Chapter 12, Problem 8P

Chemical/Bio Engineering

The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P12.8. Mass balances can be written for each reservoir and the following set of simultaneous linear algebraic equations results:

$\left[\begin{array}{cccc}13.442& 0& 0& 0\\ -13.442& 12.252& 0& 0\\ 0& -12.252& 12.377& 0\\ 0& 0& -12.377& 11.797\end{array}\right]\left\{\begin{array}{c}{c}_1\\ c_2\\ c_3\\ c_4\end{array}\right\}=\left\{\begin{array}{c}750.5\\ 300\\ 102\\ 30\end{array}\right\}$

where the right-hand-side vector consists of the loadings of chloride to each of the four lakes and $c_1,c_2,c_3,\text{and}{c}_4=$ the resulting chloride concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively.

(a) Use the matrix inverse to solve for the concentrations in each of the four lakes.

(b) How much must the loading to Lake Powell be reduced in order for the chloride concentration of Lake Havasu to be 75?

(c) Using the column-sum norm, compute the condition number and how many suspect digits would be generated by solving this system.

FIGURE P12.8

The Lower Colorado River.