%(DRAFT)
%\section{Engineering for the 14.5~T Pulsed Magnet}

\section{Mechanical Design}

A three segment, layer wound solenoid is proposed for the pulsed magnet. The conductor is half inch square, cold worked OFHC copper. The coil is inertially cooled with options for  liquid nitrogen or gaseous helium cooling between shots. Coolant flows through axial channels in the coil. The coil will be epoxy impregnated.  Wound coils of this small radius, using cold worked conductor,  retain internal elastic stresses from the winding process, and if not impregnated, require elaborate clamping mechanisms to have the coil retain it's shape.
 \begin{figure}[bht!]
\centering
\includegraphics[width=2.5in]{titus/titus-fig1.eps}
\caption{need caption here}
\label{titus-fig1}
\end{figure}
 
\begin{table}[thb!]
\caption{Proposed Operational Scenarios}
\label{titus-tb1}
\begin{tabular}{|c|ccccc|}
\hline
Case No.&
Peak Field&
Coolant &
T after pulse &
T coolant &
Start Bulk Temp\\
& (T) & & (K) & (K)& (K)\\
\hline
1&
5&
Helium Gas&
90&
66&
84\\
1a&
5&
LN2&
90&
66&
84\\
2&
10&
Helium Gas&
96&
66&
74\\
2a&
10&
LN2&
96&
66&
74\\
3&
15&
Helium Gas&
78&
22&
30\\
\hline
\end{tabular}
\end{table}

Dual operational modes require special design of the cryostat/helium can. This is discussed in the section on cooldown behavior.
Insulation design impacts the conduction cooling behavior:
\begin{itemize}
\item Kapton is the limiting element in the thermal conduction through the coil. 
\item Kapton initially was expected to be wound around the conductor. This produced the equivalent of 5 mils of Kapton between layers. 
\item To improve conduction, Kapton is used only between the layers. Turn to turn voltage is lower than layer to layer. The turn to turn voltage is less than the rule of thumb for He breakdown voltage (1 volt/mil at 1 atmosphere) for the insulation thickness proposed. Note that the He operating pressure is expected to be 15 atmospheres, the pressures inside the epoxy winding pack may be substantially lower, depending on Helium diffusion, making the 1 atm breakdown voltage for the conductor, reasonable.
\item 
The layer to layer voltage exceeds this rule of thumb, however, and would need the Kapton if there was an imperfection in the epoxy/glass insulation.  Half laps of kapton and fiberglass, similar to the CS model coil will retain some structural integrity. 
\item Once a layer of conductor is wound, a layer of Kapton/glass would be wound on  the completed layer of conductor. This produces the equivalent of 3 mils of Kapton rather than 5 if the conductor were wrapped individually.  Every 6 to 8th layer some sort of preformed channel array would be layed on, then wrapped with glass/Kapton to hold it in place, and as the layer insulation for the next layer of conductors.
\end{itemize}
\begin{figure}
\centering
\includegraphics[bb=58 180 530 620,width=2.5in]{titus/titus-fig2.eps}
\caption[Ribs that form the channels]{Ribs that form the channels may need to have sliding parting planes to allow axial and radial motion between coil segments.}
\label{titus-fig2}
\end{figure}   
Experience with Alcator C-Mod indicates that for final magnet temperatures at or below 100K, the channels will not need "turbulators" or surface trips to break-up the film boiling layer. This will have to be reviewed during the design phase. If some form of surface roughness will be required in the 2mm channels, then the method of forming the channels with removable strips will have to be re-visited. 


\section{Engineering Tasks}

    While the magnet has been conceptualized, with some analytic basis, detailed engineering will require additional work after the construction of the magnet is approved. These tasks are outlined below.
\begin{itemize}

\item Identify Voltages for All Operating Scenarios  - Choose insulation systems. Determine where Kapton is used.

\item Stress Analysis, Assess radial load on channel ligaments, consider operation with inner modules energized, and no current in outer module

\item Confirm cooldown and pressure drop calculations

\item Analyze thermal contraction/shock of channel - Avoid separation and loss of conduction

\item Design He can for 15 atm. and vacuum  operation.

\item Design mandrel and flow plenums 

\item Cryostat Design. Is this a Vacuum Cryostat with LN2 Shield, or a Gaseous N2?

\item Break-outs and Leads Penetration design - Design the required support to resist loads and torques that result from principally the end radial Field.

\item Determine if Eddy Currents in He Can represent a significant load.

\item Design Cryogenic Electrical Breaks 

\item Design Supports, Break-outs, He can and Cryostat to Allow Phased Construction
\end{itemize}

Some of the analytic work in support of the present design is now described: 


\section{Stress Analysis}

The coil is stress analyzed using  ANSYS. Fields and forces are computed using elliptic integrals in a code external to ANSYS. The model is axisymmetric. Figures~\ref{titus-fig3}~-~\ref{titus-fig7} show a 3D representation from a symmetry expansion.
\begin{figure}
\centering
\includegraphics[bb=58 180 530 620,width=4in]{titus/titus-fig3.eps}
\caption[Hoop stress, all coil segments fully energized]{Hoop stress, all coil segments fully energized. The Von Mises stress plot is similar with a peak of 165~MPa, Tresca is 166~MPa.}
\label{titus-fig3}
\end{figure}   

\begin{table}[htbp]
  \centering
  \caption{Coil Build used in the Stress Analysis}
  \label{titus-tb2}
\begin{tabular}{|ccccccc|}
\hline  
Seg No.&
r&
z&
dr&
dz&
nx&
ny\\
\hline
1&
0.15&
0&
0.098&
1.0&
16&
16\\
channel&
0.2&
0&
0.002&
1.0&
1&
16\\
2&
0.25&
0&
0.098&
1.0&
16&
16\\
channel&
0.3&
0&
0.002&
1.0&
1 &
16\\
3&
0.35&
0&
0.098&
1.0&
16&
16\\
\hline
\end{tabular}
\end{table}

For Fusion magnets the inner skin of the solenoid is allowed to reach the yield - Treating this stress as a bending stress with a 1.5*Sm allowable with Sm based on 2/3 Yield. 
\begin{table}[htbp]
  \centering
  \caption[Interpolated values:, Work  hardened  copper]{Interpolated values:, Work  hardened  copper-,  OFHC  c10100  60\% red}
  \label{titus-tb3}
  \begin{tabular}{|cccccccccc|}

%  \end{tabular}
\hline
temp deg K & 
77 & 
  90 &
100&
 125& 
 150&    
200 &
 250 & 
275 &
 292\\
yield & 
374&
369.&
365.&
356. &
347. &
328.&
 317. &
312.& 
308.\\
ultimate&
476.&
466.& 
458.& 
439.&
 420.& 
383.& 
365.& 
356.&
 350.\\
\hline
\end{tabular}
\end{table}

If the highly cold-worked copper is chosen for the winding, the conductor allowable near the inside radius of the coil would be 365~MPa. The max stress in the three segment coil is 166~MPa. With this stress level, it is expected that half hard copper could be used, simplifying the winding process.

The three segment coil has three operational modes, two of which are structurally significant. The full performance configuration is limiting in terms of hoop stress and equivalent stress. It also has some radial stresses that will have to be mitigated with parting planes at the segment boundaries, or within the winding. 

In the initial operating mode the outer coil segment is not energized. This  induces some differential Lorentz forces and differential temperatures, that cause shear stresses between segments.
\begin{figure}
\centering
\includegraphics[bb=58 180 530 620,width=4in]{titus/titus-fig4.eps}
\caption[Radial tension stress, all coils fully energized ]{Radial tension stress, all coils fully energized. There is about an MPa of tension at the boundary between the first and second module. To avoid damage to the channel ligaments, a parting plane will be incorporated in the channel detail. This needs to occur in the ligament to retain thermal connection with the coolant in the channel.}
\label{titus-fig4}
\end{figure}   
\begin{figure}
\centering
\includegraphics[bb=0 156 560 690,width=5in,clip]{titus/titus-fig5.eps}
\caption{Hoop stress with only the inner two segments energized.}
\label{titus-fig5}
\end{figure}   
 \begin{figure}
\centering
\includegraphics[bb=0 156 560 690,width=5in,clip]{titus/titus-fig6.eps}
\caption[Smeared radial-axial stress with the inner two segments energized ]{Smeared radial-axial stress with the inner two segments energized. This is a peak at the interface between the second and third modules. It must be carried across the thin ligaments between the channels, or relieved via a slip plane.}
\label{titus-fig6}
\end{figure}   

 \begin{figure}
\centering
\includegraphics[width=4in,clip]{titus/titus-fig7.eps}
\caption{write caption}
\label{titus-fig7}
\end{figure}   





It may be desirable to build only the initial two inner segments and add the outer segment at a later time. The coil was analyzed with the outer segment removed, and the same current density in the other two segments. The max stress for this case is 85.3~MPa, which  is  a bit more than with the outer segment in place, but less than for the fully energized three segment coil. In all cases the stresses are lower than the expected allowable for the  conductor. It is expected that the degree of cold work can be relaxed from the full hard condition. The final choice of the degree of cold work for the conductor will be determined during detailed design. 


\section{Cooldown Between Shots}

Phased implementation of cooling systems is expected for the project. Within the coil, two cooldown methods are being investigated: and option using liquid nitrogen and an option which uses helium gas. It is expected that design of the magnet and cooling channels will allow either working fluid.  Use of liquid nitrogen as the only coolant is contemplated for the initial operation of the magnet. It is possible to use the container around the magnet as a vacuum vessel, and sub-cool liquid nitrogen to 66~K. Helium gas operation uses liquid nitrogen to cool the helium gas in a heat exchanger, and later  allows use of liquid hydrogen to cool the helium and obtain improved fields and/or pulse lengths. The heat transfer characteristics of liquid nitrogen operation have only been conceptualized. Heat transfer calculations using gaseous helium will be presented here. 
 \begin{figure}
\centering
\includegraphics[width=5in]{titus/titus-fig8.eps}
\caption[Proposed operational modes ]{Proposed operational modes. The coil and can are designed for either cooling mode. }
\label{titus-fig8}
\end{figure}   


In the LN2 sub-cooling mode, about 50 cubic meter of nitrogen gas at standard temperature and pressure must be drawn off by the vacuum pump to lower both the magnet and the liquid nitrogen to the desired 66~K temperature.
 
\section{Cooling Time with Helium Gas as a Working Fluid}

The  solenoid has groups of  6 to 8 layers of 1/2 inch square conductors separated by  set of annular cooling channels. This could model any linear stack of .5 inch square conductors cooled from the ends of the stack whether layer wound - then there would be a layer of channels every sixth layer of conductor, - or pancake wound, where there would be radial channels every sixth pancake. The solution is a simple finite difference transient analysis. The  ground wrap or cracked conductor/Kapton tape interfaces have not been modeled.
\begin{figure}
\centering
\includegraphics[width=4in]{titus/titus-fig9.eps}
\caption{write caption}
\label{titus-fig9}
\end{figure}   

The insulation layer is modeled with five, and as a second option, three, 0.001" thicknesses of Kapton tape. The thermal conductivity of the tape is about 0.14~W/(m-K) at 100~K~\cite{titus-ref1}.
\begin{figure}
\centering
\includegraphics[bb=0 130 580 440,width=4in,clip]{titus/titus-fig10.eps}
\caption{write caption}
\label{titus-fig10}
\end{figure}   

The surface heat transfer coefficient at the channels was taken as 170~W/$m^2$ for nitrogen  gas at 100~K flowing at 40 m/s in a channel with a 2~cm hydraulic diameter~\cite{titus-ref2}.  The  helium gas coefficient has been calculated  from the Nusselt, Reynolds, and  Prandle Numbers  using the relation quoted in the Oak Ridge document~\cite{titus-ref3}, and will be verified during detailed design.
 \begin{figure}
\centering
\includegraphics[bb=100 50 560 400,width=5in,clip]{titus/titus-fig11.eps}
\caption{Temperature \textsl{vs.} time in seconds.}
\label{titus-fig11}
\end{figure}   

\begin{table}[htb!]
\centering
\caption{Present Operational Scenarios}
\label{titus-tb33}
\begin{tabular}{|cccccc|}
\hline 
Case No. & 
Peak Field &
T after pulse &
T coolant &
Start Bulk Temp &
Guesstimated Time \\
\hline
1 &
5T&
90K&
66K&
84K&
200~s\\
2&
10T &
96K &
66K&
74K &
800~s \\
3 &
15T &
78K &
22K &
30K &
1500~s\\
\hline
\end{tabular}
\end{table}
Typical results for 66~K He cooling, 0.1~kg/sec, 100~K end of pulse temperature 85~K target magnet start temperature. The cooldown time is 600~s. to reach 85~K bulk temp, but not thermal equilibrium.

\begin{figure}
\centering
\includegraphics[width=3.in]{titus/titus-fig12.eps}
\caption[\textsl{Cell} temperature profiles ]{\textsl{Cell} temperature profiles. Time progresses towards the upper right. The groups of plots are temperature profiles for the five axial stations for which the temperature are calculated.}
\label{titus-fig12}
\end{figure}   
\begin{table}
\centering
\caption{Parameters}
\label{titus-tb}
\begin{tabular}{ll}
Number of Atmospheres Operating Pressure &10\\
Enter Channel Height (mm) & 2\\
$R_{\textrm inner}$ & 0.1000000 \\
 radial build & $7.6200157\times10^{-2}$\\
inner coil start temp &100.0000\\
outer coil start temp &   100.0000\\
inner coil radius &  0.1000000\\
model cell energy (J)&   1644.685  (100 to 85~K bulk)\\
model cell volume &  $5.5099601\times 10^{-5}$\\
volume cpp &  1989954.0\\
nlength &      120 \\
naxial &  5\\
Mass flow rate &  $4.1666666\times 10^{-5}$~kg/s \\
Volume flow rate & $5.5507730\times 10^{-6}$\\
flow velocity &  2.120239 \\
Hydraulic Diameter & $2.8944151\times 10^{-3}$~m \\
Velocity Head &  1.721665    Pascal\\
Pressure Drop &   31.46283    Pascal\\
Pressure Drop & $3.1041747\times 10^{-4}$~Atmospheres\\
Helium density &   7.506462    kg/$m^3$\\
Helium viscosity &  $2.6448268\times 10^{-7}$ \\
Prandl \# & $4.0756337\times 10^{-2}$\\
Reynolds \# &  174174.1\\
Heat transfer coefficient  & 115984. 9\\
\end{tabular}
\end{table}

From $\dot{m}\times C_p\times \Delta T$  for a 20 deg inlet-outlet difference the cooldown time is about  950~s. The simulation with a finer time step ($\delta time=0.0001$ rather than 0.001) yields a 600~s cooldown . The inlet outlet $\Delta T$ ranges from 26~K to 16~K. The energy balance or difference between the conduction heat flux and the channel heat flux is good at the finer time step.

\section{30~K Coolant, Cooldown from 100~K}
\begin{figure}
\centering
\includegraphics[width=3.5in]{titus/titus-fig13.eps}
\caption[Temperature \textsl{vs.} time in seconds ]{Temperature \textsl{vs.} time in seconds. Tout1 and Tout2 are outlet temperatures.}
\label{titus-fig13}
\end{figure}   

Bulk temp is computed as the average temperature at mid axial build. It bottoms out before the down stream end. In the plot above, the channel outlet temperatures are better indicators of when the magnet reaches 30~K


\begin{table}[htb!]
\caption[Analysis to date ]{Analysis to date: Time to target  bulk temp from 100~K. $\frac{1}{2}$~inch Copper Conductor.}
\label{titus-tb10}
\begin{tabular}{|m{2cm}ccccc|}
\hline
&T after pulse&
T coolant &
Cond. Layers &
Time to 85~K  &
Time to 30~K  \\
& K & K && s & s\\
\hline
Equiv 5 Kapton 0.001~in wrap &
100 &
66 &
6 layers &
600 &
xxx \\
Equiv 5 Kapton 0.001~in wrap &
100 &
66 &
8 layers &
850 & xxx \\
Equiv 3 Kapton 0.001~in wrap &
100 &
66 &
8 layers &
450 & xxx \\
Equiv 5 kapton 0.0001~in wrap &
100 &
30 &
6 Layers &
2000 & xxx\\
\hline
\end{tabular}
\end{table}
%
% ---- Bibliography ----
%
\begin{thebibliography}{30}
%

\bibitem{titus-ref1} Rule, Smith, and Sparks, \textsl{Thermal Conductivity of Polymide Film between 4.2 and 300K with and without Alumina Particles as Filler},  NISTIR \#3948. August 1990.
\bibitem{titus-ref2} Oak Ridge CIT report \# ORNL/FEDC-85-10 Dist. Category UC20 c,d October 1986.
\bibitem{titus-ref3} Excerpts from reference~\cite{titus-ref2}

2.1.3 Convective Heat Transfer

It is important to estimate how much heat the superheated nitrogen gas $(T > 77~K)$ could absorb before exiting the cooling channel. The convective heat transfer coefficient, h, could be obtained from
$$ h=\frac{KNu}{D_e}=\frac{0.023R_e^{0.8}P_r^{0.4}K}{D_e}.$$
This coefficient is about $21 \times 10^{-3}$~W/cm${}^2$K at a vapor temperature of 200~K, vapor velocity of 40~m/s, and hydraulic diameter of 2~cm. It drops to $17\times 10^{-3}$~W/cm${}^2$K at a vapor temperature of 100~K, keeping the mass flow rate constant. It is interesting to note that the heat transfer coefficient for film boiling at 200~K from Fig.~4 is about $12\times 10^{-3}$~W/cm${}^2$K, which partially justifies the third assumption in Sec. 2.1.

\end{thebibliography}
