\chapter{Material Characterization}
\label{cha:material_characterization}

The material used to model the industrial induction heating process was a 50CrMo4 steel.


\begin{table}[htbp]
    \centering
    \caption{Chemical composition of the 50CrMo4 steel.}
    \begin{tabular}{ccccccccc}\toprule
Element & C & Mn & Cr & Mo & Si & P & S & Fe \\ \midrule
\si{\wtpercent} & 0.49 & 0.71 & 1.05 & 0.18 & 0.27 & 0.016 & \num{0.01} & \num{\sim 97.1} \\\bottomrule
    \end{tabular}
\end{table}

\section{Sample Preparation}
\label{sec:sample_preparation}

\textcolor{red}{\textbf{Bilder: Probenlage}}

\section{Thermophysical Properties}
\label{sec:thermophysical_properties}

\begin{sidewaystable}[p]
    \centering
    \begin{tabular}{ccccccrr} \toprule
         \multirow{2}{*}{\thead{Measurement\\Method}} & \multicolumn{3}{c}{\multirow{2}{*}{\thead{Physical Property}}} &  \multirow{2}{*}{\thead{Device}} &  \multirow{2}{*}{\thead{Norm}} & \multicolumn{2}{c}{\thead{Uncertainty in \\ Measurement ($\sigma \sim \qty{95}{\percent}$)}} \\
         &&&&&&at RT & at \qty{1000}{\celsius}\\ \midrule
         \makecell{Dynamic\\Differential\\Calorimetry} & Specific Heat Capacity &  $c_p$ & [\unit{\joule\per\gram\per\kelvin}] & Netzsch DSC 404 & EN 821-3 (2005) & \qty{\pm 3}{\percent} & \qty{\pm 4}{\percent} \\[20pt]
         \makecell{Laser Flash\\Method} & Thermal Conductivity & $a$ & [\unit{\mm\squared\per\s}] & Netzsch LFA 427 &  EN 821-2 (1997) & \qty{\pm 5}{\percent} & \qty{\pm 5}{\percent} \\[20pt]
         Dilatomety & Thermal Strain & $\nicefrac{\Delta l}{l}$ & [\unit{\percent}] & Netzsch DIL 402 CD & DIN 51045-1 (2005) & \makecell{\qty{\pm 0.004}{\percent}\\(at \qty{100}{\celsius})} & \qty{\pm 0.015}{\percent} \\[20pt]
         Buoyancy Method & Density & $\rho$ & [\unit{\kg\per\m\cubed}] & Sartorius ED224S & DIN EN 993-1 &  \qty{\pm 0.1}{\percent} & ---\\[10pt]
         \makecell{Four-Terminal\\Sensing} & Electrical Resistivity & $\rho_{el}$ & [\unit{\micro\ohm\m}] & ÖGI in-house setup & DIN EN 993-1 &   \qty{\pm 2}{\percent} &   \qty{\pm 3}{\percent} \\
         \bottomrule
    \end{tabular}
    \caption{Caption}
    \label{tab:characterization_summary_50CrMo4}
\end{sidewaystable}


\begin{table}[htbp]
    \centering
    \begin{tabular}{S[table-format=4]SSSS[table-format=4]SSS}\toprule
        {T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$D$} & {$D_corr$} & {$k$}\\
        {[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} \\ \midrule
        20 & \color{red}0.460 & 0.000 &{---}& 7827 & 12.47 & 12.47 & 44.9 \\
        100 & 0.488 & 0.097 & 12.085 & 7804 & 11.62 & 11.64 & 44.3 \\
        200 & 0.525 & 0.233 & 12.930 & 7773 & 10.49 & 10.53 & 43.0 \\
        300 & 0.561 & 0.380 & 13.559 & 7739 & 9.38 & 9.45 & 41.0 \\
        400 & 0.603 & 0.533 & 14.024 & 7703 & 8.30 & 8.39 & 39.0 \\
        500 & 0.655 & 0.690 & 14.377 & 7667 & 7.20 & 7.29 & 36.6 \\
        600 & 0.731 & 0.849 & 14.641 & 7631 & 6.01 & 6.11 & 34.1 \\
        700 & 0.884 & 1.010 & 14.855 & 7595 & 4.61 & 4.71 & 31.6 \\
        800 & 0.625 & 0.924 & 11.843 & 7614 & 5.46 & 5.56 & 26.5 \\
        900 & 0.632 & 1.189 & 13.507 & 7554 & 5.64 & 5.77 & 27.5 \\
        1000 & 0.639 & 1.419 & 14.480 & 7503 & 5.82 & 5.99 & 28.7 \\
        1100 & \color{red}0.645 & 1.649 & 15.264 & 7452 & 6.03 & 6.23 & 30.0 \\
        1200 & \color{red}0.652 & 1.880 & 15.932 & 7402 & 6.21 & 6.45 & 31.1 \\
         \bottomrule
    \end{tabular}
    \caption{The numbers highlighted in red denote where the specific heat capacity was extrapolated to room temperature, and to \qty{1000}{\celsius} and \qty{1200}{\celsius}.}
    \label{tab:results_50CrMo4}
\end{table}

\begin{figure}[htbp]
    \centering
    \begin{tabular}{cc}
         \subfloat[Specific Heat Capacity]{\includegraphics[width=5cm]{Abbildungen/cp_50.png}}&\quad
         \subfloat[Density]{\includegraphics[width=5cm]{Abbildungen/rho_50.png}}\\
         \subfloat[Thermal Expansion]{\includegraphics[width=5cm]{Abbildungen/expan_50.png}}&\quad
         \subfloat[Thermal Expansion Coefficient]{\includegraphics[width=5cm]{Abbildungen/epsilon_T_50.png}}\\
         \subfloat[Thermal Diffusion]{\includegraphics[width=5cm]{Abbildungen/th_diff_50.png}}&\quad
         \subfloat[Thermal Conductivity]{\includegraphics[width=5cm]{Abbildungen/htc_50.png}}\\
    \end{tabular}
    \caption{The local discontinuities at \qtyrange{700}{800}{\celsius} are caused by $\alpha$-$\gamma$ transformation happening in that range.}
    \label{fig:50crmo4-char}
\end{figure}


\section{Mechanical Properties}
\label{sec:mechanical_properties}



\section{Electromagnetic Properties}
\label{sec:electromagnetic_properties}

\subsection{Magnetization Behavior}
\label{sub:magnetization_behaviour}

A measurement setup for magnetic hysteresis behavior at high temperatures was designed in-house\autocite{jaszfi2022indirect}, but was not ready in time to deliver the material data for this project.
As a recourse, magnetic model data was taken from M. Schwenk's thesis on induction heating\autocite[111]{schwenk2012numerische}.
The subject material of a 42CrMo4 steel was deemed close enough in its characteristics to stand in for the 50CrMo4 of this work.

A recent proposal\autocite{baldan2020improving} for analytic descriptions of magnetization curves incorporates the quadratic region of the curve at the origin, but the numeric capabilities of common solvers do not support the inflection point that this increased adherence to measured data brings.
This thesis will therefore use the analytic formula described by Trutt et al.\autocite{trutt1968representation} using the arc tangent, as Schwenk did:

$$B(H,T) = \mu_0 H + \frac{2 B_{sat}}{\pi}\cdot \mathrm{atan} \left(  \frac{(\mu_{r0} - 1)\mu_0 \pi}{2 B_{sat}} H \right)\cdot {e}^\frac{T-T_C}{C}$$

