Chapter 1.1, Problem 10P

Mixing Problems. Many physical systems can be cast in the form of a mixing tank problem. Consider a tank containing a solution—a mixture of solute and solvent–such as salt dissolved in water. Assume that the solution at concentration $c_i(t)$ flows into a tank at a volume flow rate $r_i(t)$ and is simultaneously pumped out at a volume flow rate $r_o(t)$. If the solution in the tank is well mixed, then the concentration of the outflow is $Q(t)/V(t)$, where $Q(t)$ is the amount of solute at time $t$ and $V(t)$ is the volume of solution in the tank. The differential equation that models the changing amount of solute in the tank is based on the principle of conservation of mass,

$\underset{\text{rate}\text{of}\text{change}\text{of}Q(t)}{\underset{︸}{\frac{dQ}{dt}}}=\underset{\text{rate}\text{in}}{\underset{︸}{c_i(t)r_i(t)}}-\underset{\text{rate}\text{out}}{\underset{︸}{\{Q(t)/V(t)\}{r}_0(t)}}$, (i)

where $V(t)$ also satisfies a mass conservation equation,

$\frac{dV}{dt}=r_i(t)-r_0(t).$ (ii)

If the tank initially contains an amount of solute $Q_0$ in a volume of solution, $V_0$, then initial conditions for Eqs. (i) and (ii) are $Q(0)=Q_0$ and $V(0)=V_0$, respectively.

A tank initially contains $200$ liters (L) of pure water. A solution containing $1\text{g/L}$ enters the tank at a rate of $4\text{L/min}$, and the well-stirred solution leaves the tank at rate of $5\text{L/min}$. Write initial value problems for the amount of salt in the tank and the amount of brine in the tank, at any time $t$.