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\centerline{\large \bf MA 573 --- PROJECT 5}

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\noindent
{\bf Due: Monday, October 30}

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In this project, we are going to model the steady-state temperature distribution for aluminum and copper
rods heated by a soldering rod.

\begin{description}

\item{1.} For our experimental apparatus, we have rectangular, uninsulated, rods with cross-sectional dimensions $a = b = 0.95$~cm and length $L = 70$~cm.  The posted .txt files include 15 temperature measurements every 4~cm from 10~cm to 66~cm.  A heat source at $x=0$ provides a fixed,
but unknown, heat flux $\Phi$.  Derive the model
\begin{equation}
  \rho c_\rho \frac{\d T}{\d t} = \frac{\d}{\d x}\left( k \frac{\d T}{\d x} \right) 
  - \frac{2(a+b)h}{ab} \left[T(t,x) - T_{amb}\right] \ , \ 0 < x < L,
\label{eq_heat1}
\end{equation}
and boundary conditions
\begin{equation}
  k \frac{dT}{dx}(t,0) = \Phi \quad , \quad k \frac{d T}{d x}(t,L) = h [T_{amb} - T(t,L)] .
\label{eq_heat2}
\end{equation}
Here $T, \rho, c_\rho, k, h$ and $T_{amb}$ respectively denote the temperature, density, specific heat,
thermal conductivity, convective heat transfer coefficient, and ambient room temperature.   Initial conditions are specified as
\begin{equation}
  T(0,x) = T_0(x).
\label{eq_heat3}
\end{equation}

\hspace{0.2truein}
Derive the steady-state model where $T_s$ denotes the steady-state temperature.  We will use the
steady-state equations in all further computations.  Note that we will
use the thermal conductivity values $k = 2.37 \frac{W}{cm \cdot C}$ and $k = 4.01 \frac{W}{cm \cdot C}$ when modeling the aluminum and copper rods. 

\item{2.} Consider the {\em insulated} aluminum rod model with the boundary conditions
$$
  k \frac{dT_s}{dx}(0) = \Phi \ , \ T_s(L) = T_{amb}
$$
and parameter $q = \Phi$.  Determine the
analytic solution and parameter $\Phi$ that yields an optimal fit.  What do you conclude about the accuracy
of this model.  

\item{3.} Now estimate the parameters $q = [\Phi,h]$ for the {\em uninsulated} aluminum rod with the boundary
conditions (\ref{eq_heat2}).  Note that the solution is
\begin{equation}
 T_s(x;q) = c_1(q) e^{-\gamma x} + c_2(q) e^{\gamma x} + T_{amb}
\label{eq9_4}
\end{equation}
where $\gamma = \sqrt{\frac{2(a+b)h}{abk}}$ and
$$
 c_1(q) = -\frac{\Phi}{k\gamma}\left[\frac{e^{\gamma L}(h+k\gamma)}{e^{-\gamma L}(h-k\gamma) + e^{\gamma L}(h+k\gamma)}\right] \quad , \quad c_2(q) = \frac{\Phi}{k\gamma} + c_1(q).
$$
Explain why you cannot simultaneously estimate the parameter set $q = [\Phi,h,k]$.  
Plot the optimal fit and data along with the
residuals.  Do your errors appear to be iid?  Are the values for your parameters physically reasonable?

\item{4.} For the optimal fit to the aluminum rod data, determine an estimate $s^2$ for the variance
$\sigma_0^2$ of the errors and construct the covariance matrix; note that you can do this analytically. 
Determine 95\% confidence intervals for each parameter.

\item{5.} Repeat your analysis from 3 and 4 for the copper rod.

\item{6.} Finally, estimate the optimal parameters by simultaneously
using the data from both the aluminum and copper
rods.   This will yield one value of $\Phi$ and $h$ for both rods.  Discuss your results and note any limitations
of the model.  Can you obtain a better fit if you consider the parameters $q = [\Phi_a, \Phi_c, h]$, where
$\Phi_a$ and $\Phi_c$ denote potentially differing fluxes for the aluminum and copper rods?

\end{description}




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