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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^{chem} = \displaystyle\sum_{i=1}^{N} u_{i}^{chem} \\
   U = \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech} + u_{i}^{chem}) \\
   \theta_{avg} = (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\
\end{eqnarray*}                           

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