| 21 |
\label{Sec:1DHDv00} |
\label{Sec:1DHDv00} |
| 22 |
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| 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 \verb|T1| and in \textit{Block 2} is \verb|T2|. |
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\verb|T1| and \textit{Block 2} is at a temperature \verb|T2|. |
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| 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 |
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| 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: |
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\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 |
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