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\begin{document}
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Coupled Differential Equations and Heating of a Polycarbonate Block}
\markboth{Heating of a Polycarbonate Block}{Heating of a Polycarbonate Block}
\author{Mitaxi Mehta \\
School of Engineering and Applied Science\\
Ahmedabad University \\
Ahmedabad 380009 INDIA }
\makeStitlePDFLaTex
\begin{abstract}
Differential equations and Laplace transforms are an integral part of control problems in
engineering systems. However a clear explanation of the relationship of Laplace transforms
with the differential equation formalism is difficult to find for coupled differential equations.
Here we describe an example where this relationship is understood by breaking up the system of
differential equations into simpler cause-effect blocks and by relating the blocks to the the
corresponding Laplace transforms.
\end{abstract}
\section*{INTRODUCTION - THE SYSTEM}
An open system where a solid object is being heated up is a standard component
of many applications. One such solid that gets used in the design of baby warmers
is a polycarbonate sheet \cite{ACME2019,Combes2017}. See Figure~\ref{fig:7-010-BabyWarmer}.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{7-010-BabyWarmer.png}
\caption{\label{fig:7-010-BabyWarmer} Sketch of Baby warmer using polycarbonate plate.\cite{Combes2017}}
\end{figure}
Here we analyze the case of an open system made up of a
polycarbonate block being heat up by a source of temperature $s_{in}(t)$ in presence of
a proportional control of the source heat given by $c_1 ( T_h(t)- T_p(t))$.
The complete system of differential equations is the following,
\begin{eqnarray}\label{eq:model1}
T'_h(t) &=& s_{in}(t) - c_1 ( T_h(t)- T_p(t)) \\ \nonumber
T'_p(t) &=& c_1 ( T_h(t)- T_p(t)) - c_2 ( T_p(t)-T_E).
\end{eqnarray}
The first equation says that the heater temperature $ T_h(t)$ depends on the source strength $s_{in}(t)$
and on the temperature difference between the heater and the polycarbonate. Here the loss of temperature
to the environment has been neglected. The second equation expresses the fact that the temperature of
the polycarbonate $ T_p(t)$ depends on the heater temperature and on the temperature of the environment, $T_E$.
Here we assume that the constants $c_1$, $c_2$ the source term $s_{in}(t)$ are appropriately defined using
relevant dimensions.
In control problems in engineering systems, such systems of linear ordinary differential equations,
are needed to be represented in terms of simpler cause and effect blocks. The Laplace transform of the
differential equations, which connects the input variables to the output variables is calculated.
We break the problem down into the following cause and effect
reasoning for deciding the input and output of the Laplace transfer function blocks.
\begin{enumerate}
\item An input power $s_{in}(t)$ that causes the heater temperature
$ T_h(t)$ to rise. In the first transfer function block
the Laplace transform of $s_{in}(t)$ is the input and the
Laplace transform for $ T_h(t)$ is the output.
\item The heater temperature $ T_h(t)$ causes a rise in the polycarbonate temperature $ T_p(t)$ using convection.
Thus in the next block the Laplace transform for $ T_h(t)$ is the input and the Laplace transform for
$ T_p(t)$ is the output.
\item Proportional control of the source is represented by
the feedback loops that take in $ T_p(t)$ and $ T_h(t)$ and feed back
with amplification factors $c_1$ and $-c_1$ respectively to the source. In other words
the control term $c_1 ( T_h(t)- T_p(t))$ increases of reduces the effecct of the source term
to control the heater temperature $ T_h(t)$.
\end{enumerate}
$T_E = 0$ can be assumed to be zero, which is equivalent to working with the shifted temperature
scale $T-T_E$. Thus we replace $T_h(t) - T_E$ simply with $T_h(t)$ here. In this new temperature scale, the system of differential equations
becomes,
\begin{eqnarray}
T'_h(t) &=& s_{in}(t) - c_1 ( T_h(t)- T_p(t)) \\
T'_p(t) &=& c_1 ( T_h(t)- T_p(t)) - c_2 T_p(t)\, .
\end{eqnarray}
We describe the connection of this system of linear
coupled differential equations with the corresponding block diagram using Laplace transform
with cause and effect analysis.
% see that Finding separate transfer functions
% with the given steps, not only simplifies the
% problem considerably but also gives a very powerful method for % solving more complex problems.
\subsection{From the Source to the Heater}
Consider the part of the system where the input to the
system $s_in$ gives rise to the heater temperature $ T_h(t)$.
Here we neglect the polycarbonate completely. The equation
of the system is,
\[
T'_h(t) = s_{in}(t)\, .
\]
We shall consider the general case where $s_{in}(t)$ is a function of time. The Laplace transform of a
real function is defined by $${\cal L}(f(x)) = \int_0^{\infty} f(x) e^{-s x} dx = {\cal F}(s).$$ Thus we take Laplace transforms
of both sides,
\[
s {\cal T}_h(s) - T_h(0) ={\cal S}_{in}(s) \, .
\]
Assuming $ T_h(0) = 0$, i.e. the heater is at the room temperature to start with,
the transfer function for this part of the system is,
\[
{\cal G}_1(s)=\frac{{\cal T}_h(s)}{{\cal S}_{in}(s) } = \frac{1}{s}
\]
When the temperature $ T_h(t)$ is fed back into the heat source
one gets the resulting differential equation,
\begin{equation}\label{eq:tran}
T'_h(t) = s_{in}(t) - c_1 T_h(t)\, .
\end{equation}
In the diagram of the Laplace transfer function, this loop combined with the heater block can be
represented by a new transfer function block, ${\cal G}_1'(s)$, in (\ref{eq:trans}), obtained from the transform of (\ref{eq:tran}) found in (\ref{eq:tranB}). Again, we assume $T_h(0) = 0$.
\begin{eqnarray}\label{eq:tranB}
s{\cal T}_h(s) = {\cal S}_{in}(s) - c_1 {\cal T}_h(s)\,.
\end{eqnarray}
Now solving for ${\cal G}_1'(s) = \frac{ {\cal S}_{in}(s) }{ {\cal T}_h(s)}$ we obtain,
\begin{eqnarray}\label{eq:trans}
{\cal G}_1'(s) = \frac{1}{s + c_1}\, .
\end{eqnarray}
\subsection{From the Heater to the Polycarbonate}
Next let us consider only that part of the system where
the heating element temperature, $ T_h(t)$ is input to the system
and the polycarbonate temperature $ T_p(t)$ is the output.
This part of the system is modeled by,
\begin{equation}\label{eq:eqq}
T'_p(t) = c_1 ( T_h(t)- T_p(t)) - c_2 T_p(t)\, .
\end{equation}
After taking the Laplace transform of (\ref{eq:eqq}) we write the equation in $s$-space:
\[
s {\cal T}_p(s) = c_1({\cal T}_h(s)- {\cal T}_p(s) ) - c_2 {\cal T}_p(s) \, .
\]
The transfer function for this part of the system is,
\[
G_2(s) = \frac{ c_1 }
{ s + c_1 + c_2 } \, .
\]
\subsection{From the Polycarbonate to the Source}
% When the previous two systems are combined together one
% gets a linear block diagram for system as follows.
% \begin{figure}
% \centering
% \includegraphics[width=0.5\linewidth]{heat-poly-lin}
% \caption[Transfer function-linear]{}
% \end{figure}
% The differential equations for the system is,
% \begin{eqnarray}
% T'_h(t) &=& s_{in}(t) \\
% T'_p(t) &=& c_1 ( T_h(t)- T_p(t)(t)) - c_2 ( T_p(t)(t)-T_0)
% \end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{heat-poly-feedback}
\caption{The block diagram representation of the system}
\label{fig:heat-poly-feedback}
\end{figure}
The remaining system now consists of input $s_{in}(t)$, the linearly connected blocks,
$G_1'$ and $G_2$ and the feedback loop from $ T_p(t)$ to the input. Making up the complete
system of differential equations,
\begin{eqnarray}
T'_h(t) &=& s_{in}(t) - c_1 ( T_h(t)- T_p(t))\, , \\
T'_p(t) &=& c_1 ( T_h(t)- T_p(t)) - c_2 T_p(t)\, .
\end{eqnarray}
The blocks ${\cal G}_1'(s)$ and ${\cal G}_2(s)$ can be combined together again as a
block with the transfer function,
\[
{\cal G}_2'(s) = {\cal G}_1'(s) {\cal G}_2(s) = \frac{ c_1 }
{(s+c_1) (s + c_1 + c_2) }\, ,
\]
and the feedback loop
from $ T_p(t)$ to $s_{in}(t)$ gives the transfer function for the combined open system,
\[
{\cal G}_{open}(s) = \frac{ {\cal G}_2'(s)}{1 - c_1 {\cal G}_2'(s)} \, .
\]
After substitutions, the transfer function of the system can be
written as,
\[
{\cal G}_{open} = \frac{c_1}{(s+c_1)(s+c_1+c_2) -c_1^2}\, .
\]
\section*{Comparing with the eigenvalues}
Writing the system in the matrix form,
\begin{eqnarray*}
\left(
\begin{tabular}{c}
$T'_h(t)$ \\
$T'_p(t)$ \\
\end{tabular}
\right) = \qquad
\left (
\begin{tabular}{cc}
$-c_1$ & $ c_1$ \\
$c_1$ & $ -c_1-c_2$ \\
\end{tabular}
\right) \cdot \left(
\begin{tabular}{c}
$T_h(t)$ \\
$T_p(t)$ \\
\end{tabular}
\right) + \left(
\begin{tabular}{c}
$s_{in}(t)$ \\
$0$ \\
\end{tabular}
\right) \, .
\end{eqnarray*}
It can be seen that the characteristic polynomial of the matrix,
$$A = \left (
\begin{tabular}{cc}
$-c_1$ & $ c_1$ \\
$c_1$ & $ -c_1-c_2$ \\
\end{tabular}
\right) \, $$
is the same as the denominator of ${\cal G}_{open}$. Thus the singular values of the
transfer function are eigenvalues of the matrix equation, defining the growth rates
of the independent solutions.
\section*{Activities}
Consider our model (\ref{eq:model1}) with $c_2 = 1.6$ and $c_1 = 9$, $s_{in}(t) = 7\left(1 - e^{-0.01 t}\right) + 30$ and all initial temperatures at a reasonable room temperature of 20$^{o}$C, i.e. $T_h(0) = T_p(0) = T_E = 20$.
\begin{enumerate}
\item Verbally describe the situation in this model with these parameters.
\item Solve the model (\ref{eq:model1}) with these parameters using either a numerical or analytic technique (not Laplace transforms and plot on the same axes the three functions, (a) constant body temperature 37$^{o}$C, (b) temperature of the heater, $T_h(t)$, and (c) temperature of the polycarbonate block, $T_p(t)$. Explain your plot. See Figure~\ref{fig:7-010-BabyWarmer}.
\item Solve the model (\ref{eq:model1}) with these parameters using Laplace transforms and plot on the same axes the three functions, (a) constant body temperature 37$^{o}$C, (b) temperature of the heater, $T_h(t)$, and (c) temperature of the polycarbonate block, $T_p(t)$. Explain your plot. See Figure~\ref{fig:7-010-BabyWarmer}.
\item Compare your results from the previous two solutions in (2) and (3).
\end{enumerate}
\section*{Summary}
Using transfer functions for analysis of long time behavior of a linear control system
has multiple advantages, in terms of flexibility and modularity. However for coupled
systems of differential equations a direct connection between the differential equations,
the physical causes and effects, and their connection to the Laplace transfer functions may
often not be easy to see. Here
we have offered an engineering control problem in smaller steps and
connected these three ways of analyzing systems.
\section*{Acknowledgements}
I am thankful to the course ``Qualitative Engineering Analysis'' teaching team at the Olin College of Engineering
with whom I worked during the fall of 2017 on the analysis of the baby warmer project.
I especially owe my thanks to Prof.~John Geddes and Prof.~Siddhartan Govindarajan for many discussions and interest.
I am thankful to the Olin College of Engineering for hosting me and involving me in many interesting and useful
learning activities, to the Ahmedabad University for selecting me as an Argosy
Fellowship and to the Argosy Foundation for their generous fellowship. I thank Brian Winkel, Director, SIMIODE, for his editorial assistance.
\teacher{
\section*{NOTES FOR TEACHERS}
In a separate Mathematica notebook, 7-010-Text-T-Mma-CoupledSystemLaplace-TeacherVersion.nb (and an accompanying pdf version for those who do not read Mathematica files) we offer the analyses for the Activities.}
\begin{thebibliography}{98}
\bibitem{ACME2019} Acme Plastics. 2019. {\it The Application Benefits of Polycarbonate Sheets\/}. \href{https://www.acmeplastics.com/content/the-applicational-benefits-polycarbonate-sheets/}{https://www.acmeplastics.com/content/the-applicational-benefits-polycarbonate-sheets/}.
\bibitem{Combes2017} Combes, K. 2017. {\it Designing a Better Baby Warmer for Energy-Insecure Geographic Regions\/}. \href{http://content.kylecombes.com/portfolio/qea-babywarmer/QEA\_Babywarmer\_Report.pdf}{http://content.kylecombes.com/portfolio/qea-babywarmer/QEA\_Babywarmer\_Report.pdf}.
\end{thebibliography}
\end{document}