When designing a distillation column for a multicomponent mixture, a system of material and energy balances are used to determine the reflux ration (the proportion of liquid that flows into the top of the column compared to the amount of vapor that leaves the top), the number equilibrium stages, and the location of the feed. Frequently, a \"shortcut\" method is used to give a first estimate of the answers for this system of equation. This requires the use of 4 equations (Fenske, Underwood, Gilliland, and Kirkbride equations). All of these are reasonably easy to work with except for the nonlinear Underwood equation:
$1 - q = \sum_{i=1}^{N \text{ components}} \frac{\alpha_i z_i}{\alpha_i - \theta}$
A textbook by Seader, Henley, and Roper works an example for a mixture of 8 components where the values for $\alpha_i$, $z_i$, and q are given below. This problem has multiple roots. Find the root ($\theta$) between 1.00 and 0.765. Solve the equation using at least two of the techniques covered this week.
q = 0.0866
\begin{tabular}{|c|c|c|}
\hline
i & Alpha & z \\
\hline
1 & 2.43 & 0.0137 \\
2 & 1.93 & 0.5113 \\
3 & 1.00 & 0.0411 \\
4 & 0.765 & 0.0171 \\
5 & 0.362 & 0.0262 \\
6 & 0.164 & 0.0446 \\
7 & 0.072 & 0.3106 \\
8 & 0.0362 & 0.0354 \\
\hline
\end{tabular}
