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\chapter{Material Properties} \chapter{Material Properties}\label{cha:material_char}
\label{cha:material_char}
The two processes modeled in this thesis apply to stock of related but distinct materials: The two processes modeled in this thesis apply to stock of related but distinct materials:
The bars of the induction test rig are made from 50CrMo4 Steel which is widely available and well characterized in literature\autocite{vieweg2017phase, vieweg2017effects, eggbauer2018inductive, eggbauer2019situ, jaszfi2019influence}, while the crankshafts are made from a modified C38 Steel dubbed \emph{C38p}, the composition of which was given by the manufacturer. The bars of the induction test rig are made from 50CrMo4 Steel which is widely available and well characterized in literature\autocite{vieweg2017phase, vieweg2017effects, eggbauer2018inductive, eggbauer2019situ, jaszfi2019influence}, while the crankshafts are made from a modified C38 Steel dubbed \emph{C38p}, the composition of which was given by the manufacturer.
Table~\ref{tab:steel_compositions} shows the compositions for both materials in comparison to each other. Table~\ref{tab:steel_compositions} shows the compositions for both materials in comparison to each other.
Material data for Material data for
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption[Chemical compositions]{Chemical compositions of both steels examined in this chapter. } \caption[Chemical compositions]{Chemical compositions of both steels examined in this chapter.}\label{tab:steel_compositions}
\label{tab:steel_compositions} \begin{tabular}{cSS}\toprule
\begin{tabular}{cSS}\toprule & \textbf{50CrMo4} & \textbf{C38p} \\
& \textbf{50CrMo4} & \textbf{C38p}\\ & \si{\wtpercent} & \si{\wtpercent} \\\midrule
& \si{\wtpercent}& \si{\wtpercent}\\\midrule C & 0.49 & \numrange{0.36}{0.40} \\
C & 0.49 & \numrange{0.36}{0.40} \\ Mn & 0.71 & \numrange{1.30}{1.45} \\
Mn & 0.71 & \numrange{1.30}{1.45} \\ Cr & 1.05 & \numrange{0.10}{0.20} \\
Cr & 1.05 & \numrange{0.10}{0.20} \\ Mo & 0.18 & \le 0.050 \\
Mo & 0.18 & \le 0.050 \\ Si & 0.27 & \numrange{0.50}{0.65} \\
Si & 0.27 & \numrange{0.50}{0.65} \\ P & 0.016 & \le 0.025 \\
P & 0.016 & \le 0.025 \\ S & 0.01 & \numrange{0.050}{0.065} \\
S & 0.01 & \numrange{0.050}{0.065} \\ N & & \numrange{0.013}{0.017} \\
N & & \numrange{0.013}{0.017} \\ Cu & & 0.25 \\
Cu & & 0.25 \\ Al & & \numrange{0.010}{0.030} \\
Al & & \numrange{0.010}{0.030} \\ Ni & & \le 0.15 \\
Ni & & \le 0.15 \\ V & & \numrange{0.08}{0.12} \\\bottomrule
V & & \numrange{0.08}{0.12}\\\bottomrule \end{tabular}
\end{tabular}
\end{table} \end{table}
% \begin{table}[htbp] % \begin{table}[htbp]
@ -52,167 +50,152 @@ V & & \numrange{0.08}{0.12}\\\bottomrule
% \end{tabular} % \end{tabular}
% \end{table} % \end{table}
\section{Sample Preparation} \section{Sample Preparation}
\label{sec:sample_preparation} \label{sec:sample_preparation}
Crankshafts of various processing stages were supplied by the manufacturer, from which two pieces of forged blanks were chosen as sources for C38p samples. Crankshafts of various processing stages were supplied by the manufacturer, from which two pieces of forged blanks were chosen as sources for C38p samples.
These were untouched by subsequent heat treatments or surface machining, thus allowing for the characterization of a material microstructure to be used as a baseline for subsequent heat treatment and property evolution. These were untouched by subsequent heat treatments or surface machining, thus allowing for the characterization of a material microstructure to be used as a baseline for subsequent heat treatment and property evolution.
Sample locations were strategically placed in the crankshafts to allow for necessary dimensions for the different testing equipment while staying as close as possible to bearing surfaces of interest, i.e. the section closest to the center rotation axis of the crank bearings. Sample locations were strategically placed in the crankshafts to allow for necessary dimensions for the different testing equipment while staying as close as possible to bearing surfaces of interest, i.e.\ the section closest to the center rotation axis of the crank bearings.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\begin{tabular}{c} \begin{tabular}{c}
\subfloat[Crankshaft 1\label{fig:probenlage_1}]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side.png}}\\ \subfloat[Crankshaft 1\label{fig:probenlage_1}]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side.png}} \\
\subfloat[Crankshaft 2 ]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side2.png}} \subfloat[Crankshaft 2 ]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side2.png}}
\end{tabular} \end{tabular}
\caption[Crankshaft sample positions, side view]{Sample position within the two sacrificial crankshafts, side view.} \caption[Crankshaft sample positions, side view]{Sample position within the two sacrificial crankshafts, side view.}\label{fig:sample-loc-1}
\label{fig:sample-loc-1}
\end{figure} \end{figure}
Figures~\ref{fig:sample-loc-1} and~\ref{fig:sample-loc-2} show the detailed locations of the samples that were distributed as follows: Figures~\ref{fig:sample-loc-1} and~\ref{fig:sample-loc-2} show the detailed locations of the samples that were distributed as follows:
\begin{description} \begin{description}
\item[Section A] 38 dilatometer samples D\qtyproduct{4 x 10}{\mm} (black) \item[Section A] 38 dilatometer samples D\qtyproduct{4 x 10}{\mm} (black)
\item[Section B - C] 28 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red), 1 density sample D\qtyproduct{20 x 30}{\mm} (green) \item[Section B - C] 28 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red), 1 density sample D\qtyproduct{20 x 30}{\mm} (green)
\item[Section D] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 2x electrical resistivity samples D\qtyproduct{4 x 75}{\mm} (blue) \item[Section D] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 2x electrical resistivity samples D\qtyproduct{4 x 75}{\mm} (blue)
\item[Section E] 7 dilatometer samples D\qtyproduct{4 x 10}{\mm} (black), 1 thermal conductivity sample D\qtyproduct{12.55 x 40}{\mm} (cyan) , 1 heat capacity sample D\qtyproduct{6 x 40}{\mm} (purple), 2 thermal expansion samples D\qtyproduct{6 x 25}{\mm} (yellow) \item[Section E] 7 dilatometer samples D\qtyproduct{4 x 10}{\mm} (black), 1 thermal conductivity sample D\qtyproduct{12.55 x 40}{\mm} (cyan) , 1 heat capacity sample D\qtyproduct{6 x 40}{\mm} (purple), 2 thermal expansion samples D\qtyproduct{6 x 25}{\mm} (yellow)
\item[Secfion F] 14 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red) \item[Secfion F] 14 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red)
\item[Secfion G] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 3x magnetic hysteresis samples D\qtyproduct{3 x 3}{\mm} (blue) \item[Secfion G] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 3x magnetic hysteresis samples D\qtyproduct{3 x 3}{\mm} (blue)
\end{description} \end{description}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\begin{tabular}{ccc} \begin{tabular}{ccc}
\subfloat[\label{fig:sample-loc-A}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_A.png}}& \subfloat[\label{fig:sample-loc-A}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_A.png}} &
\subfloat[\label{fig:sample-loc-B}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_B.png}}& \subfloat[\label{fig:sample-loc-B}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_B.png}} &
\subfloat[\label{fig:sample-loc-C}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_C.png}}\\ \subfloat[\label{fig:sample-loc-C}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_C.png}} \\
\subfloat[\label{fig:sample-loc-D}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_D.png}}& \subfloat[\label{fig:sample-loc-D}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_D.png}} &
\subfloat[\label{fig:sample-loc-E}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_E.png}}& \subfloat[\label{fig:sample-loc-E}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_E.png}} &
\subfloat[\label{fig:sample-loc-F}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_F.png}}\\& \subfloat[\label{fig:sample-loc-F}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_F.png}} \\&
\subfloat[\label{fig:sample-loc-G}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_G.png}}&\\ \subfloat[\label{fig:sample-loc-G}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_G.png}} & \\
\end{tabular} \end{tabular}
\caption[Crankshaft sample positions, axial view]{Sample position within the two sacrificial crankshafts, frontal view at slices as indicated in figure~\ref{fig:sample-loc-1}. The material state is as-forged without any further mechanical or thermal treatments. Sample positions were chosen to allow for the required dimensions to be reached while staying close to bearing surfaces for in} \caption[Crankshaft sample positions, axial view]{Sample position within the two sacrificial crankshafts, frontal view at slices as indicated in figure~\ref{fig:sample-loc-1}. The material state is as-forged without any further mechanical or thermal treatments. Sample positions were chosen to allow for the required dimensions to be reached while staying close to bearing surfaces for in}\label{fig:sample-loc-2}
\label{fig:sample-loc-2}
\end{figure} \end{figure}
\section{Thermophysical Properties}\label{sec:thermophysical_properties}
\section{Thermophysical Properties}
\label{sec:thermophysical_properties}
A wide regimen of experiments was applied to the materials to gather all properties needed for process simulation. A wide regimen of experiments was applied to the materials to gather all properties needed for process simulation.
Table~\ref{tab:equipment_th} gives an overview of the equipment and norms used to generate direct data. Table~\ref{tab:equipment_th} gives an overview of the equipment and norms used to generate direct data.
\begin{sidewaystable}[p] \begin{sidewaystable}[p]
\centering \centering
\caption[Thermophysical test equipment]{Summary of methods and equipment used to characterize the thermophysical and thermoelectric properties.} \caption[Thermophysical test equipment]{Summary of methods and equipment used to characterize the thermophysical and thermoelectric properties.}\label{tab:equipment_th}
\label{tab:equipment_th} \begin{tabular}{ccccccrr} \toprule
\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 ($\sigma \sim \qty{95}{\percent}$)}} \\
\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
&&&&&&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{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] 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]
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] \makecell{Laser Flash \\Method} & Thermal Diffusivity & $a$ & [\unit{\mm\squared\per\s}] & Netzsch LFA 427 & EN 821-2 (1997) & \qty{\pm 5}{\percent} & \qty{\pm 5}{\percent} \\[20pt]
\makecell{Laser Flash\\Method} & Thermal Diffusivity & $a$ & [\unit{\mm\squared\per\s}] & Netzsch LFA 427 & EN 821-2 (1997) & \qty{\pm 5}{\percent} & \qty{\pm 5}{\percent} \\[20pt] Buoyancy Method & Density & $\rho$ & [\unit{\kg\per\m\cubed}] & Sartorius ED224S & DIN EN 993-1 & \qty{\pm 0.1}{\percent} & --- \\[10pt]
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} \\
\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
\bottomrule \end{tabular}
\end{tabular} \label{tab:characterization_summary_c36p}
\label{tab:characterization_summary_c36p}
\end{sidewaystable} \end{sidewaystable}
From that data some more fundamental properties were derived for use in the simulations' material data sets. From that data some more fundamental properties were derived for use in the simulations' material data sets.
Namely, Thermal conductivity $k$ was derived from thermal diffusivity $a$, specific heat $c_p$ and density $\rho$ by rearranging the diffusivity's standard formula for $k$: Namely, Thermal conductivity $k$ was derived from thermal diffusivity $a$, specific heat $c_p$ and density $\rho$ by rearranging the diffusivity's standard formula for $k$:
$$ a = \frac{k}{c_p \rho} \Rightarrow k = a c_p \rho $$ \[a = \frac{k}{c_p \rho} \Rightarrow k = a c_p \rho \]
Likewise, the thermal Expansion coefficient $\alpha$ was calculated from the thermal strain and the temperature differential to the previous data point: Likewise, the thermal Expansion coefficient $\alpha$ was calculated from the thermal strain and the temperature differential to the previous data point:
$$a = \frac {1}{l} \frac{\Delta l}{\Delta T} $$ \[a = \frac {1}{l} \frac{\Delta l}{\Delta T} \]
Data was collected for heating and cooling behaviors, but data from the heating cycle as preferred where the two disagreed due to transformation temperature varying. Data was collected for heating and cooling behaviors, but data from the heating cycle as preferred where the two disagreed due to transformation temperature varying.
Data was collected with higher resolution around the transformation temperatures \qtyrange{700}{800}{\celsius}. Data was collected with higher resolution around the transformation temperatures \qtyrange{700}{800}{\celsius}.
For use in the following simulations, all data was interpolated to a fixed set of temperature points, with $c_p$ requiring extrapolation at room temperature, and above \qty{1000}{\celsius}. For use in the following simulations, all data was interpolated to a fixed set of temperature points, with $c_p$ requiring extrapolation at room temperature, and above \qty{1000}{\celsius}.
The resulting modeal data is listed in tables~\ref{tab:results_50CrMo4} and \ref{tab:results_c38p}, and compared in figure~\ref{fig:comp-char}. The resulting modeal data is listed in tables~\ref{tab:results_50CrMo4} and~\ref{tab:results_c38p}, and compared in figure~\ref{fig:comp-char}.
Both materials behave appreciably close to one another with the biggest discrepancy in their thermal conductivities below $A_{C1}$. Both materials behave appreciably close to one another with the biggest discrepancy in their thermal conductivities below $A_{C1}$.
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\begin{tabular}{S[table-format=4]SSSS[table-format=4]SS}\toprule \begin{tabular}{S[table-format=4]SSSS[table-format=4]SS}\toprule
{T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$}\\ {T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$} \\
{[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} \\ \midrule {[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} \\ \midrule
20 & \color{red}0.460 & 0.000 &{---}& 7827 & 12.47 & 44.9 \\ 20 & \color{red}0.460 & 0.000 & {---} & 7827 & 12.47 & 44.9 \\
100 & 0.488 & 0.097 & 12.085 & 7804 & 11.64 & 44.3 \\ 100 & 0.488 & 0.097 & 12.085 & 7804 & 11.64 & 44.3 \\
200 & 0.525 & 0.233 & 12.930 & 7773 & 10.53 & 43.0 \\ 200 & 0.525 & 0.233 & 12.930 & 7773 & 10.53 & 43.0 \\
300 & 0.561 & 0.380 & 13.559 & 7739 & 9.45 & 41.0 \\ 300 & 0.561 & 0.380 & 13.559 & 7739 & 9.45 & 41.0 \\
400 & 0.603 & 0.533 & 14.024 & 7703 & 8.39 & 39.0 \\ 400 & 0.603 & 0.533 & 14.024 & 7703 & 8.39 & 39.0 \\
500 & 0.655 & 0.690 & 14.377 & 7667 & 7.29 & 36.6 \\ 500 & 0.655 & 0.690 & 14.377 & 7667 & 7.29 & 36.6 \\
600 & 0.731 & 0.849 & 14.641 & 7631 & 6.11 & 34.1 \\ 600 & 0.731 & 0.849 & 14.641 & 7631 & 6.11 & 34.1 \\
700 & 0.884 & 1.010 & 14.855 & 7595 & 4.71 & 31.6 \\ 700 & 0.884 & 1.010 & 14.855 & 7595 & 4.71 & 31.6 \\
800 & 0.625 & 0.924 & 11.843 & 7614 & 5.56 & 26.5 \\ 800 & 0.625 & 0.924 & 11.843 & 7614 & 5.56 & 26.5 \\
900 & 0.632 & 1.189 & 13.507 & 7554 & 5.77 & 27.5 \\ 900 & 0.632 & 1.189 & 13.507 & 7554 & 5.77 & 27.5 \\
1000 & 0.639 & 1.419 & 14.480 & 7503 & 5.99 & 28.7 \\ 1000 & 0.639 & 1.419 & 14.480 & 7503 & 5.99 & 28.7 \\
1100 & \color{red}0.645 & 1.649 & 15.264 & 7452 &6.23 & 30.0 \\ 1100 & \color{red}0.645 & 1.649 & 15.264 & 7452 & 6.23 & 30.0 \\
1200 & \color{red}0.652 & 1.880 & 15.932 & 7402 & 6.45 & 31.1 \\ 1200 & \color{red}0.652 & 1.880 & 15.932 & 7402 & 6.45 & 31.1 \\
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\caption[Thermophysical properties of 50CrMo4]{Thermophysical properties of 50CrMo4. 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}.} \caption[Thermophysical properties of 50CrMo4]{Thermophysical properties of 50CrMo4. 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}
\label{tab:results_50CrMo4}
\end{table} \end{table}
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\begin{tabular}{S[table-format=4]SSSS[table-format=4]SSS}\toprule \begin{tabular}{S[table-format=4]SSSS[table-format=4]SSS}\toprule
{T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$} & {$\rho_{el}$}\\ {T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$} & {$\rho_{el}$} \\
{[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} & {[\unit{\micro\ohm\m}]} \\ \midrule {[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} & {[\unit{\micro\ohm\m}]} \\ \midrule
20 & \color{red}0.470 & 0 & & 7797 & 10.27 & 37.6 & 0.283 \\ 20 & \color{red}0.470 & 0 & & 7797 & 10.27 & 37.6 & 0.283 \\
100 & 0.495 & 0.096 & 11.997 & 7775 & 10 & 38.5 & 0.332 \\ 100 & 0.495 & 0.096 & 11.997 & 7775 & 10 & 38.5 & 0.332 \\
200 & 0.528 & 0.233 & 12.926 & 7743 & 9.38 & 38.4 & 0.402 \\ 200 & 0.528 & 0.233 & 12.926 & 7743 & 9.38 & 38.4 & 0.402 \\
300 & 0.563 & 0.380 & 13.572 & 7709 & 8.67 & 37.7 & 0.482 \\ 300 & 0.563 & 0.380 & 13.572 & 7709 & 8.67 & 37.7 & 0.482 \\
400 & 0.604 & 0.534 & 14.043 & 7674 & 7.85 & 36.4 & 0.573 \\ 400 & 0.604 & 0.534 & 14.043 & 7674 & 7.85 & 36.4 & 0.573 \\
500 & 0.657 & 0.691 & 14.391 & 7638 & 6.91 & 34.7 & 0.674 \\ 500 & 0.657 & 0.691 & 14.391 & 7638 & 6.91 & 34.7 & 0.674 \\
600 & 0.733 & 0.850 & 14.660 & 7601 & 5.84 & 32.5 & 0.781 \\ 600 & 0.733 & 0.850 & 14.660 & 7601 & 5.84 & 32.5 & 0.781 \\
700 & 0.878 & 1.010 & 14.853 & 7565 & 4.51 & 29.9 & 0.905 \\ 700 & 0.878 & 1.010 & 14.853 & 7565 & 4.51 & 29.9 & 0.905 \\
800 & 0.621 & 0.920 & 11.796 & 7586 & 5.55 & 26.1 & 1.068 \\ 800 & 0.621 & 0.920 & 11.796 & 7586 & 5.55 & 26.1 & 1.068 \\
900 & 0.634 & 1.142 & 12.979 & 7536 & 5.81 & 27.8 & 1.143 \\ 900 & 0.634 & 1.142 & 12.979 & 7536 & 5.81 & 27.8 & 1.143 \\
1000 & 0.644 & 1.370 & 13.984 & 7485 & 6.02 & 29.0 & 1.175 \\ 1000 & 0.644 & 1.370 & 13.984 & 7485 & 6.02 & 29.0 & 1.175 \\
1100 & \color{red}0.660 & 1.593 & 14.753 & 7436 &6.24 & 30.6 & 1.212 \\ 1100 & \color{red}0.660 & 1.593 & 14.753 & 7436 & 6.24 & 30.6 & 1.212 \\
% 1000 & 0.644 & 1.359 & & 7488 & 6.09 & 29.4 & \\ % 1000 & 0.644 & 1.359 & & 7488 & 6.09 & 29.4 & \\
% 900 & 0.634 & 1.127 & & 7539 & 5.87 & 28.1 & \\ % 900 & 0.634 & 1.127 & & 7539 & 5.87 & 28.1 & \\
% 800 & 0.621 & 0.899 & & 7591 & 5.63 & 26.5 & \\ % 800 & 0.621 & 0.899 & & 7591 & 5.63 & 26.5 & \\
% 700 & 0.878 & 0.696 & & 7636 & 5.09 & {\color{red}transf.} & \\ % 700 & 0.878 & 0.696 & & 7636 & 5.09 & {\color{red}transf.} & \\
% 600 & 0.733 & 0.792 & & 7615 & 5.88 & 32.8 & \\ % 600 & 0.733 & 0.792 & & 7615 & 5.88 & 32.8 & \\
% 500 & 0.657 & 0.628 & & 7652 & 6.96 & 35.0 & \\ % 500 & 0.657 & 0.628 & & 7652 & 6.96 & 35.0 & \\
% 400 & 0.604 & 0.470 & & 7688 & 7.87 & 36.5 & \\ % 400 & 0.604 & 0.470 & & 7688 & 7.87 & 36.5 & \\
% 300 & 0.563 & 0.317 & & 7723 & 8.70 & 37.9 & \\ % 300 & 0.563 & 0.317 & & 7723 & 8.70 & 37.9 & \\
% 200 & 0.528 & 0.174 & & 7757 & 9.45 & 38.7 & \\ % 200 & 0.528 & 0.174 & & 7757 & 9.45 & 38.7 & \\
% 100 & 0.495 & 0.039 & & 7788 & 10.02 & 38.6 & \\ % 100 & 0.495 & 0.039 & & 7788 & 10.02 & 38.6 & \\
% 20 & \color{red}0.470 & -0.076 & & 7815 & 10.20 & 37.4 & \\ % 20 & \color{red}0.470 & -0.076 & & 7815 & 10.20 & 37.4 & \\
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\caption[Thermophysical properties of C38p]{Thermophysical properties of C38p. Red highlights again denote where the specific heat capacity was extrapolated to room temperature, and to \qty{1100}{\celsius}.} \caption[Thermophysical properties of C38p]{Thermophysical properties of C38p. Red highlights again denote where the specific heat capacity was extrapolated to room temperature, and to \qty{1100}{\celsius}.}\label{tab:results_c38p}
\label{tab:results_c38p}
\end{table} \end{table}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\begin{tabular}{cc} \begin{tabular}{cc}
\subfloat[Specific Heat Capacity]{\includegraphics[width=5cm]{Abbildungen/mat_char_cp.png}}&\quad \subfloat[Specific Heat Capacity]{\includegraphics[width=5cm]{Abbildungen/mat_char_cp.png}} & \quad
\subfloat[Density]{\includegraphics[width=5cm]{Abbildungen/mat_char_rho.png}}\\ \subfloat[Density]{\includegraphics[width=5cm]{Abbildungen/mat_char_rho.png}} \\
\subfloat[Thermal Expansion]{\includegraphics[width=5cm]{Abbildungen/mat_char_expan.png}}&\quad \subfloat[Thermal Expansion]{\includegraphics[width=5cm]{Abbildungen/mat_char_expan.png}} & \quad
\subfloat[Thermal Expansion Coefficient]{\includegraphics[width=5cm]{Abbildungen/mat_char_epsilon_T.png}}\\ \subfloat[Thermal Expansion Coefficient]{\includegraphics[width=5cm]{Abbildungen/mat_char_epsilon_T.png}} \\
\subfloat[Thermal Diffusivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_th_diff.png}}&\quad \subfloat[Thermal Diffusivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_th_diff.png}} & \quad
\subfloat[Thermal Conductivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_htc.png}}\\ \subfloat[Thermal Conductivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_htc.png}} \\
\subfloat[Electrical Resistivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_rhoe.png}} \subfloat[Electrical Resistivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_rhoe.png}}
\end{tabular} \end{tabular}
\caption[Thermophysical material data]{Graphed thermophysical data. The local discontinuities at \qtyrange{700}{800}{\celsius} are caused by $\alpha$-$\gamma$ transformation happening in that range.} \caption[Thermophysical material data]{Graphed thermophysical data. The local discontinuities at \qtyrange{700}{800}{\celsius} are caused by $\alpha$-$\gamma$ transformation happening in that range.}\label{fig:comp-char}
\label{fig:comp-char}
\end{figure} \end{figure}
\section{Mechanical Properties} \section{Mechanical Properties}\label{sec:mechanical_properties}
\label{sec:mechanical_properties}
A series of tensile tests was conducted for both materials at various temperatures. A series of tensile tests was conducted for both materials at various temperatures.
Since phase transformations are being simulated, subsets of tensile test samples were heat treated to obtain austenitic and martensitic microstructures in addition to the ferritic-pearlitic initial states, labelled "base microstructure". Since phase transformations are being simulated, subsets of tensile test samples were heat treated to obtain austenitic and martensitic microstructures in addition to the ferritic-pearlitic initial states, labelled "base microstructure".
@ -220,105 +203,94 @@ Since phase transformations are being simulated, subsets of tensile test samples
Tensile tests were conducted at overlapping temperature ranges for each material and their phases as far as transition times would allow: Tensile tests were conducted at overlapping temperature ranges for each material and their phases as far as transition times would allow:
\begin{description} \begin{description}
\item[Base Microstructure:] 25, 300, 400, 500, 600, 700 \item[Base Microstructure:] 25, 300, 400, 500, 600, 700
\item[Martensite:] 25, 100, 200, 300, 350 \item[Martensite:] 25, 100, 200, 300, 350
\item[Austenite:] 600, 650, 700, 800, 900, 1000, 1100 \item[Austenite:] 600, 650, 700, 800, 900, 1000, 1100
\end{description} \end{description}
Samples were tested to EN ISO 6892-2, with sample geometries of B4x20. Samples were tested to EN ISO 6892-2, with sample geometries of B4x20.
They were prestressed at \qty{5}{\mega\pascal} strained at $\dot{\varepsilon} = \qty{0.001}{\per\s}$. They were prestressed at \qty{5}{\mega\pascal} strained at $\dot{\varepsilon} = \qty{0.001}{\per\s}$.
Martensitic Samples were tested to failure, the others until they started necking. Martensitic Samples were tested to failure, the others until they started necking.
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption{Stress-stress parameters of base microstructure.} \caption{Stress-stress parameters of base microstructure.}\label{tab:tensile_test_base}
\label{tab:tensile_test_base} \begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule
\begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$} \\
{$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$}\\ {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} \\ \midrule
{[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} \\ \midrule 25 & 204 & 601 & 616 & 892 & 7,8 \\
25 & 204 & 601 & 616 & 892 & 7,8 \\ 300 & 191 & 511 & 551 & 856 & 9,4 \\
300 & 191 & 511 & 551 & 856 & 9,4 \\ 400 & 189 & 469 & 506 & 710 & 6,2 \\
400 &189 & 469 & 506 & 710 & 6,2 \\ 500 & 165 & 401 & 430 & 537 & 3,3 \\
500 &165 & 401 & 430 & 537 & 3,3 \\ 600 & 141 & 306 & 330 & 374 & 2,2 \\
600 & 141 & 306 & 330 & 374 & 2,2 \\ 700 & 102 & 156 & 166 & 174 & 1,6 \\
700 & 102 & 156 & 166 & 174 & 1,6 \\ \bottomrule
\bottomrule \end{tabular}
\end{tabular}
\end{table} \end{table}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_BASE.png} \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_BASE.png}
\caption{Stress-strain curves for base microstructure.} \caption{Stress-strain curves for base microstructure.}\label{fig:stress_base}
\label{fig:stress_base}
\end{figure} \end{figure}
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption{Stress-stress parameters of martensite.} \caption{Stress-stress parameters of martensite.}\label{tab:tensile_test_mart}
\label{tab:tensile_test_mart} \begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=4]S[table-format=4]S[table-format=4]SS}\toprule
\begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=4]S[table-format=4]S[table-format=4]SS}\toprule {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$} & {$A$} \\
{$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$} & {$A$}\\ {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} & {[\unit{\percent}]} \\ \midrule
{[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} & {[\unit{\percent}]} \\ \midrule 25 & 209 & 1041 & 1217 & 1994 & 3,5 & 3,5 \\
25 & 209 & 1041 & 1217 & 1994 & 3,5 & 3,5 \\ 100 & 199 & 1155 & 1362 & 2101 & 2,5 & 2,5 \\
100 & 199 & 1155 & 1362 & 2101 & 2,5 & 2,5 \\ 200 & 185 & 1141 & 1342 & 2011 & 5,0 & 6,0 \\
200 & 185 & 1141 & 1342 & 2011 & 5,0 & 6,0 \\ 300 & 180 & 926 & 1094 & 1540 & 3,0 & 5,5 \\
300 & 180 & 926 & 1094 & 1540 & 3,0 & 5,5 \\ 350 & 181 & 852 & 986 & 1284 & 2,5 & 4,5 \\
350 & 181 & 852 & 986 & 1284 & 2,5 & 4,5 \\ \bottomrule
\bottomrule \end{tabular}
\end{tabular}
\end{table} \end{table}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_MART.png} \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_MART.png}
\caption{Stress-strain curves for martensite.} \caption{Stress-strain curves for martensite.}\label{fig:stress_mart}
\label{fig:stress_base}
\end{figure} \end{figure}
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption{Stress-stress parameters of austenite.} \caption{Stress-stress parameters of austenite.}\label{tab:tensile_test_aust}
\label{tab:tensile_test_aust} \begin{tabular}{S[table-format=4]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule
\begin{tabular}{S[table-format=4]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$} \\
{$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$}\\ {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} \\ \midrule
{[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]} \\ \midrule 600 & 132 & 202 & 217 & 262 & 3,0 \\
600 & 132 & 202 & 217 & 262 & 3,0 \\ 650 & 127 & 160 & 173 & 189 & 2,0 \\
650 & 127 & 160 & 173 & 189 & 2,0 \\ 700 & 108 & 92 & 96 & 135 & 9,9 \\
700 & 108 & 92 & 96 & 135 & 9,9 \\ 800 & 105 & 69 & 72 & 100 & 11,7 \\
800 & 105 & 69 & 72 & 100 & 11,7 \\ 900 & 70 & 49 & 51 & 64 & 6,9 \\
900 & 70 & 49 & 51 & 64 & 6,9 \\ 1000 & 57 & 30 & 31 & 37 & 9,5 \\
1000 & 57 & 30 & 31 & 37 & 9,5 \\ 1100 & 56 & 19 & 19 & 22 & 9,4 \\
1100 & 56 & 19 & 19 & 22 & 9,4 \\ \bottomrule
\bottomrule \end{tabular}
\end{tabular}
\end{table} \end{table}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_AUST.png} \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_AUST.png}
\caption{Stress-strain curves for austenite.} \caption{Stress-strain curves for austenite.}\label{fig:stress_aust}
\label{fig:stress_base}
\end{figure} \end{figure}
\acrfull{rve}\autocite{fischer2000new} \acrfull{rve}\autocite{fischer2000new}
\section{Phase Transformation Behavior} \section{Phase Transformation Behavior}\label{sec:phase_transformation}
\label{sec:phase_transformation}
Phase transformation data was gathered by examination of the dilatometer experiments\autocite{antretter2002thermo} Phase transformation data was gathered by examination of the dilatometer experiments\autocite{antretter2002thermo}
\subsection{Martensite} \subsection{Martensite}
\ \
\autocite{schemmel2014modelling} \autocite{schemmel2014modelling}
\section{Electromagnetic Properties} \section{Electromagnetic Properties}\label{sec:electromagnetic_properties}
\label{sec:electromagnetic_properties}
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. 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}. As a recourse, magnetic model data was taken from M. Schwenk's thesis on induction heating\autocite[111]{schwenk2012numerische}.
@ -327,7 +299,6 @@ His subject material of a 42CrMo4 steel was deemed close enough in its character
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. 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: This thesis will therefore use the analytic formula described by Trutt et al.\autocite{trutt1968representation} using the arc tangent, as Schwenk did:
$$B(H,T) = \mu_0 H + \frac{2 B_{sat}}{\pi}\cdot \mathrm{atan} \left( \frac{(\mu_{r0} - 1)\mu_0 \pi}{2 B_{sat}} H \right)\cdot {e}^\frac{T-T_C}{C}$$ \[B(H,T) = \mu_0 H + \frac{2 B_{sat}}{\pi}\cdot \mathrm{atan} \left( \frac{(\mu_{r0} - 1)\mu_0 \pi}{2 B_{sat}} H \right)\cdot {e}^\frac{T-T_C}{C}\]
With a saturation flux $B_s=\qty{1.4}{\tesla}$, initial relative permeability $\mu_{r0}=2500$, Curie temperature $T_C=\qty{785}{\celsius}$, and a temperature scaling constant $C=\qty{60}{\celsius}$. With a saturation flux $B_s=\qty{1.4}{\tesla}$, initial relative permeability $\mu_{r0}=2500$, Curie temperature $T_C=\qty{785}{\celsius}$, and a temperature scaling constant $C=\qty{60}{\celsius}$.

81
overleaf/Kapitel/char_process.tex
View file

@ -1,7 +1,6 @@
% !TeX root = ../dissertation.tex % !TeX root = ../dissertation.tex
\chapter{Process Characterization} \chapter{Process Characterization}\label{cha:process_char}
\label{cha:process_char}
Induction heating, despite all its positive aspects, is a difficult process to control due to it's multi-physical nature. Induction heating, despite all its positive aspects, is a difficult process to control due to it's multi-physical nature.
@ -18,34 +17,32 @@ Many industrial induction ovens are however not instrumented beyond a power mete
For this reason, the \gls{mcl} commissioned an induction heating teat rig equipped with a bank of thermocouple endpoints. For this reason, the \gls{mcl} commissioned an induction heating teat rig equipped with a bank of thermocouple endpoints.
This machine (shown in figure~\ref{fig:mcl-test-rig}) is equipped with a \emph{ThermProTEC MS 30} series LC oscillator, capable of \SIrange{10}{40}{\kilo\Hz} in frequency and \SIrange{10}{50}{\kilo\W} in power. This machine (shown in figure~\ref{fig:mcl-test-rig}) is equipped with a \emph{ThermProTEC MS 30} series LC oscillator, capable of \SIrange{10}{40}{\kilo\Hz} in frequency and \SIrange{10}{50}{\kilo\W} in power.
It can be run in constant-voltage, constant-current, and constant-power mode. It can be run in constant-voltage, constant-current, and constant-power mode.
The sample can be rotated and the coil can be moved along its axis to emulate continuous induction heating. The sample can be rotated and the coil can be moved along its axis to emulate continuous induction heating.
The induction coil is water-cooled and fixed via a face plate that allows different coil geometries to be mounted, alongside a circular quenching nozzle (see figure~\ref{fig:test-rig-coil}). The induction coil is water-cooled and fixed via a face plate that allows different coil geometries to be mounted, alongside a circular quenching nozzle (see figure~\ref{fig:test-rig-coil}).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\begin{tabular}{cc} \begin{tabular}{cc}
\subfloat[Full view\label{fig:mcl-test-rig}]{\includegraphics[width=0.45\linewidth]{Abbildungen/mcl-test-rig.png}} \subfloat[Full view\label{fig:mcl-test-rig}]{\includegraphics[width=0.45\linewidth]{Abbildungen/mcl-test-rig.png}}
&\quad & \quad
\subfloat[Induction coil assembly\label{fig:test-rig-coil}]{\includegraphics[width=0.45\linewidth]{Abbildungen/test-rig-rogowski.png}} \subfloat[Induction coil assembly\label{fig:test-rig-coil}]{\includegraphics[width=0.45\linewidth]{Abbildungen/test-rig-rogowski.png}}
\end{tabular} \end{tabular}
\caption{\acrshort{mcl} induction heating test rig (model HU-VH300-MS30, manufactured by Ideal \caption{\acrshort{mcl} induction heating test rig (model HU-VH300-MS30, manufactured by Ideal Thermal Processes GmbH (ITP), Oberkirch, Germany).}
Thermal Processes GmbH (ITP), Oberkirch, Germany).}poe
\end{figure} \end{figure}
A series of induction experiments was conducted by our research group to obtain data on an arbitrary induction hardening procedure, whose temperature curve imitated that of the industrial process experienced by the crankshafts in the following section. A series of induction experiments was conducted by our research group to obtain data on an arbitrary induction hardening procedure, whose temperature curve imitated that of the industrial process experienced by the crankshafts in the following section.
These experiments on rod samples done on the \gls{mcl} in-house induction test rig were well instrumented and published under J\'aszfi \emph{et al.}\cite{jaszfi2019influence, jaszfi2022indirect, jaszfi2022residual}. These experiments on rod samples done on the \gls{mcl} in-house induction test rig were well instrumented and published under J\'aszfi \emph{et al.}\cite{jaszfi2019influence, jaszfi2022indirect, jaszfi2022residual}.
To summarize, the induction test rig was configured to approximate a linear temperature increase up to the assumed maximum of \qty{1050}{\degreeCelsius} which was held for \qty{10}{\s}, after which quenching fluid was injected in between the induction coil to achieve a cooling coefficient of $\lambda = 0.01$ or \qtyrange{800}{500}{\degreeCelsius} in \qty{1}{\s}. To summarize, the induction test rig was configured to approximate a linear temperature increase up to the assumed maximum of \qty{1050}{\degreeCelsius} which was held for \qty{10}{\s}, after which quenching fluid was injected in between the induction coil to achieve a cooling coefficient of $\lambda = 0.01$ or \qtyrange{800}{500}{\degreeCelsius} in \qty{1}{\s}.
The rod samples measured \qty{22}{\mm} in diameter and \qty{300}{\mm} in length. They had holes drilled for thermocouples at center height of the coil at \qtylist{0.5;10.5}{\mm} depth, and voltage and current were measured directly by the machine. The rod samples measured \qty{22}{\mm} in diameter and \qty{300}{\mm} in length. They had holes drilled for thermocouples at center height of the coil at \qtylist{0.5;10.5}{\mm} depth, and voltage and current were measured directly by the machine.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[height=7cm]{Abbildungen/IBA_data.png} \includegraphics[height=7cm]{Abbildungen/IBA_data.png}
\caption{Machine protocol of induction heating experiment.} \caption{Machine protocol of induction heating experiment.}\label{fig:test-rig-curve}
\label{fig:test-rig-curve}
\end{figure} \end{figure}
While the machine itself recorded the heating process (see figure~\ref{fig:test-rig-curve}), some caveats apply to collected data. While the machine itself recorded the heating process (see figure~\ref{fig:test-rig-curve}), some caveats apply to collected data.
@ -56,62 +53,60 @@ This meant that they could not be directly used as simulation inputs since the t
To alleviate this, the electrical power was measured directly across the coil was measured by a digital oscilloscope \emph{Picoscope 3404D MSO}, with a set of Rogowski coils\autocite[175]{tumanski2011handbook} used to convert current to a voltage. To alleviate this, the electrical power was measured directly across the coil was measured by a digital oscilloscope \emph{Picoscope 3404D MSO}, with a set of Rogowski coils\autocite[175]{tumanski2011handbook} used to convert current to a voltage.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\label{fig:rogoswki-coils-rod} \includegraphics[width=9cm]{example-image}
\caption{Rogowski Coils??}\label{fig:rogoswki-coils-rod}
\end{figure} \end{figure}
This gave better data, was well as the opportunity to graph the electrical wave shape as supplied by the transformer. This gave better data, was well as the opportunity to graph the electrical wave shape as supplied by the transformer.
Figure~\ref{fig:rogowski-data-rod} corroborated the power curve of the machine, but multiplied by a factor of \num{18}, somewhat lower than the reported transformation factor of the machine's transformer. Figure~\ref{fig:rogowski-data-rod} corroborated the power curve of the machine, but multiplied by a factor of \num{18}, somewhat lower than the reported transformation factor of the machine's transformer.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\subfloat[Current flow\label{fig:current-rod}]{\includegraphics[width=6cm]{Abbildungen/IBA_current_comparison.png}} \subfloat[Current flow\label{fig:current-rod}]{\includegraphics[width=6cm]{Abbildungen/IBA_current_comparison.png}}
\qquad \qquad
\subfloat[Single wave at $t=\qty{6}{\sec}$??\label{fig:waveshape-rod}]{\includegraphics[width=6cm]{Abbildungen/IBA_waveform.png}} \subfloat[Single wave at $t=\qty{6}{\sec}$??\label{fig:waveshape-rod}]{\includegraphics[width=6cm]{Abbildungen/IBA_waveform.png}}
\caption{Process data of 50CrMo4 rod being heated in the \acrshort{mcl} test rig using a helical induction coil.} \caption{Process data of 50CrMo4 rod being heated in the \acrshort{mcl} test rig using a helical induction coil.}\label{fig:rogowski-data-rod}
\label{fig:rogowski-data-rod}
\end{figure} \end{figure}
The current data used as simulation input was then reduced to a set of num?? interpolation points to increase calculation stability. The current data used as simulation input was then reduced to a set of \num{300} interpolation points to increase calculation stability.
\section{Crankshaft Hardening Process} \section{Crankshaft Hardening Process}
The crankshaft production line at the BMW Motoren Werk in Steyr, Austria, was a system constructed by \emph{SMS-Elotherm GmbH} that was powered by a transformer bank and run in a constant voltage regimen. The crankshaft production line at the BMW Motoren Werk in Steyr, Austria, was a system constructed by \emph{SMS-Elotherm GmbH} that was powered by a transformer bank and run in a constant voltage regimen.
It behaved like many industrial machines, in that it did have a controlling computer that showed the electrical characteristics of each heating cycle while they were happening and cold even plot those as needed, but it did not have the option to export that data in a table format for process control. It behaved like many industrial machines, in that it did have a controlling computer that showed the electrical characteristics of each heating cycle while they were happening and cold even plot those as needed, but it did not have the option to export that data in a table format for process control.
The same methodology of a voltage measurement across inductor contacts and a Rogowski coil for current data was applied with the added challenge of said inductor moving along with the eccentric crank bearing during rotational induction heating (see figure~\ref{fig:produciton-setup}. The same methodology of a voltage measurement across inductor contacts and a Rogowski coil for current data was applied with the added challenge of said inductor moving along with the eccentric crank bearing during rotational induction heating (see figure~\ref{fig:produciton-setup}).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[width=0.66\linewidth]{Abbildungen/crank-rogowski.png} \includegraphics[width=0.66\linewidth]{Abbildungen/crank-rogowski.png}
\caption{Experimental setup at the crankshaft hardening line.} \caption{Experimental setup at the crankshaft hardening line.}\label{fig:produciton-setup}
\label{fig:produciton-setup}
\end{figure} \end{figure}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\subfloat[Power flow\label{fig:current-crank}]{\includegraphics[width=0.45\linewidth]{Abbildungen/elotherm-hl1.png}} \subfloat[Power flow\label{fig:current-crank}]{\includegraphics[width=0.45\linewidth]{Abbildungen/elotherm-hl1.png}}
\quad \quad
\subfloat[Normalized single wave at $t=\qty{1}{\sec}$\label{fig:waveshape-crank}]{\includegraphics[width=0.45\linewidth]{Abbildungen/Elotherm_sine.png}} \subfloat[Normalized single wave at $t=\qty{1}{\sec}$\label{fig:waveshape-crank}]{\includegraphics[width=0.45\linewidth]{Abbildungen/Elotherm_sine.png}}
\caption{Process data of C38p crankshaft bearing being heated by a \ang{120} arc shaped inductor at the BMW production line. As per publication~\ref{apx:pub1}, the slight over regulation of the waveform can be ignored during simulation.} \caption{Process data of C38p crankshaft bearing being heated by a \ang{120} arc shaped inductor at the BMW production line. As per publication~\ref{apx:pub1}, the slight over regulation of the waveform can be ignored during simulation.}\label{fig:rogowski-data-crank}
\label{fig:rogowski-data-crank}
\end{figure} \end{figure}
For the simulation, the electrical current data was again simplified. For the simulation, the electrical current data was again simplified.
the fluctuating current was set to switch between two set levels of \qtyrange{00}{00}{A} with a timing of \qty{00}{\s} high and \qty{00}{\s} low. the fluctuating current was set to switch between two set levels of \qtyrange{00}{00}{A} with a timing of \qty{00}{\s} high and \qty{00}{\s} low.
\section{Signal Quality} \section{Signal Quality}
As seen in figures~\ref{fig:waveshape-rod} and~\ref{fig:waveshape-crank}, the electrical signal driving the inductors of this thesis are not sinusoidal. As seen in figures~\ref{fig:waveshape-rod} and~\ref{fig:waveshape-crank}, the electrical signal driving the inductors of this thesis are not sinusoidal.
This presents a problem for the harmonic solution of the electromagnetic field problem (discussed in section~\ref{sub:sota-fem-em}): This presents a problem for the harmonic solution of the electromagnetic field problem (discussed in section~\ref{sub:sota-fem-em}):
While the principle of \emph{superposition} states that different subharmonic components can be summed up to solve an arbitrary oscillation\autocite{??}, this stops being the case when materials respond in a non-linear fashion\autocite{??}. While the principle of \emph{superposition} states that different subharmonic components can be summed up to solve an arbitrary oscillation\autocite{??}, this stops being the case when materials respond in a non-linear fashion\autocite{??}.
Harmonic solutions with non-linear materials therefore must therefore assume a simple sinusoidal electric input, which will lead to some systemic error of the simulation. Harmonic solutions with non-linear materials therefore must therefore assume a simple sinusoidal electric input, which will lead to some systemic error of the simulation.
Publication~\ref{apx:pub1} presents the \acrfull{thd} as a method of characterizing the deviation of input signals from a sinus wave, based on the amplitudes of the Fourier transformation: Publication~\ref{apx:pub1} presents the \acrfull{thd} as a method of characterizing the deviation of input signals from a sinus wave, based on the amplitudes of the Fourier transformation:
\begin{equation} \begin{equation}
\label{eq:thd} \label{eq:thd}
\textrm{THD} = \frac{\sqrt{\sum_{k=2}^{N} I_k^2}}{I_1} \textrm{THD} = \frac{\sqrt{\sum_{k=2}^{N} I_k^2}}{I_1}
\end{equation} \end{equation}
Resulting in a \acrshort{thd} of \num{0.19} for the induction test rig, and \num{0.10} for the industrial furnace. Resulting in a \acrshort{thd} of \num{0.19} for the induction test rig, and \num{0.10} for the industrial furnace.

3
overleaf/Kapitel/introduction.tex
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@ -1,5 +1,4 @@
\chapter{Introduction and Problem Statement} \chapter{Introduction and Problem Statement}\label{cha:intro}
\label{cha:intro}
A crankshaft's function within a motor is to combine the force of the firing cylinders and transform linear into rotational motion. A crankshaft's function within a motor is to combine the force of the firing cylinders and transform linear into rotational motion.
In essence crankshafts are a set of offset bearings and counterweights along a primary axis that terminates in a gearwheel. In essence crankshafts are a set of offset bearings and counterweights along a primary axis that terminates in a gearwheel.

14
overleaf/Kapitel/state_of_the_art.tex
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@ -2,8 +2,7 @@
This chapter compiles the state of the art for the fundamental topics necessary for the simulation of an inductive heat treatment process and its validation. This chapter compiles the state of the art for the fundamental topics necessary for the simulation of an inductive heat treatment process and its validation.
\section{Inductive Surface Hardening} \section{Inductive Surface Hardening}\label{sec:sota_induction}
\label{sec:sota_induction}
In modern physics, Maxwell's Equations are the basis for understanding electromagnetic phenomena. In modern physics, Maxwell's Equations are the basis for understanding electromagnetic phenomena.
Their differential forms, names, and approximate meaning are as follows: Their differential forms, names, and approximate meaning are as follows:
@ -60,21 +59,24 @@ The skin depth for conductive materials can be calculated using the frequency (o
\end{equation} \end{equation}
The geometry of the induction coil is the most difficult aspect of inductive hardening, where it's broad range of application leads to a multitudes of designs. The geometry of the induction coil is the most difficult aspect of inductive hardening, where it's broad range of application leads to a multitudes of designs.
Continuous heating of rods may require several conscutive soilenoids, where gears and sprockets are briefly held in a single loop. Continuous heating of rods may require several conscutive solenoids, where gears and sprockets are briefly held in a single loop.
In general only the most simple geometries allow for analytical solutions of the heated volume and so inductor design still greatly relies on the experience of process engineers. In general only the most simple geometries allow for analytical solutions of the heated volume and so inductor design still greatly relies on the experience of process engineers.
Many iterations of experiments are needed to dial in the geometry if the hardened zone and arrive at a usable product. Many iterations of experiments are needed to dial in the geometry if the hardened zone and arrive at a usable product.
Numerical simulation has been a boon in this regard as it allows initial experiments to be run digitally, freeing valuable production equipment and saving the cost of bespoke copper inductors in exotic geomtries. Numerical simulation has been a boon in this regard as it allows initial experiments to be run digitally, freeing valuable production equipment and saving the cost of bespoke copper inductors in exotic geomtries.
\section{Finite Element Method} \section{Finite Element Method}\label{sec:sota_fem}
\label{sec:sota_fem}
The \acrfull{fem} is one of several numerical approaches to subdividing a volume into smaller chunks in which a simplification of physics can be calculated numerically. The \acrfull{fem} is one of several numerical approaches to subdividing a volume into smaller chunks in which a simplification of physics can be calculated numerically.
It can achieve solutions to differential equations for complex volumes, for which analytical solutions are all but impossible. It can achieve solutions to differential equations for complex volumes, for which analytical solutions are all but impossible.
\acrshort{fem} in particular is mostly used for the continuum mechanics problems, such as stress/strain calculations, heat transfer problems, mass transport, and electromagnetic potential. \acrshort{fem} in particular is mostly used for the continuum mechanics problems, such as stress/strain calculations, heat transfer problems, mass transport, and electromagnetic potential.
Its method of subdivision are the eponymous finite elements, that (in a 3D case) can take the shape of tetrahedrons, pyramids, wedges and cuboids which are in turn defined through their vertex points. Its method of subdivision are the eponymous finite elements, that (in a 3D case) can take the shape of tetrahedrons, pyramids, wedges and cuboids which are in turn defined through their vertex points, also called nodes.
Each finite element can be transformed through a matrix to ideal elements, in which the all properties such as stress, movement or temperature only exist at the nodes.\textcolor{red}{?? IMAGE}
Inbetween nodes, simple helper-functions interpolate the properties.
These can be linear for elements consisting of only vertex nodes, or polynomes if nodes are also located at the element's edges.
Since touching elements share nodes, and nodes of an element can describe the property continuum inside an element via a matrix, the matrices of all elements can be combined to a global matrix which describes the dependencies of inputs and results for one of the model's properties (e.g.\ stresses and strains).
\subsection{Electromagnetic Analysis} \subsection{Electromagnetic Analysis}