$^{(h)}$} \\ \\ {\bf 96.} & {\bf Project Contingency (10\% of TDC)} \\ \\ {\bf 97.} & {\bf Interest during Construction (IDC)} \\ \\ {\bf 98.} & {\bf Escalation during Construction (EDC)} \\ \\ {\bf 99.} & {\bf Total Capital Cost (TCC)} \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Gross electric power, $P_{ET}$, net electric power, $P_{E}$, and total thermal power, $P_{TH}$,}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$}are given in MW. Volumetric, $V$, (m$^3$) or corresponding mass, $M$, (tonne) unit costs}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$}for the FPC and related items are given as follows:}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$} \ \ Reactor building, $V_{RB} = 4 \, (R_T + r_s + 9)^2 (6 r_s) + 1.55\X 10^5$ (m$^3$);} \\ \multicolumn{3}{l}{\phantom{$^{(a)}$} \ \ Blanket structure (5\%), $M_{BL}$ (tonne); Shield, $M_{SHD}$ (tonne);}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$} \ \ Magnet coils, $M_C$ (tonne); Structure, $V_{STR}$ (m$^3$);}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$} \ \ Vacuum tank, $M_{VAC} = (0.07) (7.8) 2 \pi [(R_T + r_s + 3)^2 + 4r_s (R + r_s + 3)]$ (tonne);}\\ \multicolumn{3}{l}{\phantom{$^{(a)}$} \ \ Divertor-plate surface area, $A_{DIV}$ (m$^2$).}\\ \multicolumn{3}{l}{$^{(b)}$See Ref.~\cite{Naj90} for detailed TITAN design cost summaries.}\\ \multicolumn{3}{l}{$^{(c)}$1980 costs are multiplied by 1.424 to yield 1988 costs.}\\ \multicolumn{3}{l}{$^{(d)}$Liquid metal, $M_{LM}$ (tonne):\ $^6$Li enriched, $0.075 < f_{^6Li} < 0.90$~\cite{Bad82}.}\\ \multicolumn{3}{l}{$^{(e)}$Applicable to CRFPR.}\\ \multicolumn{3}{l}{$^{(f)}$Applicable to \TI.}\\ \multicolumn{3}{l}{$^{(g)}$Applicable to \TII.}\\ \multicolumn{3}{l}{$^{(h)}$Not applicable to ARIES-I.}\\ \multicolumn{3}{l}{$^{(i)}${\it cf.}, Eq.~(\ref{e-2-3-4}).}\\ \end{tabular} \end{table} For purposes of costing in the systems code, the reactor building is divided into a variable-volume reactor cell (housing the FPC and vacuum tank) and a fixed-volume region (housing the primary heat-transport loops). The volume of the latter portion is estimated to be \hbox{$1.55\X 10^5\,$m$^3$} and is similar to that of STARFIRE~\cite{Bak80} after escalating costs. The reactor room is modeled as a rectilinear enclosure extending horizontally 9$\,$m beyond the FPC with a height approximately three times that of the FPC, such that \hbox{$V_{RB}$ (m$^3$) $ = 16 {\,}(R_T + r_s + 9)^2 \, (6 r_s) + 1.55\X 10^5$}. The basic building structure (\hbox{Acct.~No.~21.2.1}) is priced at \$300/m$^3$, a value between that of STARFIRE~\cite{Bak80} and MARS~\cite{Log84} designs. To this value is added 2~M\$ for building services (\hbox{Acct.~No.~21.2.2}), \$40~M for containment structures (\hbox{Acct.~No.~21.2.3}), and \$10.1~M for architectural costs (\hbox{Acct.~No.~21.2.4}). The ARIES FPC is represented by most of the reactor equipment (\hbox{Acct.~No.~22.1}) items. The reactor-torus replacement-cost estimate applies a factor of 2 to the direct cost of these components to allow for the handling and replacement of the spent reactor-torus modules. For an assumed first-wall fluence life, \hbox{$I_w \tau = 20\,$MWy/m$^2$} at a cost-optimized neutron wall loading of \hbox{$I_w \simeq 2.5\,$MW/m$^2$} and a plant factor \hbox{$p_f \simeq 0.76$}, routine FPC replacement occurs every six years. \hbox{Account No.~50} represents $\sim$9\% of the base-case COE for \AI\ and is distinct from the nominal annual operations and maintenance charge (Acct.~Nos.~40--47, 51), previously estimated~\cite{Sch78,Sch79} to be $\sim$2\% of the direct cost, or scaled by~\cite{Del89a} % \begin{eqnarray} \label{e-2-3-5} C_{OP} \, \mbox{(M\$/y)} & = & 69.2 \, (P_E/1200)^{0.5} \end{eqnarray} % for the ARIES study. \begin{table}[b] \caption{} \centering{\bf Evolution of Financial Parameters } \vspace{1.cm} \renewcommand{\arraystretch}{1.2} \begin{tabular} {l c c c c c }\hline \multicolumn{1}{c}{Reference} & $(a)$ & $(b)$ & $(c)$ & $(d)$ & $(e)$ \\ \hline \\ Fixed charge rate (constant dollars) & 0.10\Z\Z & 0.10\Z\Z & 0.08\Z\Z & 0.0844 & 0.0966 \\ Fixed charge rate (then-current dollars) & 0.15\Z\Z & 0.15\Z\Z & 0.136\Z & -- & 0.1638 \\ Inflation/cost escalation rate (y$^{-1}$) & 0.05\Z\Z & -- & 0.06\Z\Z & 0.06\Z\Z & 0.05\Z\Z \\ Effective cost of money (y$^{-1}$) & 0.05\Z\Z & -- & 0.042\Z & 0.09\Z\Z & 0.0605 \\ Interest during construction$^{(f)}$ & 0.1290 & 0.1303 & 0.1124 & -- & 0.1652 \\ Indirect costs ($\X$ TDC) & 0.35\Z\Z & 0.23\Z\Z & 0.35\Z\Z & 0.375\Z & 0.35\Z\Z \\ Project contingency costs ($\X$ TDC) & 0.0\Z\Z\Z & 0.0\Z\Z\Z & 0.10\Z\Z & 0.20\Z\Z & 0.10\Z\Z \\ \\ \hline \multicolumn{6}{l}{$^{(a)}$Ref.~\cite{Sch79}.}\\ \multicolumn{6}{l}{$^{(b)}$STARFIRE~\cite{Bak80}.}\\ \multicolumn{6}{l}{$^{(c)}$TITAN~\cite{Naj90,Ham85}.}\\ \multicolumn{6}{l}{$^{(d)}$ESECOM/GENEROMAK~\cite{Del88a,Hol89}.}\\ \multicolumn{6}{l}{$^{(e)}$ARIES-I~\cite{Del89b}.}\\ \multicolumn{6}{l}{$^{(f)}$Design and construction lead time, $\tau_c = 6$~y.}\\ \end{tabular} \label{t-2-3-8} \end{table} Some historical drift in the financial parameters used to estimate the indirect, interest, and contingency costs are summarized in \hbox{Table~\ref{t-2-3-8}}. This drift complicates making cost comparisons between published fusion-reactor results if adjustments for differing assumptions are not made. Updates reflect, for example, changes in U.S.\ tax rates~\cite{Del88b,Del89b}. Comparisons with competitive fossil and fission energy sources are facilitated by a common set of modern financial guidelines. \section{DESIGN POINT DETERMINATION} \label{s-2-4} \subsection[Safety Factor Variation: Beta {\it vs} Magnetic Field] {Safety Factor Variation: Beta vs Magnetic Field} \label{s-2-4-1} Based on considerable theoretical and experimental effort in recent years, confidence has increased that tokamaks may be operated continuously with the equilibrium toroidal current, $I_\phi$, generated by RF waves~\cite{Fis87}. Furthermore, recent studies have indicated that RF current drive is marginally acceptable for a commercial reactor~\cite{Ehs87} in agreement with earlier estimates~\cite{Fis78}. Emphasis is placed now on improving the performance of large, current-driven tokamaks. The effort in the current-drive theory has been generally aimed at maximizing the local ratio of driven parallel current density to absorbed RF power, $j_\|/p_{RF}$. A more relevant parameter for a reactor is the plasma $Q_p$ (ratio of the fusion power, $P_F$, to the absorbed RF power, $P_{RF}$). Recent work~\cite{Ehs90} has shown that the RF current-drive power can be lowered by adjusting the MHD equilibrium in order to maximize the bootstrap-current contribution. This work departs from previous studies in two respects: (1)~the RF current drive is calculated self-consistently for specified equilibria that have been explicitly shown to be stable and (2)~MHD equilibrium is chosen to optimize the bootstrap-current contribution while retaining stability. One way of modifying the MHD equilibrium to reduce the current-drive power would be simply to reduce $I_\phi$ for a fixed toroidal beta, \hbox{$\beta \equiv 2\mu_o p / B^2_\phi$} ($B_\phi$ is the vacuum magnetic field at the major radius, $R_T$; and $p$ is the volume-averaged pressure). However, the theoretical and experimental efforts~\cite{Tro84} in the past decade have shown that the minimum ratio of current to beta is limited to % \begin{eqnarray} \label{e-2-4-1} \frac{I_\phi}{\beta} & = & R_T \, B_\phi \, (A \, C_T) \: , \end{eqnarray} % where $A$ is the aspect ratio, and the Troyon coefficient, $C_T$, is on the order of 0.03~T$\,$m/MA. It should be noted that theoretical MHD studies predict a ``second stability'' region~\cite{Sab89} as the poloidal beta increases, in which case $I_\phi$ can be reduced below the limit of \hbox{Eq.~(\ref{e-2-4-1})}. Here, poloidal beta is taken to be \hbox{$\beta_I \equiv 2\mu_o p /\left(\mu_o I_\phi A/2\pi R_T S\right)^2$}, where \hbox{$S \equiv c/(2\pi a)$}, $a$ is the plasma minor circumference, and \hbox{$a = R_T/A$}. Because of a lack of experimental data for second stability with high $\beta$, this regime of operation was ruled out for the \AI\ design. The alternative approach is to simultaneously reduce $I_\phi$ and $\beta$ while keeping the $I_\phi/\beta$ ratio fixed at the value given by \hbox{Eq.~(2.4-1)} and holding the plasma pressure constant. This can be done by increasing $q_o$, the safety factor at the magnetic axis, which is, in fact, another way of increasing $\beta_I$. As $q_o$ increases, the bootstrap contribution to the current increases and can exceed 80\% even for the relatively flat density and pressure profiles that might characterize a reactor. Even though raising $q_o$ in an ohmically driven device is difficult, because the current tends to peak sharply in the center where the temperature and conductivity are highest, increased $q_o$ is feasible in an RF-driven device. Several non-inductive current-drive \hbox{experiments~[\citelist{Sol87}\,--\,\citelist{Luc88}]} have, in fact, demonstrated this ability to modify the safety factor. In addition, more experimental evidence that shows that bootstrap currents do contribute substantially to the total current at high values of $\beta_I$ is becoming \hbox{available~[\citelist{Sim88}\,,\,\citelist{Zar88}\,--\,\citelist{Bec89}]}. A disadvantage of this approach is that the magnetic field must increase as $\beta$ decreases in order to keep the plasma pressure and fusion power constant. However, the reductions in current-drive power can be dramatic because a smaller amount of current needs to be driven. The relative costs of magnetic field and current drive for a particular device determine the attractiveness of this option. Indications exist that the trade-off is favorable as long as the field is not too large~\cite{Mil89}. In order to illustrate these points, equilibria were compared for tokamak reactors having the same $P_F$ and the same physical dimensions. The pressure, temperature, density, and impurity content were all held constant when comparing equilibria and calculating the current-drive requirements. Holding the plasma pressure constant implies that this set of equilibria satisfy% % \begin{eqnarray} \label{e-2-4-2} \beta \, B_\phi^2 & = & K^2 \: , \end{eqnarray} % where $K$ is constant. Several degrees of freedom are left in the equilibrium calculation, and parameters had to be chosen carefully in order to achieve MHD stability. A simple, broad pressure profile, \hbox{$p(\tilde{\psi}) = p_o \tilde{\psi}^{\alpha}$}, where $\tilde{\psi}$ is the normalized poloidal-flux function, was used with \hbox{$\alpha = 1.4$}. The diamagnetism was specified as \hbox{$RB \equiv F(\tilde{\psi}) = R_o B_o \left(1 - \delta \tilde{\psi}^{\alpha_F}\right)^{1/2}$}. These choices give conventional, peaked, current profiles and ensure that the current density vanishes at the limiter boundary (\hbox{$\tilde{\psi} = 0$}). The peak pressure, $p_o$, is determined by $P_F$, and the parameters $\delta$ and $\alpha_F$ were chosen to produce a series of equilibria for each value of $q_o$. The edge safety factor, $q$, was adjusted from a low value until ideal \hbox{$n = 1$} stability (with the conducting wall far from the plasma) and \hbox{$n \rightarrow \infty$} ballooning stability were both achieved. The kink stability was calculated using the \hbox{PEST-2} code~\cite{Gri83}, and the ballooning stability was calculated with the Phillips ballooning code~\cite{Phi88}. These equilibria were constrained to satisfy \hbox{Eq.~(\ref{e-2-4-1})} with \hbox{$C_T =0.030$~T$\,$m/MA}, as expected for highly elongated plasma cross sections~\cite{Phi88}. \begin{table}[tb] \caption{} \centering{\bf MHD Parameters and Current-Drive Requirements \\ of Three Representative Equilibria$^{\bf (a)}$} \vspace{1.cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{1.cm}} c c c}\hline & \hspace{3.truecm} & \hspace{3.truecm} & \hspace{3.truecm} \\ $q_a$ & 1.07 & 1.60 & 2.07 \\ $q_b$ & 3.56 & 6.24 & 8.47 \\ $I_o$ (MA) & 11.3 & 8.81 & 7.58 \\ $\beta$ (\%) & 3.23 & 1.98 & 1.51 \\ $B_o$ (T) & 10.5 & 13.4 & 15.3 \\ $\beta_I$ & 2.00 & 3.27 & 4.40 \\ $P_{RF}$ (MW) & 348. & 269. & 233. \\ $\gamma\ (10^{20}$~MA/MW-m$^2$) & 0.34 & 0.38 & 0.37 \\ $P^B_{RF}$ (MW) & 223. & 97.7 & 46.8 \\ $\gamma^B\ (10^{20}$~MA/MW-m$^2$) & 0.59 & 1.05 & 1.99 \\ $Q^B \equiv P_F/P^B_{RF}$ & 11. & 24. & 50. \\ \\ \hline \multicolumn{4}{l}{$^{(a)}$For equilibria stable to \hbox{$n = 1$} kink and $n \rightarrow \infty$ ballooning modes. Values with a}\\ \multicolumn{4}{l}{\phantom{$^{(a)}$}superscript $B$ include the bootstrap contribution. All cases having $R_o = 6.0$~m,}\\ \multicolumn{4}{l}{\phantom{$^{(a)}$}$A = 6.0$, elongation = 2.25, $S = 1.69$, $C_T = 0.30$~T$\,$m/MA, and the same}\\ \multicolumn{4}{l}{\phantom{$^{(a)}$}plasma parameters: $\bar n_e = 1.9\X 10^{20}$~m$^{-3}$, $n_{eo} = 2.7\X 10^{20}$~m$^{-3}$, $\bar T_e = 16$~keV,}\\ \multicolumn{4}{l}{\phantom{$^{(a)}$}$T_{eo} = 42$~keV, and $Z_{eff} = 1.5$, resulting in $P_F = 2360$~MW with DT fuel.}\\ \end{tabular} \label{t-2-4-1} \end{table} The three equilibria with parameters given in \hbox{Table~\ref{t-2-4-1}} were chosen as representative of the issues involved and are near the limit for \hbox{$n = 1$} kink stability. The case with \hbox{$q_o = 2.07$} was found to be stable to \hbox{$n = 1.5$}, 2, and 3 toroidal modes as well. The equilibrium with \hbox{$q_o = 1.07$} was in the first stability regime for ballooning, but near the limit (shown in \hbox{Fig.~\ref{f-2-4-1}}). For the other two equilibria, the unstable region between the first and second stability regimes (\hbox{Fig.~\ref{f-2-4-1}}) has shrunk to zero, so a stable access to higher $\beta$ exists for these cases with regard to ballooning modes. All three equilibria are robust in the sense that nearby equilibria with similar values of the safety factor are also stable for the value of $C_T$ assumed. However, the \hbox{$n = 1$} kink instability can arise when $q$ is in a narrow range around an integer~\cite{Phi88}; this result underscores the desirability of carefully controlling the safety factor with the RF current drive. \begin{figure}[tb] \vspace*{4.0in} \caption[Fig-2-4-1] {Phillips diagram~\protect~\cite{Phi88} for the case \hbox{$q_o = 1.07$}. The ballooning-unstable region for each flux surface in the plasma is predicted (shaded). As long as $-p^\prime (\tilde{\psi})$ is below this region, the equilibrium is stable in the first stability regime. The second stability regime is above this region. No ballooning-unstable regions were found for the other two cases shown in \hbox{Table~\ref{t-2-4-1}}.} \label{f-2-4-1} \end{figure} The existence of MHD equilibria with desirable stability properties was established and checked to determine whether they could be created with RF current drive. Simulations based on experimentally calibrated transport models~\cite{Sin88} suggest that density profiles may be rather flat in a reactor. Electron densities of the form \hbox{$n_e(\tilde{\psi}) = n_{eo} \tilde{\psi}^{\alpha _n}$} and temperature profiles of the form \hbox{$T_e(\tilde{\psi}) = T_{eo} \tilde{\psi}^{\alpha_T}$} were used with \hbox{$\alpha_n = 0.3$} and \hbox{$\alpha_T = 1.1$}. The RF current drive was provided by two different waves: (1)~the lower-hybrid slow wave at 8.0~GHz, which suffers strong Landau damping and thus generates current only near the plasma edge and (2)~the fast wave, which is known to couple well to electrons at high density and \hbox{temperature~[\citelist{Yam88}\,--\,\citelist{Eri89}]} and can provide the current density in the plasma interior. Following the method of Ref.~\cite{Ehs87}, a set of rays was selected for each equilibrium and the required RF power in each ray was calculated; a least-squares fit to \hbox{$G(\tilde \psi) \equiv \langle j_\| B \rangle/\langle B^2 \rangle$} from the equilibrium was made in order to duplicate the cases in \hbox{Table~\ref{t-2-4-1}}. Seven rays were used for these calculations, which were able to match the important equilibrium quantities such as the current density and safety factor to about 5\% or better; additional rays would give a closer match at the expense of more computer time. The current drive was calculated with and without the bootstrap contribution, and the results are summarized in \hbox{Table~\ref{t-2-4-1}}. When the bootstrap contribution is not included, $P_{RF}$ decreases modestly in proportion to $I_\phi$ as $q_o$ increases. This trend is expected from the relationship % \begin{eqnarray} \label{e-2-4-3} P_{RF} & = & \frac{n_e \, R_T \, I_\phi}{\gamma} \: , \end{eqnarray} % where previous work~\cite{Ehs87} has shown that $\gamma$ is a function only of $\bar T_e$ and $Z_{eff}$. \begin{figure}[tb] \vspace*{3.2in} \caption[Fig-2-4-2] {Contributions to the current density for the case \hbox{$q_o = 1.60$}. The solid line is the required \hbox{$G(\tilde \psi) \equiv \langle j_\| B \rangle/\langle B^2 \rangle$} for the specified equilibrium. The chain-dashed curve is the bootstrap contribution, and the other curves are the contributions for each of the RF waves. The dotted curve is the total, which matches the required $G(\tilde{\psi})$. This example had 16~MW of slow waves with \hbox{$\eta_{\|} = 1.5-3.2$} and 82~MW of fast waves with \hbox{$\eta_{\|} \approx 1.2$}.} \label{f-2-4-2} \end{figure} \begin{figure}[tb] \vspace*{4.2in} \caption[Fig-2-4-3] {Fractional RF current-drive-power reduction resulting from the bootstrap effect as a function of $\beta_I$ (normalized). Parameter $C$ is a function of the plasma profiles. Open symbols are former results for \hbox{$q_o \approx 1.1$}. Solid symbols are new results for equilibria tested for MHD stability and correspond to flat density profiles with \hbox{$\alpha_n = 0.3$}. Solid circles correspond to equilibria in \hbox{Table~\ref{t-2-4-1}} and the solid diamond is an additional case for \hbox{$A = 4.7$}, \hbox{$\kappa = 1.6$}, and \hbox{$q_a = 1.4$}.} \label{f-2-4-3} \end{figure} The full benefit of the high safety factor is evident when the bootstrap current is included~\cite{Hsi89}. As shown on \hbox{Fig.~\ref{f-2-4-2}}, the bootstrap contribution to $G(\tilde{\psi})$ for the case \hbox{$q_o = 1.60$}, is a substantial portion of the total current requirement. The RF-driven currents only need to supply the remaining fraction of the total, thereby greatly reducing the RF power requirements. The large decrease of bootstrap-aided current-drive power, $P^{B}_{RF}$, as $q_{(o)}$ increases is shown in \hbox{Table~\ref{t-2-4-1}} and in \hbox{Fig.~\ref{f-2-4-3}}. This decrease may be explained heuristically by rewriting $P_{RF}$ from \hbox{Eq.~(\ref{e-2-4-3})} along with \hbox{Eqs.~(\ref{e-2-4-1})} and \hbox{(\ref{e-2-4-2})} as \hbox{$P_{RF} = {n_e} \, R^2_T \, K \sqrt{\beta}/(A \, C_T \, \gamma)$}. This formula shows the benefit of increasing $C_T$ or reducing $\beta$ and can be modified to display the influence of the bootstrap effect by expressing $\beta$ in terms of $\beta_I$ and by defining \hbox{$\Gamma \equiv 1 - \left({\gamma}/{\gamma^B}\right) = 1 - \left( {P^B_{RF}}/{P_{RF}}\right)$} as the bootstrap ``fraction'' of the total current. The result is % \begin{eqnarray} \label{e-2-4-4} P_{RF}^B & = & \frac{5\X 10^{6} \left({\bar n_e} \, R_T^2 \, S \, K\right) \left(1 - \Gamma\right)}{A \, \gamma \, \sqrt{\beta_I}} \: . \end{eqnarray} % Since the bootstrap current remains constant for fixed density and temperature profiles, $Q$ increases as the required equilibrium current, $I_\phi$, decreases. The dependence of $Q_p$ as a function of $\beta_I$ is plotted in \hbox{Fig.~\ref{f-2-4-3}}. Calculations~\cite{Hsi89} with \hbox{$q_o \approx 1.1$} \hbox{(Fig.~\ref{f-2-4-3})} have previously quantified the monotonic increase of $Q_p$ with $\beta_I$. Based on \hbox{Eq.~(\ref{e-2-4-4})}, the high $\beta_I$ values associated with a high safety factor lead to large reductions in $P^B_{RF}$. It can be seen that \hbox{$\Gamma \geq 0.8$} can be achieved even with rather flat density profiles, given a sufficiently large safety factor. This result can lead to tokamak reactors with \hbox{$Q_p \approx 25-50$} compared to \hbox{$Q_p \approx 7-15$}~\cite{Ehs87} without the bootstrap contribution. In conclusion, specified MHD equilibria with desirable stability properties were reproduced numerically by launching properly tailored RF power spectra. With RF current drive, it is not necessary to constrain the safety factor to profiles typical of ohmic current drive (\hbox{$q_o \approx 1.0$}). Increasing the safety factor while maintaining stability can provide large bootstrap contributions to the current, even assuming flat density profiles. This approach greatly reduces the current-drive requirements. A possible drawback is that the toroidal field would have to increase to maintain the same fusion power for a reactor. \subsection[Temperature Variation: Current-Drive Power {\it vs} Fusion Power] {Temperature Variation: Current-Drive Power vs Fusion Power} \label{s-2-4-2} Key issues in optimizing steady-state tokamak-reactor design concern the variation of fusion power and current-drive power with the plasma temperature. In order to quantify the trade-offs, it was necessary to know the dependence of the normalized current-drive efficiency, $\gamma$, on plasma temperature. The values of $\gamma\left(\bar{T_e}\right)$ and $\gamma^B\left(\bar{T_e}\right)$ are calculated with self-consistent fast-wave currents and \hbox{2-D} MHD equilibria (superscript $B$ indicates quantities in the presence of bootstrap currents). The RF current-drive model described in \hbox{Sec.~4.4} includes magnetic trapping effects and also includes the Alfv\'en-wave-type damping associated with low-frequency ion-cyclotron range-of-frequency (ICRF) fast waves. Also, the bootstrap current is calculated with the best available model, as is described in Sec.~4.3. These new results are thus more accurate than previous estimates of $\gamma$ and $\gamma^B$~\cite{Hsi89,Ehs87}. The geometry is the reference \AI: \hbox{$R_T = 7.245$~m}, \hbox{$a = 1.555$~m} (95\% flux surface), \hbox{$\kappa = 1.6$}, \hbox{$\delta = 0.5$}. The equilibrium is also the reference: pressure, \hbox{$\beta = 1.9\%$}; axis safety factor, \hbox{$q_o \approx 1.3$}; and \hbox{$Z _{eff} = 1.6$}, with a broad density and narrow temperature profile (Sec.~3.5). The first task in this study was to calculate the power balance, plasma density, and fusion power for a range of temperatures (\hbox{$\bar{T_e} = 4-22$~keV}) with a fixed magnetic field (\hbox{$B_\phi = 11.6$~T}), and plasma current (\hbox{$I_\phi = 11.0$~MA}). Confinement times were adjusted such that \hbox{$T_i \approx T_e$}, the $\alpha$-particle density was set at 10\% of the fuel ions, and oxygen content was adjusted so \hbox{$Z_{eff} = 1.6$}. In addition, the electron confinement time was adjusted until the auxiliary heating power was roughly consistent with the current-drive power. For fixed beta and magnetic field, the fusion power density peaks at \hbox{$\bar{T_i} \approx \bar{T_e} \approx 6$~keV}~\cite{Ehs79}. However, the electron density is very high at low temperatures (\hbox{$n_e = 5\X 10^{20}$~m$^{-3}$ at 6~keV}), which presents a problem for noninductive current drive. \begin{figure}[tb] \vspace*{4.5in} \caption[Fig-2-4-4] {Calculated normalized current-drive efficiency (points) in the absence of any bootstrap effect, including electron trapping for low-frequency fast waves with transit-time-magnetic pumping, for \AI. The solid curve is the prediction of earlier theory~\protect\cite{Ehs87}.} \label{f-2-4-4} \end{figure} The second step in this study, therefore, made an explicit calculation of $\gamma$ and $\gamma^B$ at several points over the temperature range. An additional penalty to the high density operation at low temperature is the inherent dependence of $q$ on $\bar T_e$. Results are shown in \hbox{Fig.~\ref{f-2-4-4}} as discrete points and are compared with the simple scaling of $\gamma(\bar{T_e})$ previously developed~\cite{Ehs87}. Whereas the Alfv\'en-wave/transit-time magnetic-pumping current-drive mechanism raises $q$ relative to the Landau damping mechanism, the electron-trapping effect reduces $q$ relative to the simple field result. These effects tend to offset each other, with the fortunate result that the simpler scaling of $\gamma(\bar T_e)$ is an acceptable approximation to the improved calculation. \begin{figure}[tb] \vspace*{4.4in} \caption[Fig-2-4-5] {Fractional current-drive-power reduction due to the bootstrap effect. Triangles are present calculations with finite aspect-ratio theory~\protect\cite{Hir88}; circles are predictions of previous work~\protect\cite{Hsi89} but assuming banana regime (high-$T_e$) operation.} \label{f-2-4-5} \end{figure} The same series of equilibria were generated with the bootstrap current included. The fractional reduction in current-drive power, $\Gamma$, resulting from bootstrap effects~\cite{Hir88} is plotted in \hbox{Fig.~\ref{f-2-4-5}}; this result is compared with the previous predictions~\cite{Hsi89}. At all temperatures $\bar T_e$ above 15~keV, both calculations agree (\hbox{$\Gamma \simeq 0.65$}) and depend only on profiles and poloidal beta and, to a smaller extent, on $Z_{eff}$ and aspect ratio. The calculations show, however, a sharp decrease in $Q_p$ as the temperature is reduced below $\sim$10~keV. This decrease results from increased collisionality since the bootstrap-current coefficients diminish as more of the plasma enters the plateau regime. \begin{figure}[tb] \vspace*{4.2in} \caption[Fig-2-4-6] {Plasma performance and current-drive power at different temperatures for \AI\ (\hbox{$R_T = 7.3$~m}, \hbox{$B_\phi = 11.6$~T}, \hbox{$\beta = 0.019$}, \hbox{$I_\phi = 11.0$~MA}, and \hbox{$Z_{eff} = 1.6$}). Plotted are fusion power, $P_F$; average 14-MeV-neutron wall load, $I_w$; bootstrap-aided current-drive power, $P^B_{CD}$; and net electric power, $P_E$.} \label{f-2-4-6} \end{figure} Using the above results for the current-drive efficiency and the bootstrap fraction, \hbox{Fig.~\ref{f-2-4-6}} shows the plasma performance and current-drive power as function of the plasma temperature. The bootstrap-aided current-drive power, $P_{CD}^B$, is modest at the reference \AI\ temperature ($\sim$100~MW) but grows quickly as $\bar{T_e}$ is reduced ($\sim$1000~MW at 6~keV). Hence, the circulating power fraction is minimized at high temperature~\cite{Ehs82} (\hbox{$Q_p \equiv P_F/P^B_{CD} \geq 20$} at \hbox{$\bar{T_e} = 18$~keV}, while \hbox{$Q_p \leq 10$} at \hbox{$\bar T_e = 10$~keV}). However, the net electric power is at a maximum at the lower temperature, \hbox{$\bar T_e \leq 10$~keV}, using parameters similar to those of the final \AI\ design: current-drive system efficiency, \hbox{$\eta_{CD} = 0.71$}, blanket neutron-energy multiplication, \hbox{$M_N = 1.28$}, and thermal conversion efficiency, \hbox{$\eta_{TH} = 0.47$}. There are additional engineering penalties associated with low temperature operation, specifically a high neutron wall load (\hbox{$I_w > 4$~MW/m$^2$}) and high heat load on the divertor ($\alpha$-particle power plus large current-drive power). In fact, the STARFIRE study demonstrated~\cite{Bak80,Ehs82} that the minimum cost of power will occur at some intermediate temperature between the maxima in $P_{E}$ and $Q_p$. The reference temperature for \AI\ (\hbox{$\bar{T} \simeq 17$~keV}) also reflects this trade-off. \subsection{Parametric Results} \label{s-2-4-3} The results of \hbox{Secs.~\ref{s-2-4-1}} and \hbox{\ref{s-2-4-2}} establish the plasma engineering basis of the \AI\ design point on a self-consistent modeling framework. Using interim \AI\ parameters, an extensive parametric survey was performed and summarized in Ref.~\cite{Mil89}. General trends of this analysis also remain valid for the reference \AI\ design. At that point in the \AI\ study, an advanced \NAG\ \hbox{TF-coil} conductor with a \hbox{46-T} critical field was under consideration, leading to a design point with \hbox{$B_{\phi c} = 24$~T} at \hbox{$A = 4.5$}. The advanced \NAG\ superconductor is presently available only in the form of tape conductors. Losses in the tape conductor were found to be large during a disruption such that the TF coils would probably quench. Therefore, subsequent effort in the ARIES study has considered using a ternary \NB\ {\hbox{TF-coil} conductor with an \hbox{$\sim$26-T} critical field, leading to a design point with \hbox{$B_{\phi c} = 21$~T}, again at \hbox{$A = 4.5$}. Generally speaking the cost of the tokamak reactor (for any $A$) increases as its size increases ({\it i.e.,} larger $a$ or $R_T$ with fixed $A$). Small values of $a$ are associated with higher values of $B_{\phi o}$ and $B_{\phi c}$, resulting in a thicker build of the \hbox{TF-coil} inboard leg and a thicker bucking cylinder. Low values of $A$ yield higher values of $\beta$ and reduced $B_\phi$. The correspondingly higher values of $I_\phi$ result in higher $P_{CD}$ and, together with a lower bootstrap-current fraction (\hbox{$f_{BC} = I_{BC}/I_\phi$}), yield lower values of $Q_E$. Higher values of $B_{\phi c}$ at higher $A$ are associated with a more massive coil set and lower MPD values. The costs of more massive coils at higher $A$ trade off with the dominant costs of current drive at low $A$ to minimize the COE near \hbox{$A \simeq 4.5$}. The trade-off between FPC cost, reflected in MPD, and recirculating power costs associated with current drive, reflected in $Q_E$, is a recurrent theme in optimizing the tokamak reactor that, for the unit costs assumed, generally favors high MPD at the expense of reduced $Q_E$ for a given $A$. Low values of $Q_E$ also result in increased thermal-cycle and electric-power-generation costs, not in the FPC itself, but in the balance of plant. Economies of scale~\cite{Loo80,Spi83} built into the cost model (\hbox{Sec.~\ref{s-2-3}}) suggest some incentive for larger plant sizes ({\it e.g.,} 1200~MWe {\it cf.,} STARFIRE~\cite{Bak80}). Conversely, economic considerations of utility ``coverage ratios'' and other technical considerations ({\it e.g.,} divertor performance) suggest smaller plant sizes. A nominal value of \hbox{$P_E = 1000$~MWe} was selected for \AI\ similar to \TI~\cite{Naj90} (actually \hbox{$\sim$970~MWe}). \begin{figure}[tb] \vspace*{4.4in} \caption[Fig-2-4-7] {Projected COE as a function of plasma aspect ratio. The net plant output is fixed at \hbox{$P_E = 1000$~MWe}, peak \hbox{TF-coil} field at \hbox{$B_{\phi c} = 21$~T}, and \hbox{TF-coil} allowable stress at \hbox{$\sigma_{all} = 1000$~MPa}. The COE exhibits a shallow minimum (\hbox{$\sim$65~mill/kWh}) for \hbox{$A \simeq 4.5$}. The \AI\ design point is denoted by the filled circle.} \label{f-2-4-7} \end{figure} Figure~\ref{f-2-4-7} displays the \AI\ projected COE as a function of aspect ratio, $A$, for a fixed net electric power output, \hbox{$P_E = 1000$~MWe}. The peak \hbox{TF-coil} allowable stress is \hbox{$\sigma_{all} = 1000$~MPa}. Mass power density (MPD) is typically lower at higher $A$ as the required higher toroidal field leads to more massive \hbox{TF-coils} at lower $j_{\phi c}$ (MA/m$^2$) and the engineering $Q$-value is higher, reflecting lower current-drive power as plasma current decreases. Divertor-plate plasma temperature and peak heat flux are monotonically lower as $A$ increases. Average neutron wall load increases as $A$ increases. The COE exhibits a shallow minimum near $\sim$65~mill/kWh for \hbox{$A = 4.5$}. The sensitivity of the \AI\ design point to plasma and plasma engineering assumptions and technological performance were also investigated (\hbox{Fig.~\ref{f-2-4-8}} through \hbox{\ref{f-2-4-11}}). \hbox{Figure~\ref{f-2-4-8}(A)} displays the COE as a function of the Troyon coefficient, $C_T$ (\hbox{$\beta \propto C_T$}). It may be possible to increase $C_T$ for fixed aspect ratio, $A$, and safety factor, $q$, by optimizing and controlling the plasma profiles. The bootstrap-current fraction, $f_{BC}$, and current-drive efficiency, $\gamma$, are fixed at the reference values. The peak field on the TF coil is also fixed at its nominal value, \hbox{$B_{\phi c} = 21$~T}. Lower values of $C_T$ ({\it i.e.,} lower $\beta$) result in higher COE values. The gain to be obtained from higher $C_T$ values is significant, but less pronounced. \hbox{Figure~\ref{f-2-4-8}(B)} plots projected COE as a function of FWCD wall-plug efficiency, $\eta_{CD}$. As $\eta_{CD}$ increases, there is a direct reduction in the cost of current-drive power as well as less-obvious cost reduction in the overall thermal cycle as $Q_E$ increases. However, the leverage on COE available to incremental changes in $\eta_{CD}$ is fairly modest because the \AI\ overall design approach has been successful in reducing the plasma current by operating at high $A$, by increasing the bootstrap fraction, and by using high magnetic field. For the reference \AI\ design, the current-drive cost is $\sim$\$100~M and plasma $Q$-value is 20. \begin{figure}[p] \vspace*{7.1in} \caption[Fig-2-4-8] {Projected COE as a function of (A)~Troyon coefficient, and (B)~FWCD system ``wall-plug'' efficiency (\hbox{$P_E = 1000$~MWe}, \hbox{$B_{\phi c} = 21$~T}, \hbox{$\sigma_{all} = 1000$~MPa}, \hbox{$f_{BC} = 0.68$}, \hbox{$\gamma^\ast = 0.33\X 10^{20}$~A/W-m$^2$}, and \hbox{$A = 4.5$}). The \AI\ design point is denoted by the filled circle.} \label{f-2-4-8} \end{figure} \begin{figure}[p] \vspace*{7.1in} \caption[Fig-2-4-9] {(A)~The MPD as a function of peak \hbox{TF-coil} field strength; and (B)~Projected COE as function of MPD (\hbox{$P_E = 1000$~MWe}, \hbox{$B_{\phi c} = 21$~T}, \hbox{$\sigma_{all} = 1000$~MPa}, \hbox{$f_{BC} = 0.68$}, \hbox{$\gamma^\ast = 0.33\X 10^{20}$~A/W-m$^2$}, and \hbox{$A = 4.5$}). The \AI\ design point is denoted by the filled circle.} \label{f-2-4-9} \end{figure} \begin{figure}[tb] \vspace*{4.0in} \caption[Fig-2-4-10] {Projected COE as a function of peak \hbox{TF-coil} field strength (\hbox{$P_E = 1000$~MWe}, \hbox{$\sigma_{all} = 1000$~MPa}, \hbox{$f_{BC} = 0.68$}, \hbox{$\gamma^\ast = 0.33\X 10^{20}$~A/W-m$^2$}, and \hbox{$A = 4.5$}). The \AI\ design point is denoted by the filled circle.} \label{f-2-4-10} \end{figure} Figure~\ref{f-2-4-9}(A) illustrates the interaction of peak field on the TF coil, $B_{\phi c}$, and MPD. The MPD increases as a function of $B_{\phi c}$ from $\sim$60~kWe/tonne at 13~T to $\sim$100~kWe/tonne at \hbox{$B_{\phi c} \simeq 21$~T}, representative of the \AI\ design point. Access to higher fields using more advanced conductors or structures than invoked for \AI\ would continue this improvement. Of course, increased MPD is not an end in itself and improvements in MPD are limited by the peak heat load and other constraints. Rather, increased MPD is used as a predictor of improved COE performance~\cite{Kra88,Con85}, as indicated in \hbox{Fig.~\ref{f-2-4-9}(B)}. The COE decreases as a function of increasing MPD values for \AI\ as well as for many other fusion reactor concepts. Finally, displaying the same information in \hbox{Fig.~\ref{f-2-4-10}}, the COE is plotted as a function of $B_{\phi c}$. \begin{figure}[p] \vspace*{7.3in} \caption[Fig-2-4-11] {The direct costs of major reactor components as functions of the MPD figure of merit, reflecting higher peak \hbox{TF-coil} field at higher MPD (\hbox{$P_E = 1000$~MWe}, \hbox{$\sigma_{all} = 1000$~MPa}, \hbox{$f_{BC} = 0.68$}, \hbox{$\gamma^\ast = 0.33\X 10^{20}$~A/W-m$^2$}, and \hbox{$A = 4.5$}).} \label{f-2-4-11} \end{figure} The impact of MPD on the direct cost of major components of the \AI\ is shown in \hbox{Fig.~\ref{f-2-4-11}}. Consistent with \hbox{Fig.~\ref{f-2-4-9}}, MPD increases as peak \hbox{TF-coil} field increases and the FPC size (mass) decreases for fixed \hbox{$A = 4.5$}. The magnet cost, dominated by the \hbox{TF-coil} set, remains unchanged. First wall, blanket, and shield costs do drop for smaller reactors at higher MPD. The current-drive and main heat-transport costs are decoupled from MPD to first order. The cost contribution of reactor equipment (Acct.~No.~22.1) to reactor plant equipment (Acct.~No.~22) is $\sim$74\%. The ratio of reactor plant equipment to total direct cost is RPE/TDC $\simeq$63\%. Total direct cost (Acct.~No.~90) is projected to be $\sim$\$2.1~B (1988) and the total capital cost (Acct.~No.~99) is $\sim$\$3.6~B, for a ``tenth-of-a kind'' plant, including future learning-curve cost credits. \begin{figure}[tb] \vspace*{4.0in} \caption[Fig-2-4-12] {The variation of \AI\ confinement-time enhancement factor for a range of proposed, semi-empirical scaling relations~\protect\cite{Ite89,Uck88}. The \AI\ design point at \hbox{$a = 1.50$~m}, \hbox{$I_\phi = 10.2$~MA}, and \hbox{$B_{\phi c} = 11.3$~T} is denoted by the set of filled circles.} \label{f-2-4-12} \end{figure} In terms of confinement physics extrapolation, the \AI\ design point is displayed in \hbox{Fig.~\ref{f-2-4-12}} for a range of confinement-time enhancement factors, \hbox{$H^j \equiv \tau_E/\tau^j_E$}, obtained by applying a number of proposed semi-empirical scaling relations~\cite{Ite89,Uck88}. Included are the relations proposed by Goldston, Kaye and Goldston, Kaye (Kaye-Big, Kaye-All), Rebut and Lallia, Riedel and Kaye, Odajima and Shimomura (JAERI), together with ITER power law (ITER89-P) and offset linear (ITER89-OL) scaling~\cite{Ite89}. The required enhancement factor ranges from $\sim$1.8 to 3.3 as shown for the nominal \AI\ confinement time, \hbox{$\tau_E = 2.4$~s} at \hbox{$f_{RAD} = 0.5$}. It should be noted that several of the relations predict more required enhancement at larger values of $a$ ({\it i.e.,} lower $B_\phi$). The \AI\ study explores one of the possible approaches for improving conventional DT-tokamak reactor performance and in order to reduce the current-drive power requirements of steady-state operation~\cite{Bro87,Sch89b}. It is not possible to present here a comprehensive exposition of every parametric study performed that led to the selection of the reference \AI\ conceptual design point. Nor is it claimed that all potential sensitivities have been exhaustively explained. Rather, an attractive design window for first-stability tokamak reactor operation has been identified. Tokamaks in this design window operate with a relatively high aspect ratio, low plasma current, and high magnetic field. Future technical development and physics optimization can exploit this approach. \subsection{ARIES-I Reference Design} \label{s-2-4-4} \begin{table}[p] \caption{} \centering {\bf Parameters of the \AI\ First-Stability Tokamak Reactor} \vspace{0.5cm} \renewcommand{\arraystretch}{0.95} \begin{tabular}{l @{\hspace{1.truein}} c}\hline \\ Aspect ratio, $A = R_T/a$ & 4.5 \\ Major toroidal radius, $R_T$ (m) & 6.75 \\ Plasma minor half width, $a$ (m) & 1.50 \\ Plasma vertical elongation, $\kappa = b/a$ & 1.8 \\ Plasma-edge safety factor, $q$ & 4.50 \\ Troyon coefficient, $C_T$ & 0.032 \\ Plasma toroidal beta, $\beta$ & 1.92\% \\ Plasma temperature, $T_e \simeq T_i$ (keV) & 20 \\ Plasma ion density, $n_i$ ($10^{20}{\!}$/m$^{3}$) & 1.24 \\ Plasma electron density, $n_e$ ($10^{20}{\!}$/m$^{3}$) & 1.45 \\ Lawson parameter, $n_i\tau_E$ ($10^{20}$~s/m$^3$) & 2.9\Z \\ Goldston confinement-time multiplier, $H_G$ & 2.2 \\ ITER-89P confinement-time multiplier, $H_{89P}$ & 2.6 \\ Plasma $Q$-value, $Q_p = P_F/P_{CD}$ & 19.9 \\ On-axis toroidal magnetic field, $B_{\phi o}$ (T) & 11.3 \\ Peak TF-coil magnetic field, $B_{\phi c}$ (T) & 21.0 \\ Stored magnetic energy, $W_{B \phi}$ (GJ) & 132 \\ Plasma toroidal current, $I_\phi$ (MA) & 10.2 \\ Bootstrap-current fraction, $f_{BC}$ & 0.68 \\ Current-drive efficiency, $\gamma~(10^{20}$~A/W-m$^2$) [A/W] & 0.33 [0.034] \\ Absorbed current-drive power, $P_{CD}$ (MW) & 97 \\ 14.1-MeV-neutron wall loading, $I_w$ (MW/m$^2$) & 2.5 \\ Fusion power, $P_F$ (MW) & 1,925 \\ Thermal power, $P_{TH}$ (MW) & 2,544 \\ Gross electric power, $P_{ET}$ (MW) & 1,246 \\ Net electric power output, $P_E$ (MW) & 1,000 \\ Recirculating power fraction, $1/Q_E$ & 0.198 \\ Magnet costs (M\$) & \D392 \\ Current-drive costs (M\$) & \D104 \\ First-wall/blanket/shield costs (M\$) & \D338 \\ Total direct cost, TDC (M\$) & 2,140 \\ Unit direct cost, UDC (\$/kWe) & 2,140 \\ Total cost (M\$) & 3,617 \\ Cost of electricity, COE (mill/kWh) & 65.3 \\ Mass power density, MPD (kWe/tonne) & 99 \\ \\ \hline \end{tabular} \label{t-2-4-2} \end{table} The interaction of the parametric modeling described in \hbox{Sec.~\ref{s-2-4-3}}, and the detailed design activities of the ARIES project as a whole, led to the selection of the \AI\ reference design, as summarized in \hbox{Table~\ref{t-2-4-2}}. The nominal net electrical power output is \hbox{$P_E = 1000$~MWe}; the plasma aspect ratio is \hbox{$A = 4.5$} at a major toroidal plasma radius of \hbox{$R_T = 6.75$~m}; the peak \hbox{TF-coil} magnetic field is \hbox{$B_{\phi c} = 21$~T}; and the Troyon-limited toroidal beta is \hbox{$\beta \simeq 1.9\%$} leading to a modest plasma current of \hbox{$I_\phi = 10.2$~MA} and current-drive power, \hbox{$P_{CD} \simeq 97$~MW}. The recirculating power fraction is \hbox{$\epsilon = 1/Q_E \simeq 20\%$}; the average neutron wall load is \hbox{$I_w \simeq 2.5$~MW/m$^2$}; the MPD is 99~kWe/tonne; and the projected COE is \hbox{$\sim$65~mill/kWh}, using ``standard'' financial assumptions. For comparison, the corresponding values for ``median-experience'' and ``better experience'' fission pressurized-water reactors (\hbox{$P_E = 1100$~MWe}) are 78 and 46~mill/kWh, respectively, developed on the same cost-accounting basis. Coal-fired plants (\hbox{$P_E = 2\X$~550~MWe}) costs are projected at 50~mill/kWh~\cite{Del88b,Del88c}. The cost of the \AI\ reactor are summarized on \hbox{Table~\ref{t-2-4-3}}. \begin{table}[p] \caption{} \centering {\bf ARIES-I Systems-Code Cost Summary (1988 \$)} \vspace{1.0cm} \renewcommand{\arraystretch}{0.8} {\small \begin{tabular}{l l c}\hline \\ Acct.\ No. & \hfil Account Title \hfil & Million Dollars \\ \\ \hline \\ 20. & Land and land rights & \D\Z\Z5.3 \\ 21. & Structures and site facilities & \D339.3 \\ 22. & Reactor plant equipment (RPE) & 1,362.8 \\ 22.1.1 & \ \ Blanket and first wall & \D260.3 \\ 22.1.2 & \ \ Shield & \D\Z77.3 \\ 22.1.3 & \ \ Magnets & \D392.3 \\ 22.1.4 & \ \ Supplemental heating systems (current drive) & \D103.9 \\ 22.1.5 & \ \ Primary structure and support & \D\Z63.6 \\ 22.1.6 & \ \ Reactor vacuum systems (unless integral elsewhere) & \D\Z32.2 \\ 22.1.7 & \ \ Power supply, switching, and energy storage & \D\Z51.3 \\ 22.1.8 & \ \ Impurity control & \D\Z\Z7.9 \\ 22.1.9 & \ \ Direct energy conversion system & N/A \\ 22.1.10 & \ \ ECRH breakdown system & \D\Z23.0 \\ 22.1 & \ \ Reactor equipment & 1,011.8 \\ 22.2 & \ \ Main heat transfer and transport system & \D229.7 \\ 23. & Turbine plant equipment & \D245.3 \\ 24. & Electric plant equipment & \D137.1 \\ 25. & Miscellaneous plant equipment & \D\Z50.5 \\ 26. & Special materials & \D\Z\Z0.6 \\ 90. & Total direct cost & 2,140.7 \\ 91. & Construction services and equipment (10\%) & \D214.1 \\ 92. & Home office engineering and services (10\%) & \D214.1 \\ 93. & Field office engineering and services (10\%) & \D214.1 \\ 94. & Owner's costs (5\%) & \D107.0 \\ 96. & Project contingency (10\%) & \D214.1 \\ 97. & Interest during construction (IDC) & \D512.8 \\ 98. & Escalation during construction (EDC) & \D\Z\Z0.\Z \\ 99. & Total capital cost & 3,616.9 \\ \\ & & \us{Constant Dollars} \\ % have to keep [90] etc in {} or LATEX will screw up. {[90]} & Unit direct cost, UDC (\$/kWe) & 2,140.8 \\ {[94]} & Unit base cost, UBC (\$/kWe) & 3,104.1 \\ {[99]} & Unit total cost, UTC (\$/kWe) & 3,616.9 \\ \\ & Capital return (mill/kWh) & 52.6 \\ {[40-47,51]} & O\&M (1.4\%) (mill/kWh) & \Z6.7 \\ {[50]} & First-wall/blanket replacement (mill/kWh) & \Z5.6 \\ & Decomissioning allowance & \Z0.5 \\ {[02]} & Deuterium fuel (mill/kWh) & \Z\Z0.04 \\ & Cost of electricity, COE (mill/kWh) & 65.3 \\ \\ \hline \end{tabular} } \label{t-2-4-3} \end{table} Standard assumptions (Sec.~\ref{s-2-3}) regarding construction time (\hbox{$\tau_c = 6$~y}), plant availability (\hbox{$p_f = 0.76$}), economies of scale, and operation and maintenance charges are used to estimate the constant-dollar (1988) COE. The sensitivity of \AI\ costs to assumed construction time, $\tau_c$, is summarized in \hbox{Table~\ref{t-2-4-4}}. Without developing a detailed construction schedule, the \AI\ lead time is taken to be six years, in common with the times of the STARFIRE~\cite{Bak80}, MARS~\cite{Log84}, and TITAN~\cite{Naj90} studies. \begin{table}[p] \caption{} \centering{\bf Impact of Construction Lead Time$^{\bf (a)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.1} \begin{tabular}{c c c c}\hline Lead Time (y) & $f_{IDC}$ & Total Cost (M\$) & COE (mill/kWh) \\ \hline & \hspace{1.5truein} \\ 4 & 0.1118 & 3,451 & 62.9 \\ 5 & 0.1381 & 3,533 & 64.1 \\ \phantom{$^{(b)}$}6$^{(b)}$ & 0.1652 & 3,617 & 65.3 \\ 7 & 0.1931 & 3,703 & 66.6 \\ 8 & 0.2219 & 3,793 & 67.9 \\ 9 & 0.2515 & 3,885 & 69.2 \\ 10\Z & 0.2821 & 3,980 & 70.6 \\ \\ \hline \multicolumn{4}{l}{$^{(a)}$For all cases, the total direct cost is \$2,141~M (1988).}\\ \multicolumn{4}{l}{$^{(b)}$ARIES-I base case.}\\ \end{tabular} \label{t-2-4-4} \end{table} \begin{table}[p] \caption{} \centering{\bf Impact of Safety-Assurance Cost Credits on COE (mill/kWh)$^{\bf (a)}$} \vspace{1.0cm} \renewcommand{\arraystretch}{1.1} \begin{tabular}{l c c c}\hline & Base Case$^{(b)}$ & Option-1 & Option-2 \\ \hline & \hspace{1.1truein} & \hspace{1.1truein} & \hspace{1.1truein} \\ Capital return & 52.6 & 37.7 & 43.5 \\ Operation and maintenance & \Z6.7 & \Z6.7 & \Z5.1 \\ First-wall/blanket replacement & \Z5.6 & \Z2.8 & \Z5.0 \\ Decomissioning & \Z0.5 & \Z0.5 & \Z0.5 \\ Fuel & \Z\Z0.04 & \Z\Z0.04 & \Z\Z0.04 \\ \\ Total COE & 65.3 & 47.7 & 54.1 \\ \\ \hline \multicolumn{4}{l}{$^{(a)}${\it cf.,} Table~\ref{t-2-3-6}.}\\ \multicolumn{4}{l}{$^{(b)}$\AI\ without passive-safety credits.}\\ \end{tabular} \label{t-2-4-5} \end{table} \begin{figure}[p] \vspace*{6.9in} \caption[Fig-2-4-13] {The COE as a function of plasma minor radius for fixed plasma aspect ratio, \hbox{$A = 4.5$} and the other indicated fixed parameters. Curves of constant net electrical power output, $P_E$, are shown. Contours of fixed average neutron wall load, $I_w$, are included. The boundary with peak \hbox{TF-coil} field strength, \hbox{$B_{\phi c} = 21$~T}, is also included, restricting access to small values of $a$.} \label{f-2-4-13} \end{figure} Figure~\ref{f-2-4-13} summarizes the \AI\ design space and exhibits the economy of scale associated with plant size. Reactors to the left of the chain-dashed line are physically smaller and have higher neutron wall load and MPD, but require technology beyond that of \AI\ to achieve \hbox{$B_{\phi c} \geq 21$~T}. The incremental COE savings at higher $P_E$, taking advantage of the economies of scale, implies more severe divertor-plate loads. In the cost estimates for \AI, no safety-assurance cost credits~\cite{Hol89} have been taken, which may be expected to result from the use of low-activation materials. These credits lower the reported COE by 20\% to 25\%, as suggested in \hbox{Table~\ref{t-2-4-5}}, by using the passive-safety cost credits summarized in \hbox{Table~\ref{t-2-3-6}}. A more restrictive application of the passive-safety cost credits (which retains N-stamp costs for the TF coils, blanket, and main heat-transport system) results in a more modest $\sim$14\% reduction in projected COE. \begin{figure}[p] \vspace*{7.3in} \caption[Fig-2-4-14] {Dependence of COE on peak \hbox{TF-coil} magnetic-field strength, for \AI\ assuming various \hbox{TF-coil} unit costs, $C_{TF}$. A conceptual ITER-reactor extrapolation and a preliminary second-stability \AII\ case are also included.} \label{f-2-4-14} \end{figure} To summarize \AI\ costs, \hbox{Fig.~\ref{f-2-4-14}} displays estimated COE as a function of peak \hbox{TF-coil} magnetic-field strength, $B_{\phi c}$. The nominal \AI\ TF-coil unit cost is \$90/kg. The impact of different TF-coil unit costs are also shown. For comparison, a conceptual \hbox{1000-MWe} ITER extrapolation (\hbox{$A \simeq 3$}) using nearer-term 14-T \hbox{TF coils}, consistent with Refs.~\cite{Dev89} and \cite{Hen89}, are shown with \hbox{COE $= \simeq 90$~mill/kWh}. \AI\ represents a significant improvement of the tokamak approach, given the availability of high field-coil technology and physics performance with high radiation fraction, \hbox{$f_{RAD} = 0.5$}, and large bootstrap-current contribution, \hbox{$f_{BC} = 0.68$}. Also, a preliminary result for the second-stability-regime \AII\ (\hbox{$\beta \simeq 16\%$}) is included, although work on this tokamak approach is ongoing~\cite{Mil90} and COE values may change. \section{CRITICAL ISSUES} \label{s-2-5} In addition to the engineering feasibility of various subsystems, the following issues impact the selection of cost-optimized design points: \begin{enumerate} \item The cost of fabricating large SiC-composite components (assumed to be \$400/kg for \AI\ blanket and \$50/kg for bulk SiC shield, as described in \hbox{Sec.~8.2.5}); \item The cost of balance of plant for achieving the gross thermal efficiency of 49\% (650\degC\ and 10-MPa helium coolant with a primary heat-transport cost of \$0.09/Wt, and 650\degC\ steam-turbine plant equipment at \$0.20/We); \item The cost of zirconium isotope separation needed for \LZ\ breeder (\$2000/kg as described in \hbox{Sec.~10.4.1}) and for cost implication of other low-activation breeders ({\it e.g.}, Li$_2$O, Li$_4$SiO$_4$); \item The economic impact (from resource, safety, and environmental viewpoints) of using large quantities of Be; \item Realization of plant availability of 0.76. \end{enumerate} The \AI\ design reflects major improvements when compared with past ``conventional'' tokamak reactor studies: The reactor cost is reduced by a factor two and, at the same time, significant advances in physics realism are achieved and dramatic improvement in safety and environmental attributes are made. This reduction in cost is mostly due to the advanced fusion-power core of \AI\ (advanced high-field superconducting magnet and SiC-composite structural material). The \AI\ cost of electricity (COE) is projected at 65.3 mill/kWh, which is comparable to median-experience fission reactors but more expensive than best-experience advanced-fission or coal power plants (not including external, hidden costs associated with the use of both fission and fossil fuels). It is expected that potential advances in tokamak physics, such as the second stability operation, can improve the economics of tokamak reactors. The \AII\ design will explore this region of tokamak operation. \section{SUMMARY AND CONCLUSIONS} \label{s-2-6} The \AI\ cost-optimized design point is the result of a symbiotic integration of physics, engineering, and power economics. The systems model has received continual input and updating from the conceptual detailed engineering-design activity. While many of the physics and engineering models are simplified, the result, nevertheless, has been the first self-consistent tokamak power-reactor study to include the quantitative interdependence of equilibrium/stability, current drive, impurity control, transport in high-radiation tokamak plasmas, {\it etc.,} while simultaneously providing realistic and detailed cost projections using a modern methodology that is presently being used to assess a broad spectrum of nuclear and fossil energy sources. The attractiveness of the first-stability \AI\ tokamak reactor, as measured by COE and related figures of merit ({\it e.g.}, MPD and $Q_E$) has been achieved through \begin{enumerate} \item Operation at high aspect ratio (\hbox{$A = 4.5$}) and low plasma current (\hbox{$I_\phi = 10.2$~MA}) to increase the poloidal beta and obtain a high bootstrap-current fraction of 68\%. This approach minimizes the current-drive power and maximizes the engineering $Q$-value (\hbox{$Q_E = 5.1$}). \item Trade-off of plasma aspect ratio with current to limit the required confinement enhancement factor (\hbox{$H = 1.8-3.3$}, depending on the specific scaling considered). \item Use of advanced TF coils (ternary Nb$_3$Sn operated at 21~T) with a stabilizer that carries structural load and unique structural cup to carry overturning moments. 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