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\begin{eqnarray*}

   U^{cond} = \displaystyle\sum_{i=1}^{N} u_{i}^{cond} \\ 

   U^{mech} = \displaystyle\sum_{i=1}^{N} u_{i}^{mech} \\

   U = \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech}) \\

   \theta_{avg} = (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\

\end{eqnarray*}                           

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$$

   u_{i} = u^{mech}_{i} + u^{cond}_{i} = C_{V} \theta_{i}

$$

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