Tutorial Week 6

Worksheet Questions, February 23, 2015

1. Salt water with concentration of 1 g/L flows into Tank A at a rate $a$ L/min. The mixed solution in Tank A flows into Tank B at a rate $b$ L/min. Another pipe takes the solution in Tank B back into Tank A at a rate $c$ L/min. Finally, solution drains out of Tank B at a rate $a$ L/min. Note: the volume in each tank is the same and $a$+$c$=$b$ (to ensure that the volumes in the tanks are constant).
   1. Write down the system that gives the amount of salt in each tank at any given time.
   2. Show that it is impossible for this system to have oscillations (damped or otherwise).
2. Plot the phase plane for the following system of equations and explain:

\begin{eqnarray} x_1 &=& x_1 -8x_2 \\ x_2 &=& 8x_1+x_2 \end{eqnarray}

Solutions:

1. 1. \begin{eqnarray} Q_1^\prime &=& (1)\cdot a -b\frac{Q_1}{V} + c\frac{Q_2}{V} \\ Q_2^\prime &=& b\frac{Q_1}{V} -(a+c)\frac{Q_2}{V} \end{eqnarray}
   2. Eigenvalues of the homogeneous equation are: $-b \pm \sqrt{bc}$. Since b,c $\ge$ 0, the eigenvalues are real and the system can not have oscillations.
2. Critical point is at (0,0) and the phase plane is a spiral moving outwards in the counter-clockwise direction.