21 
\label{Sec:1DHDv00} 
\label{Sec:1DHDv00} 
22 


23 
The first model consists of two blocks of isotropic material, for instance granite, sitting next to each other. 
The first model consists of two blocks of isotropic material, for instance granite, sitting next to each other. 
24 
Initially, \textit{Block 1} is of a temperature 
Initial temperature in \textit{Block 1} is \verbT1 and in \textit{Block 2} is \verbT2. 

\verbT1 and \textit{Block 2} is at a temperature \verbT2. 

25 
We assume that the system is insulated. 
We assume that the system is insulated. 
26 
What would happen to the temperature distribution in each block over time? 
What would happen to the temperature distribution in each block over time? 
27 
Intuition tells us that heat will transported from the hotter block to the cooler until both 
Intuition tells us that heat will be transported from the hotter block to the cooler until both 
28 
blocks have the same temperature. 
blocks have the same temperature. 
29 


30 
\subsection{1D Heat Diffusion Equation} 
\subsection{1D Heat Diffusion Equation} 
31 
We can model the heat distribution of this problem over time using the one dimensional heat diffusion equation\footnote{A detailed discussion on how the heat diffusion equation is derived can be found at \url{http://online.redwoods.edu/instruct/darnold/DEProj/sp02/AbeRichards/paper.pdf}}; 
We can model the heat distribution of this problem over time using one dimensional heat diffusion equation\footnote{A detailed discussion on how the heat diffusion equation is derived can be found at \url{http://online.redwoods.edu/instruct/darnold/DEProj/sp02/AbeRichards/paper.pdf}}; 
32 
which is defined as: 
which is defined as: 
33 
\begin{equation} 
\begin{equation} 
34 
\rho c\hackscore p \frac{\partial T}{\partial t}  \kappa \frac{\partial^{2} T}{\partial x^{2}} = q\hackscore H 
\rho c\hackscore p \frac{\partial T}{\partial t}  \kappa \frac{\partial^{2} T}{\partial x^{2}} = q\hackscore H 