Chapter 3.2, Problem 30P

Mixing Problems.

Each of the tank shown in Figure 3.2.9 contains a brine solution. Assume that Tank $1$ initially contains $30$ gallons (gal) of water and $55$ ounces (oz) of salt, and Tank $2$ initially contains $20\text{gal}$ of water and $26\text{oz}$ of salt. Water containing $1\text{oz/gal}$ of salt flows into Tank $1$ at a rate of $1.5\text{gal/min}$, and the well –stirred solution flows from Tank $1$ to Tank $2$ at a rate of $3\text{gal/min}$. Additionally, water containing $3\text{oz/gal}$ of salt flows into Tank $2$ at a rate of $1\text{gal/min}$(from outside).The well-stirred solution in Tank $2$ drains out at a rate of $4\text{gal/min}$ of which some flows back into Tank $1$ at a rate of $1.5\text{gal/min}$, while the remainder leaves the system. Note that the volume of solution in each tank remains constant since the total rates of flow in and out of each tank are the same: $3\text{gal/min}$ in Tank $1$ and $4\text{gal/min}$ in Tank $2$.

(a) Denoting the amount of salt in Tank $1$ and Tank $2$ by $Q_1\left(t\right)$ and $Q_2\left(t\right)$, respectively, use the principle of mass balance (see example 1 of section 2.3) to show that

$$ $\frac{d{Q}_1}{dt}=-0.1Q_1+0.075Q_2+1.5$,

$\frac{d{Q}_2}{dt}=0.1Q_1-0.2Q_2+3$,

$Q_1\left(0\right)=55,Q_2\left(0\right)=26$.

(b) Write the initial value problem (i) using matrix notation.

(c) Find the equilibrium values $Q_1{}^E$ and $Q_2{}^E$ of the system.

(d) Use a computer to draw component plots of the initial value problem (1), and the equilibrium solutions , over the time interval $0\le t\le 50$.

(e) Draw a phase portrait for the system cantered at the critical point.