HOMEWORK$\_$CEN$2236\_$CHE$236$ Numerical Methods$\_2024$

Figure given below shows three chemical mixing tanks linked by pipes. As indicated, the rate of transfer of chemicals through each pipe is equal to a flow rate $,$ with units of cubic meters per second$)$ multiplied by the concentration of the other tanks from which the flows originate $($c represents the certain substance concentration with units of milligrams per cubic meter$).$ If the system is at a steady state, the transfer into each mixing tank will balance the transfer out. $($These tanks are well mixed.$)$

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Write mass balances for each reactor at steady state. Determine the coefficient matrix inverse. Use the coefficient matrix inverse to determine the concentration in the three mixing tanks at steady state. Check your results by verifying that $[A][A{]}^{-1}=[I].$

Data are given below for three cases.

A$-25$p$)$

$Q_{01}{C}_{01}=120m\frac{g}{s}$

$Q_{03}{C}_{03}=300m\frac{g}{s}$

$Q_{33}=120\frac{m^3}{s}$

$Q_{13}=40\frac{m^3}{s}$

$Q_{12}=90\frac{m^3}{s}$

$Q_{23}=60\frac{m^3}{s}$

$Q_{21}=30\frac{m^3}{s}$

$B-25p$

$Q_{01}{C}_{01}=240m\frac{g}{s}$

$Q_{03}{C}_{03}=600m\frac{g}{s}$

$Q_{33}=120\frac{m^3}{s}$

$Q_{13}=40\frac{m^3}{s}$

$Q_{12}=90\frac{m^3}{s}$

$Q_{23}=60\frac{m^3}{s}$

$Q_{21}=30\frac{m^3}{s}$

$C-25p$

D$-25$p$)$ Compare the concentrations obtained from A$,$ B$,$ and C and explain the results with proportionality and superposition properties.